Production Note Cornell University Library produced this volume to replace the irreparably deteriorated original· It was scanned using Xerox software and equipment at 600 dots per inch resolution and compressed prior to storage using CCITT Group 4 compression. The digital data were used to create Cornell's replacement volume on paper that meets the ANSI Standard Z39.48-1984. The production of this volume was supported in part by the Commission on Preservation and Access and the Xerox Corporation. Digital file copyright by Cornell University Library 1992.AN fetEMENTARY INVESTIGATION OF THE Theory of numbers.ELEMENTARY INVESTIGATION OF 'i’HB theory of Numbers, WITH Its APPLICATION TO THE INDETERMINATE AND DIOPHANTINE ANALYSIS, The ANALYTICAL AND GEOMETRICAL DIVISION OF THE CIRCLE, AND SEVERAt OTHEft CURIOUS ALGEBRAICAL AND ARITHMETICAL PROBLEMS. BY PETER BARLOW, OF THE ROYAL MILITARY ACADEMY, LONDON: PRINTED FOR J. JOHNSON AND CO.* ST. PAUt’s CHURCH-YARD, 1811· ίΉ/Printed hj C Wood, Foppirts Courtt Fleet Street^ London*PREFACE, THE Theory of Numbers is a subject which has engaged the attention and exercised the talents of many celebrated mathematicians, both ancient and modern; under the first of which classes, may be reckon ed Pythagoras and Aristotle, the former of whom is said to hare invented our present multiplication table, or the Abacus Pythagorieus of the ancients; though what is alluded to under this designa- tion was probably a much mope extensive table than that now in common use: Pythagoras also attributed tonumbers certain mystical pro- perties, and seems first to have conceived the idea of what are now termed magic squares. Aristotle, amongst other numerical specula- tions, noticed the uniformity in almost ail nations of dividing numbers into periods of tens, and attempted an explanation of the cause of this singular coincidence upon phi-, losophical principles. But the earliest regular system of numbers is that given by Euclid in the 7tli, 8th, 9th, antiVI PEEPACE. 10th books of his Elements, which, notwith·; standing the embarrassing notation of the Greeks, and the inadequacy of geometry to the investigation of numerical propositions, is still very interesting, and displays, like all the other parts of the same celebrated work,, that depth of thought and accuracy of demonstration for which its author is so eminently distinguished, Archimedes likewise paid particular atten- tion to the powers and properties of numbers, as may be seen by consulting his tract entitled “ Arenarius,” in whieh some modern writers have thought they could perceive inculcated the principles of our present system of lo- garithms; but all that can be allowed op this head is, that the method by which he performed his multiplications and di- visions bears a considerable analogy to that which we now commonly employ in the multiplication and division of powers; that is, by the addition and subtraction of their indices. Before the invention of analysis, however, no very extensive progress could be made in a subject, which required so much generality of investigation; and, accordingly, we find but little was effected in it till the time of JDiophantus, whose treatise of algebra con- tains many interesting problems in the morePREFACE, Vll abstruse parts of this science. But here, also, its author had to encounter the difficulties of a complicated notation, and a very deficient analysis, when compared with that of the present period; and, therefore, it cannot be expected that his work should contain a com- plete investigation of the theory of numbers. After Diophantus, the subject remained unnoticed, or at least unimproved, till Bachet, a French analyst of considerable reputation, undertook the translation of the abovementioned work into Batin, retaining also the Greek text, which was published by him in 1621, interspersed with many marginal notes of his own, and which may be con-, sidered as containing the first germ of our present theory. These were afterwards con- siderably extended by the celebrated Fermat, in his edition of the same work, published, after his death, in 1670, where we find many of the most elegant theorems in this branch of analysis; but they are generally left without demonstration, an omission which he accounted for by stating, in one of his notes, page 180, that he was himself preparing a treatise on the theory of numbers, which would contain many new and interesting numerical propositions; but, unfortunately, this work never appeared, and most of hisTill PREFACE. theorems remained without demonstration for a considerable time. At length, the subject was again revived by Euler, Waring, and Lagrange, three of the most eminent analysts of modern times. The •former, besides what is contained in the second volume of his Ci Elements of Alge- bra/- and his “ Analysis Infinitorum/’ has several papers in the Petersburg Acts, in which are given the demonstrations of many of FermaPs theorems. What has been done by Waring on this subject is comprised in chap. v. of his “ Meditationes Algebraicee; ” and Lagrange, who has greatly extended the theory of numbers, by the invention of many newT propositions, has several interestr ing papers on this head, in the Memoirs of Berlin, besides what are contained in his ad 7 ditions to Euler’s Algebra, It is, however, but lately that this branch of analysis has been reduced into a regular system, a task that was first performed by Legendre, in his “ Essai sur la Τΐιέρπβ des Nombres; ” and nearly at the same time Gauss published his “ Disqujsit'iones Arith- meticee: ” these two works eminently display the talents of their respective authors, and con- tain a complete development of this interesting « theory. The latter, in particular, has openedPREFACE. ix a new field of inquiry, by the application of the properties of numbers to the solution of binomial equations, of the form £· —1-0, on the solution of which depends the division pf the circle into n equal parts, as was before known from the Gotesian theorem. This solution he has accomplished in several partial cases, whence the division of the circle into a prime number of equal parts is performed, by the solution of equations of inferior dimensions; and when the prime number is of the form 2" + 1, tlie same may be done geometrically, a problem that was far from being supposed possible before the pub- lication of the abovementioned performance. From the foregoing historical sketch, it appears that the writers on this subject are far from being numerous; but the well esta-r blished celebrity of those, who have investk gated its principles, would be of-itself suf- ficient to stamp it with a degree of importance, and to render it worthy of attention. Few persons, it is conceived, will be disposed to consider that a barren subject, which has so much engaged the attention of the above named celebrated writers; in fact, there is no branch of analysis that furnishes a greater variety of interesting truths than the theoryX PREFACE. of numbers, and it is therefore singular that it should have been so little attended to by English mathematicians. With the exception of what is contained iii vol. ii. of Euler’s Algebra, and the notes added to the second English edition of that work, there is nothing on this subject to be found in our language. This circumstance, it is conceived, will be deemed a sufficient apology for the appear- ance of the present volume; in which, if I have, in certain theorems, availed myself of what others have done on the same subject, yet it is presumed, that it will be found to possess a sufficient degree of novelty, both in matter and arrangement, to exempt me from the imputation of being a mere copyist. With the exception of a few theorems, what is contained in the first six chapters may be considered as new: in the latter of which will he found a demonstration of Fermat’s general theorem, on the impossibility of the indeter- minate equation sen±yn = zn, for every value of n greater than 2; the leading principle of which I first demonstrated in the Appendix to Euler’s Algebra, and afterwards completed in vol. xxvii. of the Philosophical, Journal. I also consider as original what is, contained in chapter x„ with the exceptionPSEF1P1. xi pf that part relating to the arithmetic of the Greeks, for which I have been indebted to the Essay of Delambre, subjoined to the French translation of the works of Archi- medes. The methods of solving indetermi- nate equations pf the first degree, and of as- certaining, a priori, the number of possible solutions, have likewise some claim to novelty. In the other parts of the work, there will also be found several new theorems, and many former ones differently demonstrated, where simplicity and perspicuity could be attained by such alteration;, this is particularly the ease in the last chapter relating to Gauss’s celebrated theorem on the division of the circle. Perspicuity has, indeed, been one of my principal objects; for this treatise being intended for the instruction and amusement of those who may not possess a very ex- tensive knowledge of analysis, it became necessary to make it as clear and intelligible as possible: but how far I may have suc- ceeded in my design, or what merit may be otherwise due to the performance in general, must be left tp the decision of the public. It only remains now for me to mention a circumstance, that may probably be thought to stand in need of some explanation: it will be perceived that I have introduced two newPREFACE. XU symbols, the necessity for which, however, will, 1 trust, appear upon a slight inspection of the work itself: the words of the form of recur very often, and the repetition of them would have been tedious and irksome to the reader, for which reason the double f (Jf ) is intro- duced instead of them, hut, for the sake of uniformity, it is placed lengthwise thus, ; this, therefore, can scarcely be considered as an innovation of a very important rule laid down by modern analysts, “ Not to multiply without necessity the number of mathematical symbols.” And the same apology may be made for the introduction of the other sign, for the words divisible by. These characters were adopted on the suggestion of Mr. Bonnycastle, Professor of Mathematics in the Royal Mili- tary Academy, to whose judgment and ex- perience I have been greatly indebted for many important remarks relating to the pre- sent performance, and on various other oc·. casions, PETER BARLOW. Royal Military Academy, Woolwich, October 1, 1811.CONTENTS. PART ί. CHAP. I. pAbE On the Sums, Differences-9 mid‘Products of Numbers in General .............. I CHAP. II. On Divisors, and the Theory of Perfect, Amicable, and Polygonal Numbers 3:2 CHAP. III. On the Lineal Forms of Prime Numbers, mid their most simple Properties 5 2 CHAP. IV. On the possible mid impossible Forms of Square Numbers, and their Application to Numerical Propositions- ....... ...... 7$ CHAR V. On the possible and impossible Forms of Cubes, and ■Higher Powers * . 123 CHAP. VL· On the Properties of Powers in General * c . *155 CHAR VII. On the Products and Transformations of certain Algebraical Formulae ... . . . . . . .17$ CHAR VIII. On the Quadratic Divisors of certain Algebraical Formulae . » . . . . ... ... . *186 CHAP. IX. On the Quadratic Forms of Prime Numbers, with Mules for determining them in certain Cases . . 200iff C6tiTENT& CHAP. X. ft,*. On the different Scales bf Notation, and their Appli- cation to the Solution of Arithmetical Problems . 220 Notation of the Greeks...........................245 Miscellaneous Propositions 257 PART II. CHAP. Ii Continued Fractions^ and their Application to va- rious Problems . . . * . e> . . * . * .261 CHAP. II. On the Solution of Indeterminate Equations bf the First Degree 317 CHAP. ΙΙΪ. On the Solution of Indeterminate Equations of the Second Degree * * . * 4 * ; · . * * , 345 CHAP. IV, On the Solution of Indeterminate Equations of the Third Degree, and those of Higher Dimensions . 396 CHAP, V, On the Solution of Indeterminate Equations bf the Form xn— 1=m(«) ............................434 Jfable of Indeterminate Formula .452 CHAP. VI. On the Solution of Diophantine Problems . „ . . 4GO Miscellaneous Problems ..**.**.. . 476 CHAP. VIL On the Analytical and Geometrical Division of the Circle ...................4*9 Table of Prime Numbers to 4000 ...... 506 Table containing the least Values of p and q in the Equation pz — xq2t=: 1, for every Value of n, from 2 to 102 ............. . 507RRRATA. iPage 20, line ύ, for “ factor/’ read fraction. 24-, 1. 2 from bottom, for “ cor. 4/’ read cor. 2. 25, 1. iO, for “ of/’ read as. 35, 1. 5, /or “ 2=4-/’ read 2=5. 35, 1. 9, /or “ 2.3a. 44=4608,” read 2.3®.54i±I1250. 5m+1-l „ ftw+1— i 37, 1. 19, /or “------, read—--------, ό— 1 1 40, 1.7, add the sign =. 54, 1. 22, for “ 10000/’ read 4000. 61, 1.8,-9, 10, 11, for “ If n=± Ι/’ &c., read If o=l, 75, 1. 5, place a comma between a and Hh l. 85, 1. 9, for ie to the squares,” read as the squares. 89, 1. 18, for “ — /’ read —. J * 90, 1. 5, for ee is not a complete nth power/5 read is a com- plete ntk power. 90, 1. 8, for “ 2t2,” read {3p+2)t2. . 92, 1. 19 and 20, for “ z^qu%” read ^5qu2. 95, 1. 2, for “ 7η'-6/' read 7η'+β. 118, 1. 3 from bottom, for “ 2r2,” read 2s2. 126, 1. 11, for “ of the same,” read of the same form, 158, 1. 2, for tf xn—yn,” read xn-}-yn. 178, 1. 2 from bottom, for “ art, 89,” read art. 91. 179, 1. 23, for “ annunciation,” read enunciation. 185, 1. 21, for “ read x2-f-oxY. 185, 1. 24, for “ - b%” read +52. 187, 1. 1 of prop. 1, read If in the, Sec. 192, 1. 15, for “pr—y2f read pr~q2. 260, I. 9 from bottom, for “ (n — 2},” read (n— 2}®* £84> \.&, for “ uf read uf. Explanation of the new Character - To be read “ divisible by.” ist To be read " of the form of.”DEFINITIONS. 1. An Unity or Unity, is the representation of any thing considered individually, without regard to the parts of which it is composed. 2. An Integer, or Integral Number, is an unit, or an assemblage of units. 3. A Fraction, is any part, or parts, of an unit. 4. Factors, are those numbers from the multi- plication of which another number is produced. 5. A Product, is that number, which is pro- duced by the multiplication of two, or more, factors. 6. A Multiple of a number, is the product of that number by some integral factor. 7. An Even Number, is that, which can be· di- vided, or separated into two equal integral parts. 8. An Odd Number, is that, which cannot be divided into two equal integral parts ; being greater or less than some even number by unity. 9. A Composite Number, is any number that is produced, by the multiplication of two, or more, factors; or, it is a number which may be divided into two, or more, equal integral parts, each greater than unity. 10. A Prime Number, is that which cannot be produced by the multiplication of any integral fac- tors.; or it is a number, that cannot be divided into any equal integral parts, greater than unity. B ·2 Definitions. ί 1. Commensurable Numbers, are such, as have each the same common divisor; or that may be each exactly divided into the same number of equal integral parts. 12. Incommensurable Numbers, or Numbers prime to each other, are such as have no common divisor. 13. A Square, or 2d Power, is the product of two equal factors. , 14. A Cube, or 3d Power, is the product of three equal factors. 15. The nth Power of a number, is the product of n equal factors, n representing any integral number whatever. 16. The Exponent, or Index of a Power, is that number by which the power is expressed: thus, an represents a raised to the nth power, where n is said to be the Exponent, or Index of the Power. 17- The Root of a Power, is that factor from the continued multiplication of which, a certaiq number of times into itself, the power is pro- duced. 18. A Perfect Number is that, which is equal to the sum of all its divisors, or aliquot parts: thus, 6 = ^ ^ and is, therefore, a perfect «umber. ip, Amicable Numbers, are those pairs of in-'· tegers, each of which is equal to all the aliquot parts of the other: thus, 284 and 220 are a pair of amicable numbers, for to 4 184 284· i 284 ΥΓ 'r Ϊ42 + ■ 284 284' 20, and3 Definitions. 220 220 220 220 220 220 1_ L t.—l· ■ J- 4" 2 4 5 10 11 22 220 220 220 4“ 4~ = 284 55 110 220 ~ 220 220 ----1--—— +* 20 44 20. Figurate Numbers, are all those, that fall tmder the general expression n.(n + l) («+ 2) (» + 3) &c. n+ m Γ~. 2 3~. 4 Scc- tn + V and they are said to be of the 1st, 2d, 3d, &c.· order, according as in — 1, 2, 3, 4, &c.: thus, Nat. series, 1 2 3 4 5 6, &c. 1st ord. 13 6 10 15 21, &c. 2d ord. 1 4 10 20 35 56, &c. 3d ord. 1 5 15 35 70 126, &c. General term, n n.n + 1 1. 2 n.n'+ 1 .n + 2 T71l n.n + 1 m + 2,n + 3 1 . 2 3 . 4. 21. Polygonal Numbers, are the sums of dif- ferent and independent arithmetical series, and are termed Lineal or Natural, Triangular, Qua- drangular or Square, Pentagonal, Hexagonal, See. Numbers, according to the series from which they are generated. 22. Lineal, or Natural Numbers, are formed from the sum of a series of units ; thus, Units, . . . .1.1 1 1 1 1 1, &c. Natural numbers, 1 2 3 4 5 6 J, &c. 23. Triangular Numbers, are the successive sums of an arithmetical series, beginning with unity, the common difference of which is 1; thus, b 24 Definitions, Arith. series, 1 2 3 4 5 6 f, kc. Triangular I, , ( ,j 31 28,&c. numbers, J 24. Quadrangular, or Square Numbers, are the sums of an arithmetical series, beginning with unity, the common difference of which is 2 ; thus, Arith. series, Quadrang. or 7 '>· J 3 4 7 16 9 25 11, &C. 36, kc. square numb. 25. Pentagonal Numbers, are the sums of an arithmetical sei’ies, beginning with unity, the com- mon difference of which is 3; thus, Arith. series, 1 4 7 1° 13 l6, &C. Pentagonal } , s ls SJ 35 M &c. numbers, j 26. And universally, the m-gonal Series of Num- bers, is formed from the successive sums of an arithmetical progression, beginning with unity, the common difference of which is m— 2. 27. The Forms of Numbers, or Formulas, are certain algebraical expressions, under which those numbers are contained. Thus, 17 is of the form 4-1, that is, when divided by 4, the remainder is 1; and, for the same reason, 19 is of the form An 4- 3, or 4ft — 1; and this is expressed by the character tfc: thus17 ^ 4 w + 1, 19^. 4 n —1. 28. A Modulus, is that number by which the forms of numbers are compared, thus, the forms 4ft± 1, are compared by modulus 4, and 6n± l, by modulus 6. 29. Numbers of the same Form, are all those that are contained under the same algebraical e,\~Axioms. 5 pression, by changing the value of the indetermi- nate letter, or letters, that enter therein. Thus, 19 and 27 are of the same form, with respect to modulus 4; being each expressed by the same for- mula, An — 1. 30. A Function of a Quantity, is any algebraical expression, into which that quantity enters; and it is said to be rational, irrational, or integral, ac- eording as the expression is of either of those kinds. AXIOMS. 1. Every Even Number is of the form 2n. 2. Every Odd Number is of the form 2n ± 1. 3. The Sum, Difference, and Product of any number of integer numbers, are integers. 4. A number cannot be a divisor of another number less than itself. 5. Any number (a) taken once, or one time, is equal to unity, or 1, taken a times. Or, the pro- duct 1 x a = A x 1.AN ELEMENTARY INVESTIGATION, £$c. 8$c. CHAP. I, On the Sums, Differences, and Products of Numbers in general, PROPOSITION I. 1. The sum, or difference, of any two even num- bers, is an even number. For all even numbers are pf the form (ax. 1), and, therefore, the sums and differences of even numbers will be represented by 2» + 2n' = 2 (n ± n') sfc 2n"; which is evidently an even number, being of the form 2n.—■ a. e. d. Cor. The sum of any number of even numbers, is an even number. PROP. II. 2. The sum, or difference, of two odd numbers, is even ; but the sum of three odd numbers, is odd. For all odd numbers are of the form 2n + 18 Numbers in General. (ax. 2.); and, therefore, the sum, or difference, of two odd numbers, will be represented by (2tt± 1)± (2»'± l) = 2(η±n'± 1) sfc2n", which is evidently an even number, being of the form 2 n. But the sum of three odd numbers will be ex- pressed by (2« ± 1) + (2n' ± 1) + {2n" ± 1) = 2(n + n' + n"+ l) ± 1 m2n'"± 1, and is, therefore, an odd number. — a. e. d. Cor. 1. The sum of any even number of odd numbers is even; but the sum of any odd number of odd numbers is odd. Cor. 2. The sum or difference of an even and an odd number is an odd number, for 2n^(2n'± l) = 2(rc=p»')± l^2n"± 1, which is, therefore, an odd number. prop. xir. 3. The product of an even and an odd number, or of two even numbers, is even. For 2n x (2m' + l) = 2(2nml + n) 2n", and 2n x 2n' = 2(2mi') tfc 2n"; which are both even forms. — a. e. r>. Cor. 1. If an even number be divisible by an odd number, the quotient is an even number. Cor. 2. The product of any number of factors is even, if any one of them be even. Cor. 3. An odd number cannot be divided by an even number. For if 2n± 1 ~ 2n' ’~m’ artf Ulteger number, then 2n' x m— 2 η ± 1;Numbers in General. i± l), if 2n be so. — a. e. n. Cor. If an even number be divisible by an odd number, it will also be divisible by double that number. PROP. VI. (?. If a number, p, divide each of two other numbers, a and b, it will also divide the sum and difference of those numbers; or the sum and dif- ference of any multiples of them. For let -=<7, and o', then will p 1 p 1 a b a±b — + — —-----= q + q j p p p and since q and q' are each integers by the hypo- thesis, therefore, q ± q' is also an integer (ax. 3) ; and, consequently, a±b is divisible by p. Again, if ma, and nb, represent any multiples of a and . b ■ ma then, since- — 9, and ~~9> therefore, — = mq, na ■ ’ ma nb ma ± nb and — = nq ; consequently — = — = --------— = p P P p mq±nq'. And because both q and q', as also m and n} are integers, therefore mq + nq' is likewise a»Numbers in General. n integer (ax. 3), and, consequently, ma± ?ib, is di- visible by p. — a. e. D. Cor. 1. Hence, if a number divides the whole of another number, and a part of it, it will also divide the other part. Cor. 2, Hence, also, if a number consists of many parts, and each of those parts have a com- mon divisor p; then will the whole number, taken collectively, be divisible by PROP. VII. 7· If β and δ be any two numbers prime to each other, then will their sum, a -f δ, be also prime to each of them. For if (a-l· b) and n, had any common divisor p, then (a+b)~a and a, would also have the same common divisor (prop, vi.); that is, a and h would have a common measure, which is contrary to the supposition, because they are prime to each other; therefore (a -f b) and a, cannot have a com- mon measure. And in the same manner it may be shown, that (a -f b) and δ, can have no common measure ; and, consequently, if a and b be any two numbers prime to each other, their sum a -f δ is prime to each of them. — a. e. ix Cor. 1, In the same manner it may he demon- strated, if a and b be any two numbers prime to each other, that their difference (a-~b) is prime to each of them, if (a—b) > X» Cor. 2. If a number, consisting of two parts, m (a -f δ), be prime to one of its parts a, it will also be prime to the other part δ, if b > 1. Cor. 3 , If of two numbers, a and δ, one of them12 Numbers in General. be divisible by a third number p, but the other part not divisible by it, then neither the sum nor difference of those numbers (a± b) is divisible by p. Cor. 4. If a number, consisting of many parts, have all those parts but one divisible by another number p, but the other part not divisible by it, then the whole number, taken collectively, is not divisible by p. PROP. VIII. 8. If a and b be any two numbers prime to each other, then will their sum and difference, (a + b) and (a—b), he also prime to each other, or they will only have the common measure 2. For if (a + b), and (a—b), have any common measure, their sum and difference, 2a and 2b, will have the game (prop, vi.); but since a and b are prime to each other, 2a and 2b can only have the common measure 2; therefore (a + b) and (a — b), can only have the same common measure 2: and, consequently, if one of those quantities, a or b, be even, and the other odd, then (a + b) and (a — b), being both odd, cannot have the common measure 2; therefore, in this case, they are prime to each other. — a. E. D. PROP. IX. q. The product of any two numbers is the same, which ever of the two is the multiplier; or n'x b = bx a. First, if a = b, then it is evident that ab—ba\ but if these factors are not equal, one of them must be the greatest: let, then, a > b, and make a=b + a',Numbers in General\ 13 in which a' is necessarily less than a, then ab = hb + a'b, and ba = bb + ba'; if, therefore, ab is different from ha, so also is a!b different from ba', for the -equality of the products ab and ha, depends upon the equality of the.pro- ducts a'b and ba'. Now if in this last, a' = b, the equality is established, as it is also if &'= 1, because, 1 x b = b.x 1 (ax. 5). But if neither o' = b nor a'=l, then one of these factors is the greatest; let then b > aand make b = a' 4- b'., where b' < b, and we have a'b— a'a' + a'b' ba' ~a'a' + b'a', and the equality of the products a'b and ba! now depends upon the equality of the products a'b' and b'a', and, therefore, the equality of the first products ab and ba, depends also upon the equality of these last; and if in .this, b' == a', or b'=l, the equality is established; but if not, by proceeding as above, we may show, that the equality of these products de- pends upon others still less, and so on. Now it is evident, that in the products jib, ba ; a'b, ba'; a'b', b'a', &c., in which we have a'< a, a" < a', If < b, b" < b&c.; we must necessarily arrive at a case in which the terms of the product are equal, or in which one of them is equal to unity; and in either case the identity of the original products is established, from what is said above and by ax. 5; and, consequently, the product ab = ba. — a. e. d. Cor. In a similar m anner it maybe shown, that the product of any number of factors, a b e d, &e., is the same, in whatever order they are multiplied together.14 Numbers in General. PROP. X. 10. If a and p be any two numbers prime to* each other, then may either of these numhers, as a, be expressed by the formula , a—np-^r, in which r shall be less than p% and also prime to it.. For, first, if ap, let a be divided by p, giving a quotient n, and remainder r, which makes a — np + r; and since r is the remainder arising from a divisor p, it is evident that we may have r -ip, then (p — r) = r' < p; and it is evident thatNumbers in General. IS a~ up + r— (η + 1 )p — (p — r) = n'p — r', tjy making η + 1 = n', and (p — r) = r', in which, as we have seen above, p—r, or r' < \p: therefore, i» the formula a = np ± r, n may always be so assumed, that r <±p; and that it is prime to it, is evident from what has been ob- served in the proposition. Cor. 2. This formula, a — np ± r, is equally true of numbers not prime to each other, except that here p and r are not necessarily prime to each other, and r may also, in this case, become zero, which, cannot be in the former. PROP. XI. 11. If a and p be any two numbers prime to each other, there cannot be another number b, prime to p, that renders the product ah divisible by p. Or if a number, p, be prime to tw o other numbers, a and δ, it will also be prime to their product ah. First, it is evident, that if there be such a number δ, prime to p, it is either the only one, or there are others besides itself; in either of which cases, we may suppose δ to he the least of all those numbers that am prime to p, and that renders the product ah divisible by p, Now since b is prime to /j, we may make ρ — nb -f δ', in which formula, b' < δ, and also prime to both p and b (prop, x.); also b' cannot be o, because is prime to δ. Again, multiplying both sides of thig formula by a> we have ap — nab + ah'f or ' ap — nab — ab'.Ιίϊ Numbers in General* And here it is evident, that if the product ah be di- visible by p, the first side of the above equation, ap — nab, will be so likewise (prop, vi.); and, con- sequently, the equal quantity ab', will be divisible by p: but V < b, and b is the least of all numbers that renders the product ab divisible by p; whereas we have now found a less, which is absurd. There cannot, therefore, be a number, which is the least of all those, that render the product ab divisible by p ; but if there were any such numbers, one of them must necessarily be the least; therefore, there are no such numbers; and, consequently, if p be prime to two other numbers, a and b, it is also prime to their product ab. — a. e. d. Cor. 1, If a number p be prime to any number of quantities or factors, a, b, c, d, &cy it is also prime to their product, abed, &c. And if p be prime to any number a, it is prime to every factor of a. Cor. 2. Hence a product can only be divided by those numbers, or factors, from the multiplication of which it is produced; and, therefore, if ab be divisi- ble by p, it is divisible by every factor of p. Cor. 3. If in cor. 1 the factors are equal, that is, if a— b = c = d, &c., then the continued product, abed, becomes some power of a, as (jHfyU :^·/«Ιι Also n must be of one of the forms 4/?'* An' + Ϊ* An' 4- 2* or An' + 3, and, therefore* one of the first four terms is of the form An' \ that is* one of those tetms is divisible by 4; and in the same manner it may be shown* that one of the first five terms is divisible by 5* and so on; and* there being as many terms in the numerator as in the denominator* and each term in the denominator being a divisor of some one term in the numerator* it follows* that the whole product of the former terms is a divisor of the whole product of the latter ; and* conse- quently* hs the coefficient of each term is of the above form* it is an integer. — a. E. d. Cor. 1; If n be a prime number* then each of the coefficients* except the first and last* in the ex- panded binomial (a±b)n* is divisible by n. For n being a prime number* it is prime to every factor in the denominator* these being all less than ?z* and, consequently* n is prime to the whole product in the denominator; that is* calling the product of all the terms in the numerator* except the first* n ; and those in the denominator D; we have — = <7* an in- d teger: but since n is prime to d* the quotient — = r/* is divisible by n (cor. 8* prop, xi.); that is* each of the coefficients* except the first and last* is divisible by w* when n is a prime number. And* therefore* in any case* where the nature of the investigation requires such a substitution* we may put for the co- efficients of the expanded binomial (x ~h y)n, when n is a prime** the series 1, /?* net, nbi &c. nb, na9 η, I ,Number's in General, 19 and in which we shall always hare a, b, c, &c., in- tegral numbers ; but such a substitution cannot he generally made, if n be a composite number. Mop. xnf. 13. Neither the sum nor the difference of two fractions, which are in their lowest terms, and of which the denominator of the one contains a factor not common with the other, can be equal to an in- teger number^ Let and —, be aliv two fractions in their lowest A ΒΓ J terms; and of which the denominator of the one* as —r, contains a factor t, that is not contained in a ? at then I sav, that. * ' a h ~± —7 A B t an integer, is, in all such cases, impossible. For, a b ant±Ab A— B t AB t ’ and if this expression can be eqhal to an integer, the numerator «Βί± Mb, must be divisible by the de- nominator ABi; and, consequently, by each of its factors a, B, and t (cor. 2, prop, x.); but it can- not be divisible by t, for if it was, + Ab must be di- visble by t, because the other part ant is divisible by it (cor. 1, prop,' vi.): but since b_ nt is in its lowest terms, b is prime to B/, and, therefore, necessarily prime also to t, and a is likewise prime to t by the hypothesis.; consequently ab is not divisible by t20 Numbers in General. (prop. xi.). Since, then, the numerator «Βt± ab is not divisible by the denominator, the factor aBt±Ab . .a b —------—, or its equal - H— ABt ^ a~ bt cannot be equal to an integer. — a. e. d. Cor. 1. The same is also true, if the first fraction -, be not in its lowest terms, providing A the other fraction — be in its lowest terms, and the B t factor t, of this last, not common with a ; for we should still have, in this case, t prime both to a and b, and, consequently, also to the product ab, and likewise to the numerator aBt + Ab; and, therefore, aBt±Ab ,a b -■------, or its equal —H--= e, ABt A Bt an integer, is impossible. Cor. 2. In the same manner it may be shown, if there be several fractions, as abed a’ bP c’d’ &c., and one of them, as —be in its lowest terms, and b t contain a factor t, in its denominator, that is not common to all the other denominators ; that these -fractions cannot, however they may be combined, be equal to an integer; that is, abed ~±~r±~ + — = ise~ ’ P which is evidently a square. And, with respect to the product, it is obvious that/?2 y.pn—jrpH, is a square, its root beingpp'. Again, if the multiplier p' be not a square, then I say, that pip' is not a square, for if pl y.p'~ri,24 Numbers in General. then ~^—pf would be a square by the foregoing part of the proposition, ;which is contrary to the supposition; therefore, }fp' is not a square. Neither is “ a square; for if ^ = ν'1, then would p‘ — r9p', «* »® . or, p' = —, but —· is a square, therefore p' is a square, which is contrary to the supposition, and, consequently, is not a square. — a. E. D. Cor. In the same manner it may be shown, p5 that p* x p’% and ^ are both cubes: but ps x p', and p5 — are not cubes, if p' be not a cube. And generally, P if' J vn ' .. pn x p' and £7- are both nth powers, if p' be an nth power, but otherwise they are not. PROP. XVI, 16. If the square of a number, as p*, can be di- vided once, by some other number, as p', and after that, neither by p', nor by any factor pf p', then will p' itself be a complete square, For let p be resolved into its· factors, making p — anbmc'1, &c., or p* — (fnbimc^; and since p* is divisible by p', p' must contain some one, or more, of the prime factors of p (cor. 4, prop, xiv.); that is, p' must be of the form arb% &c.; and henceNumbers in General. 25 f V' dl“lr!aclq, &c. arb\ &c. pn - rJjZm - s^lq which quotient will evidently be still divisible by some one of the factors of p% unless r=2w, and s~2m; and since by the supposition this quotient is not again divisible, either by pf or by any factor of p', therefore it follows, that pf — cfnb~m = (anbnY; that is, p' is itself a complete square. — q,. e. d. Cor. In the same manner it may be shown, if p3 be divisible once by some other number pf pand after that, neither by p' nor by any factor of p\ that p' is a complete cube. And generally, if pn be divisible once by p', and after that, neither by // nor by any factor of //, then will p' be a complete nth power. prop. xvn. 17. The product arising from two different prime numbers cannot be a square number. For let p and q represent any two prime numbers, that are not equal to each other, and, if possible, pq let also pq — nP; or — = m. But a .number can m only be divided by those prime factors from the multiplication of which it is produced (cor. 2, prop, xi.); therefore, pq being divisible by m, either , pq pq 7n = p, or m = q: let m==», then— = , that is not a square, may be represented by the formula p^jCcd, c and d being prime numbers (cor. 5, prop, xiv.): let, there- fore, p and q be two numbers prime to each other, and not both squares, and make p — A°cd, and τγΰ’ m is prime to then — = e, and, consequently, rrf must be divisible by if; but since m is prime to 71, the product, or square, nf is also prime to ri‘ (cor, 7? prop. χΐ·); therefore, nf cannot be di- visible by vf, or — ==«, is impossible, and conse- quently so is also ya=—. — a. e, d.28 Numbers in General. Cor. 1. In the same manner it may be shown, that the cube root of an integer, that is not a com- plete cube, can never be represented by any rational fraction; and generally, if a be an integer, and Va cannot be represented by an integer, it is also impossible to represent it by any rational fraction. Cor. 2. The product of the square root of two numbers prime to each other cannot be expressed by any rational fraction. For if p and q were two numbers prime to each other, and */Px VQ — —> ?Yt we should have also ^/pq = —, which is impossible by the above proposition, because the product joy is not a square (cor. 2, prop. xvii.). Cor. 3. In the same manner it may be shown, that the product of the cube roots of two numbers prime to each other, and not both cubes, cannot be repre- sented by any rational fraction; and generally, if p and q be any two integer numbers, prime to each other, then the product l/p x l/q, cannot become equal to any rational quantity, unless p and q bfe each complete nth powers, and in this case the product is an integer. Cor. 4. All that has been demonstrated in the above two corollaries, of two quantities prime to each other, is equally true of any number of quam titles under the same restrictions. PROP. XIX. 19. Neither the sum, nor difference, of the square roots of two quantities, prime to each other, can be represented by any rational quantity; nor by theNumbers in General. 29 square root of any rational quantity; unless each of those numbers be a complete square. For let p and q be any two such numbers, and, if it be possible, let also vp± vq=c, c representing any rational quantity, or the square root of any rational quantity; then, by squaring, we have (vp± vqy=c*=p + q±2 vpq; and since c is either rational, or the square root of a rational quantity, c1 is in either case rational; and, consequently, also c*—p — q— ± 2 vpq, or Vpq- c2 ~p — q + 2 ’ is also a rational fraction, which is impossible (cor. 2, prop, xviii.); and, consequently, VP± Vq-c, or = vTj is also impossible. — a. e. d. . Cor. It may likewise be demonstrated, by means of this proposition, p and q, being prime as be- fore, and not both squares, that neither the sum nor difference of their roots can be represented by the sum or difference of the roots of any other two integral quantities whatever, that are prime to p and q; that is, vp± vq= vt± vs' is impossible. For, by squaring, we have p + q±2 vpq — T + s± 2 V'rs, or _ r+s—p—q ± vpq + vrs~------------ ,230 Numbers in General. Now either ^/rs is rational, or it is not; if it he rational", so also must be r+s—p—q ± =------2------- which it cannot be (cor. 2, prop, xviii.); therefore, vrs cannot be rational. Again, if +/rs be not rational, then We should have vpq± Vrs — r+s—p— 2 9 a rational quantity; which is also impossible by the above proposition, because pq and rs are prime to each other, and, consequently, Vp± vq— λfr± vs is impossible, under the specified limitations.—&.E.D., prop, xx. 20. If in any equation whatever,· higher than the first degree, the coefficient of the first term he unity, and those of the other terms integral num- bers, then no one of the roots of such an equa- tion can be equal to a rational fraction. For let xn± axn~' ± bxn~'z + , &c., ± r=o, represent a general form of equation; and, if it be possible, P P leta=-, the fraction ~ being supposed already in it? lowest terms ; then, bv substituting - for os, the equation becomes jf apn~1 bp · + cpn qv~* ql~3 jr , &C,y + V = 0 ; OYpNumbers in General. p" + aqpn~1 + bq*pn~~± c(fpn~s +, &c., _ —- pn± q{apn~l + bqpn~*± cq%pn~*±, &c.) 31 + r; or, + r. Now as this is equal to r, an integer, it follows that the numerator is divisible by qn; and, therefore, also by q; but since the part q(ap“~' ± bqpn~*± cq%pK~3 ±, &c.) is divisible by q, it is evident that p* must also be divisible by q, if the whole quantity be so (cor. 1, prop, vi.); but this is impossible, because p is prime to q, therefore the numerator is not divisible by q; p and, consequently, a; is impossible.—a. E. d.32 CHAP. II. On Divisors, and the Theory of Perfect, Amicable, and Polygonal Numbers. PROP* i. 21. Any number n being reduced to the form N= amhncvd\ &c., the number of its divisors will be expressed by the formula (m+ l) x (n+ l) x (p+ l), &c. For it is evident, that n will be divisible by a. and every power of a3 to am; that is, by each of the terms 1, a, a*, a?, &c., am. Also by by and every power of by to bn ; that is, by every term in the series 1, by b*y b\ &cbn. And, in the same manner, n is divisible by c, and every power of c, to cp; by dy and every power of dy to dqy &c*; and also by every possible combina- tion of the respective terms of the above serieS; that is, by every term of the continued product, (I + a+ +, &c., am) x (1 -f b-f b*4-, &c., bn) x (1 4-c-f c*-f, &c., cv) x (1 + d+d*-l· y &c., dq). But the number of terms of this product, since tio two of them can be the same, is truly expressed by the formula (m-f l) x (n-f 1) x (p + l) x (9 + l), &c.;Divisors and Figurative Numbers. 33 therefore, the number of the divisors of N is also expressed hy the same formula. — a. E. D. Remark. It will readily be observed, that, in the above formula, n is considered as a divisor of itself. Ex. 1. Having given the number 3βθ, to find the number of its divisors. First, 36θ = 2’.32.5'; therefore, n — 3, m = 2, and p = 1. Hence, (3+ 1) x (2+ 1) x (1 + l) = 4 x 3 x 2=24,; the number of its divisors. Ex. 2. It is required to find how many numbers there are, by which 1000 is divisible. First, 1000= 23. 53; or m — 3, and n = 3; there- fore, (3 + l) x (3 + l) =4 x 4= 16: that is, there are l6‘ numbers by which 1000 is divisible. Cor. 1. As the number of divisors of any given number, n = « mbnc?, &c., is represented by the formula (m + 1) x (n + 1) x (p + l), &c,, it is evident, that the number of ways, in which the given number N may be resolved into two factors, will be represented by “X (tn + 1) x (n-F 1) x (p + l), &c., λ ♦ because every divisor has its reciprocal factor; and, therefore, the number of ways, in which a number may be resolved into two factors, is equal to half the formula, that expresses the number of its divi* sors. But if the given number n be a square, then all the exponents, m, n, p, &c., will be even, and, therefore, (m+ l) x (n+ 1) x (p+ I), &c.j vyill be odd; and the formula, representing the num- jn34 Divisors and Figurative Numbers. ■■ ϊ**+·\ fi$U 1 **#··# »* % *;·' \ *#%i her of ways that n may be resolved into two factors, will be (wM-l) x (n+l) x (jp+ l), &c., 4-1 because, in this case, two of the factors are equal. Cor. 2. If it be required, in how many ways a number, N = aw//cp, &c., may be resolved into two factors prime to each other, it is evident, that this number no longer depends upon the value of the exponents m, n, p> &c., but will be the same as if n was simply resolved into the factors a, b, c, &c.; and is, therefore, equal to (l 4- 1 ).(l t l).(l 4 l), &C.^ 2 hence, if h represents the number of prime factors, a, by c, d, &c., then will 2k'1 be the number of ways in which n may be resolved into two factors prime to each other. Thus, for example, 360 has twenty-four divisors (example l), and, con- sequently, may be resolved into factors twelve different ways (cor. 2); but it has only three prime factors, 2, 3, and 5, and can, therefore, be resolved into factors prime to each other only, 2* =4, different ways. PROP. II. 22.N To find a number that shall have any given number of divisors. Let tv represent the given number of divisors, and resolve tv into factors, as iv = x x y x £. Take m—x ~ 1, n—y — I, p — z — 1, &c.; so shall ambvcv. &c., be the number required, as is evident from the t35 FHvisors and Figurative ^Numbers. foregoing proposition, where a, δ, c, &c., maybe taken any prime numbers whatever. Ex. Find a number that shall have thirty di- visors. First, 30 = 2 x 3 x 5; that is, a? = 2, i/ = 3, * — 4 ; or, m~ I, n—2^ .p^=4; therefore, aJfc4 is the! number, having thirty divisors, a,$ required. If a = 2, b = 3, c— 5; then 2 . 32 4βθ8. If ά= 5, i == 3, c = 2tlien 5 . 3s. 24 = 720. If a =5, /;=2, c = 3; then 5 . 2°; 34== 1620. Each of which numbers has thirty divisors, and it is evident, that various other numbers might be obtained that would have the same property, by only changing the value of a, &, and c* Remarki If it were required to find the least number of all those that have a given number of divisors* it is manifest, that we must proceed with more caution, as our formula would not then have that unlimited form that is given above. It will, therefore, be proper to enter, here, into an investi- gation of this particular case. First, it is evident, that the value of our number depends upon two different operations: 1 st, the resolution of w into its factors, which in the fore- going problem is 'arbitrary; and, 2dly, on the as- sumption of the quantities <7, b9 c, &c. * Now if to be a prime number, it cannot be re- solved into any other factors than Ixtc; and, therefore, aw~l is the only form a number can have when the number of its divisors- is a prime number; and, therefore, the less value we give to α > I, the . d 236 Divisors anil Figurative Numbers. less the number will be: and, consequently, the least of all is when a = 2 ; thus, the least number that has seven divisors is 26 = 64, and the least having eleven divisors is 21U= 1024, and so on. Again, when tv is not a prime number, but equal to the product of two prime factors, as w = xy, then the only variation in its resolution will be w = x x y, and tv = 1 x xy; and, consequently, the only two forms of numbers will be ax~'by~\ and a"J~\ that have the required number of divisors, and it is obvious that the first form is that which gives the least value of the number required; because a may, in both forms, be equal to 2, and b may be taken equal to 3; and, whatever be the numbers x and y, it is evident, that 21-1 x 3S~' < 2zy~': for if even w7e make b — 4, in win eh case 2I'1x4,'1 = (2)'+!,"s, it is still less than 2xy"‘, for (x + 2y — 2) < xy, because both x and y are > 1, and x may also be supposed the greater of the two, that is, x>y. Therefore, a’-'b*-1 is that form wilich produces the least num- bers, and it is manifest, that the least numbers in this form are those in which a = 2, and b=3, x— 1 being supposed the greatest exponent; and, in the same manner, it may be shown, that if tv be re- solvible into any number of factors, that will be the least form, in which tv is resolved into the greatest number of factors, and the lowest number of any such form, as amb"cT, &c., is obviously that in which the greatest exponent has the least root, the next greater exponent the next less root, and so on. Therefore, when it is required to find the least number that has a given number of divisors, we must resolve the given number into its greatestDivisors and Figurative 'Numbers. 37 number of factors, and then proceed in given values to a, b, c, &c., according to the rule above given; viz. to the greatest exponent, the least root, &c. Suppose, for example, it were required to find the least number that has twenty divisors. The greatest number of factors is when 20 = 2 x 2 x 5; therefore, a'b'c4 is the least form: and by making c = 2, b — 3, a — 5, we have 24.3.5 = 240, which is the least number of all those that have twenty divisors. If we had resolved 20 into the factors 4 and 5, then a4b3 would have been the form, and, by making a = 2 and b = 3, we should have had 243s = 432, for the least number in that form; and the same for all others. PROP. IH. 23. If n = amb’lcp) &c., represent any integer, then will the sum of all the divisors of n be ex- pressed by the formula For, by art. 21, every divisor of n is contained in the product (1 + a + a®, &c., a”) x (l + b + έ2, &c., b") x (1 + c + c®, &c., cp) x (l + d+ d?, &c, dq). And, by .the common rule for summing a geome- trical progression, we have . a + — 1 1+α + α — - a =-----------, a— 1 J»+1 _ 1 1 + &+P-------.&· =-----Γ-, & press the sum of the divisors of n. prop. vi. 26. To find a pair of amicable numbers κ and m, or such a pair of numbers, that each shall be re- spectively equal to all the divisors of the other. Make n = «"'feV, &c., and Μ = α."βίγ7Γ, &c.; then, from the definition of amicable numbers, and what has been demonstrated (art. 23), it follows that we must have /am+l — l\ /cp+'~ l\ ~)xK^^)x\^r)=*+m’and / au+l — l\ / βν+ι — l\ /V+1-l\ \ )*\ β-i A r-1 ;-M + N· Because these formulae include the whole·.numbers n and m, as divisors of themselves (Remark, art. 23), whereas, in amicable numbers, they are excluded by the definition. We see, therefore, in order that the numbers N and m may be amicable, that these formulae must be equal to each other, and each equal to n + m. Now this is accomplished by finding such a power of 2 as 2r, that 3.2r — 1, 6.2r—1, and 18.2*' — 1, may be all prime numbers; or, making 2r = tv, we must have 3 w — 1 — b, 6ιν — 1 = c, and 1 Siv2 — 1 =*d, all prime numbers; then will N = 2r+ld, and m = 2r+lbc, be the pair of amicable numbers sought. For, if we represent 2(4?ϋ — 1) — 2w( 18 uf — QU) + 1) = 2 wx 18-10*+ 1 = 2T+'d— N. Therefore, the sum of the divisors of m = n ; and we have seen, that the sum of the divisors of n = m: consequently, n and M are a pair of amicable num- bers. — a. e. d. Scholium. By making r — 1, or 2T — w — 2, we have 3tv —1 = 6 = 5, 6w — 1 = c = 11, and \8w“— 1 = f/=71 ; and, consequently, 2r+id=4 x 71 =284 = n, 2r+,6c = 4x 5 x 11=220 = m. which are the least pair of amicable numbers, the next two pair being 17296 and 18416, 9363583 and 9437056.46 t)ivisors and Figurative iVumbers. The difficulty, therefore, of finding amicable numbers, is connected with that of finding the above specified conditions of tv, and for which ήο rule has yet been discovered. The reason for taking w some power of 2 is, that 2 is the only even prime; for if w was the power of any odd number, then Άιϋ— I, 6w— 1, 18m/2— 1, would not be primes, and if w was the power of an even number, not a prime, as 2 m, then (^HKMS) would not be the true expression for the sum of the divisors of M: arid, therefore, 2 is the only number that can be employed for this purpose* prop. vii. 27. If the nth term of any order of figurate numbers be added to the n -f 1 term of the next in- ferior order, the sum will be the η + 1 term of the same order as the forriier. Arid the nth term of any order is ecpial to the n first terms of the preceding order* In our definition of figurate number's, we have given the form of each of the orders, because it is more simple to deduce the generation of figurate numbers from the form of them, than to deduce tfieir forms from their generation. We have, there- fore, to demonstrate, that the numbers falling un- der the forms we have given are generated one from another, as announced in this proposition; and this will be manifest immediately, by repeatingDivisors and Figurative Numbers. A’J here again those series of figurate numbers, and their general terms, as given at definition 26; viz. Nat. series, 1 2 3 4 5, &c, lstord. 13 6 10 15, &c. 2d ord. 1 4 10 20 35, &c. 3d ord. 1 5 15 35 70, &c. General term, n n.(n+ 1) 1 . 2 n.(n + l).(n+ 2) 1.2. 3 n.{n + !).(» + 2 ).(« + 3) mth order, -— n.(n + l).(w+ 2).(n + 3), &c., (n + m) 4, &c., (m+l)’ Now it is evident, that n +1, which is the n+lth ~ i 1 . n.(n +1) term of the natural series, being added to ------, 1 · Jk which is the nth term of the first order, gives ».(»?.+l) (n + l).(»4- 2) + »+l- τ— which is the n+lth term of the first order. . . n.(n+ l).(n + 2) , , , Again, to —----------—, the nth term 2d order, ... (w+l).(« + 2) - . . adding -— --------—, the n+lth term 1st order, (iM-1').(» + 2).(» + 3) , . gives--------------—-—, the n+ 1 r/iterm 2d order. 1 » 2r · 3 And, generally, to n.(n + l).(n + 2).(n + 3) - - - (n + m) . ---- —:------—-— ------------------■■-—nth term 1 . 2 . 3 . 4 - - - (m +1) mth order, adding 148 Divisors and Figurative Numbers. (n+ l)-(w + 2).('/M-3).(n + 4)--------(n + m) 1 . 2 . 3 . 4 - - ~ m = n +1th term m — 1 th order, gives in-f l).(n +2).(n + 3).{?i +4) - - (n+ 1 + m) 1.2 .3.4------------- m + 1 = n-h li/f term mth order. Hence, then, we have the general law of forma- tion ; namely, to the nth term of any order, add the n+\ term of the inferior order, and the sum will be the succeeding term of the former order. And, there- fore, since the second term of any order is equal to the sum of the two first terms of the next in- ferior order, the third term will be equal to the sum of the three first terms of the preceding order; and, generally, the nth term of any order is equal to the sum of the first n terms of the next inferior order. — a. e. d. Scholium. In this proposition we have, after Legendre, inverted the order of proceeding, by de- ducing the law of generation from the forms of the successive series, instead of ascertaining the forms of those series from their law of generation, which has the advantage of greater simplicity, and leads us at once to the demonstration of one of Fermat’s theorems, that he considered as one of his princi- pal numerical propositions, arid which is this: If the nth term of the natural series he multiplied by the n+\th term of any order m, the product will be equal to niA 2 times the nth term of the order m + 1... Thus, taking the fifth and sixth term,, of the fore- going series, we have 5= X 6 = 2 x 15 ; 5 x 25 = 3 X 35 v 5 x 56 = 4 X 70, &c.Divisors and Figurative Numbers. 49 This property is deduced immediately from our forms, for . ... M.(«+l) n(n + 1) = 2 x —- : v ’ 1.2' (n + l).(« + 2) n.(n + l).(n+ 2) Tt X ~ : τ — ό X —————— ■ * 1.2 1.2.3 and so on for any other order. This theorem, which is so remarkable simple iii the way that we have considered these numbers, is very difficult to demonstrate by any other method; and that Fermat considered it as one of his most in- teresting propositions is evident, from what he says after the annunciation of it: “ Nee existimo pul- ehrius aut generalius in numeris posse dart theorema, cujus detnonstrationem margini inserere nec vacat nee licet" (Notes on Diophantus, page 16). PROP. VIII. 28. Every polygonal number of the denomination m, or every m-gonal number; is expressed by the formula (m — 2))f — (m — 4)ti --XT— V " £ 2 For, by definition 26, every ra-gonal number is a sum of an arithmetical progression, beginning with unity, the common difference of which is m — 2; or, making m — 2 = d, it will be the sum of any number n terms of the series 1 + (1 + d) + (i + 2d) + (1 + 3d), &c., (1 + (η-1 )d); which, by the common rules, is equal to VOL. 1. E50 Divisors and Figurative Numbers. (2 + (ft—l)d)ft - , or, since d=ni — 2, if we substitute for d, we shall have (2 + (ft — 1). d)n _ 2« + («* — ft). (m — 2) _ ; - ; ~ (m — 2)n~ — (m — 4)n ^ ^ ^ ~~ 1 " — ,i" £L E. D. 2 Cor. 1. Hence, by making successively m=3, 4, 5, &c., we shall have the following results. All fl* -j- γι Triangular numbers tt; - ·-■- . 2 ft* —oft -------— = ft*. 2 3«2 — ft sfc ------. 2 4ft® — 2ft tfc ------. 2 &c. Cor. 2. By means of the general form (m — 2)«* — (m — 4)« 2 * any polygonal number, of which the root ft, and de* nomination in, is given, may be readily ascertained. Thus, by making m=3, m = 4, m — 5, &c.; and in each series ft = 1, 2, 3, 4, &c., we have Series of triang. numb. 1 3 6 10 15 21 28, &c. Series of squares - - 14 9 l6 25 36 49> &c. Series of pentagons - 1 5 12 22 3551 7°, &c. Series of hexagons *· 1 6 15 28 45 66 'Ql, &c. &c. &c. Squares - - Pentagonals - - Hexagonals - - &c.51 Divisors and Figurative Numbers. Also any polygonal number, and, its denomina- tion being given, the root of the polygon is readily obtained, For let (m .-i- 2)n2 — (m — 4)ti P_ _ represent any given polygonal, of which the deno- mination m is. known ? then, (m — 2)n2 — (m — 4)n = 2p ; or, / m — 4\ \m—2) n — 2P m — 2 Whence _ rn~ 4 + V 2p{m — 2) + (m — 4y: 2m — 4 * /which is a general form for the root of atiy po- lygottal number. Remark. Fermat has given, at page 15, in one of his notes to the ninth proposition of Diophantus on Multangular Numbers, particular rules for finding the roots of given polygonais, without the extraction of the square root, but, as they are of little or no use, we have not inserted them.S2 CHAP. III. On the Forms of Prime Numbers, and their most simple Properties. PROP. i. 29. If a number cannot be divided by some other number, which is equal to, or less than, the square root of itself, that number is a prime. For every number p, that is not a prime, may be represented by p — ah. Now if a — b, then a. and b are each equal to yp; and, consequently, p, which is ,not a prime, is divisible by vp. Again, if a > vp, then will h < Λ/ρ; for otherwise, we should have ax h = ab>p, which is con- trary to the supposition; therefore, if a > jp, then will b < yp; and if b > yp> then will a < yp; and, consequently, since p is divisible both by a and b, it is divisible by a number less than the square root of itself: and this is evidently true of all numbers that can be resolved into the form p = ab; that is, of all numbers that are not primes: therefore, if a number cannot be so divided, that number is a prime. — a. E. D. Cor. Hence, in order to ascertain whether a given number be a prime number or not, we must attempt the division of it, by all the prime numbers less than the square root of itself; and if it be not divisible by any of them, it is a prime. It is obvious,Prime Numbers. 53 that we need only essay the division by prime num- bers, for if it be divisible by a composite number, it is evidently also divisible by the prime factors of that divisor. This method, however, although it admits of some contractions, is, notwithstanding, extremely laborious for large numbers ; nor has any easy, practical rule been yet discovered, for ascertaining whether a given number be a prime or not. Scholium. The problem of finding prime numbers, was agitated so far back as the time of Eratosthenes, who invented what he called his κοκκίνον, or sieve, for excluding those numbers that are not prime, and, consequently, thus discovering those that are. The principle of this method, which is the same that has since been employed by modern writers for ascertaining those numbers, is as follows: Having written down in their proper order all odd numbers, from 1 to any extent, required; as 1 3 5 7 9 11 13 15 17 19 hi 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99· We begin with the first prime number 3, and over every third number, from that place, we put a point, because all those numbers are divisible by 3 ; as 9, 15, 21, &c. Then, from 5, a point is placed over every fifth number, all these being divisible by 5; such are 15, 25, 35, &c.54 Prime Numbers. Again, from 7 every seventh number is pointed in; the same manner, such as 21, 35, 49, &c. And, having done this, all the numbers that now remain without points are prime numbers; for there is no prime number between 7 and Λ/100; and it is obvious, from what is said in the preceding corollary, that it is useless trying any composite number; adding, thei’efore, to the above, the prime number 2, which is the only even prime, we have 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 6l 67 71 73 79 83 89 97, which are all the prime numbers under 100. By this means, assisted, probably, by some me- chanical contrivance, Vega has computed, and pub- lished, a table of those numbers from 1 to 400000, And as it will be sometimes useful in the following part of this work, to know whether a given number be prime or not, without the trouble of computing it, a table is given at the end of this volume, exhi- biting all the prime numbers from 2 to 10000, PROP. XI. 30. Every prime number, greater than 2, is of one of the forms 4»,+ 1, or An — 1. For every number whatever is either divisible by 4, pr, when the division by 4 is carried on as far as possi- ble, there will be a remainder, 1, 2, or 3 ; that is, every number whatever is included under one or other of the four forms 4n, 4n+ 1, 4w + 2, 4« + 3.Prime Numbers. 55 Bnt the first and third pf these forms are even num- bers, being of the form 2n, and, therefore, cannot contain any prime number > 2; and, consequently, all prime numbers greater than 2, are contained in the other two forms; and, therefore, every prime, number is found in one or other of the two forms 4m 4- l,or4m + 3; but4M + 3 = 4(n + l) — 1, tfe.4η — 1; therefore every prime number is of one of the forms An± 1. — a. e. d. Cor. 1. Every prime number being of one of the forms 4m +1, or An — 1, and as m, in these forms, must be either even of odd, we may subdivide them into the four following forms : 1. If n be even, or of f An + 1 -I-1; the form 2»', f 4m — 1 tfc 8n' — 1 m8n" + 7· 2. If n be odd, or of f An + 1 tfc8n' + 5; the form 2n' +1, \ An— 1 tfe8n' + 3. Hence all prime numbers, greater than 2, with regard to 8 as a modulus, are of one of the forms 8m+1, 8m+ 3, 8m+ 5, or 8m+ 7* Cor. 2. The two forms 4m + 1, separate prime numbers into two principal divisions, that are found to possess very distinct properties, which will be the subject of our future investigations ; we shall, there- fore, in this place, only give a few of those numbers, under each form, in order to render this classifica- tion the more familiar to the reader. Primes OzAn+1, are 1 5 13 17 29 37 41 53, &c. Primes as4M-1, are 3 7 11 19 23 31 43, &c. ^ The other four fprms, also, which are obtained56 Prime Numbers. in the foregoing corollary, and in which the above two are necessarily involved, lead, also, to another principal classification of prime numbers, each class possessing properties exclusively its own. Primes divided according to this modulus are as follows: 8n+1 ----- 1 17 41 73 89 97 113. 8« + 3 - - - 3 11 19 43 59 67 83. 8«+5 - - - 5 13 29 37 53 6l 101. 8« + 7 - - - 7 23 31 47 71 79 103. Which are evidently numbers distinct from each other, and, as we observed above, possess very dis- tinct properties, highly curious and interesting, many of which are demonstrated in the following chapters. PROP. III. 31. Every prime number, greater than 3, is of one of the forms 6n + 1, or 6n — 1. For every number whatever is either exactly di- visible by 6, or when the division is carried on as far as possible, there will be a remainder 1, 2, 3, 4, or 5; that is, every number whatever is of one of the forms 6n, 6n+ 1, 6n + 2, 6n + 3, 6« + 4, 6n + 5. But the first, third, and fifth, of those numbers, are divisible by 2, and the fourth is divisible by 3, and, therefore, no one of these forms can contain prime numbers greater than 3; and, consequently, all prime numbers, greater than 3, must belong to one of the other two forms 6w+l, or 6n + b; but 6n + 5 tfc6n' — 1; therefore, every prime number, greater than 3, is of one of the forms 6n + 1, or 6»— 1. — a. E. D. Cor. Since every prime number is of one of thePrime Numbers. 57 forms 6ft + 1, or 6ft — 1, and as ft, in these forms, must be either even or odd, these may be subdivided into the four following forms: 1, If ft322ft', 2. If ft322ft'-j- 6ft+1 32l2ft'-f 1; 6ft — 1 3212ft' — 1 3212 ft + 11. 6n+i32i2n/ + 7; 6n— 1 3212ft' + 5. Hence every prime number, with regard to mo- dulus 12, is of one of the four forms 12ft -f 1, 12ft *f 5, 12ft + 7? or 12ft 4-11: or, which is still the same, every prime number is of one of the forms 12*± 1, or 12ft± 5. Scholium. It should be observed here, that al though all prime numbers are contained in one or other of the foregoing forms, derived from art. 30 and 31, we must not conclude that the, converse of the proposition is true also; namely, that all num- bers contained in these forms are prime numbers: this is evidently not the case in the form 4ft + 1, if ft = 6; nor in An — 1, if ft = 4; and similar exceptions may be found for every other form that may be de- vised: in fact, no formula can be found that shall express prime numbers only, as appears from the following proposition. PROP. IV. 32. No algebraical formula can contain prime numbers only. Let p + qx -f- nr + sj? +, &c., represent any general algebraical formula, and it is to be demon- strated, that such values maybe given to #, that the formula in question shall not produce a prime number, whatever values are given to pP q, r, s, &c«58 Prime Numbers. - For suppose, in the first place, that, by makings χ—m, the formula p=p + qm + mi2 + sms +, &c., is a prime number. And, if now we assume x—m + 3p3. But this last quantity is divisible by p; and, conse- quently, the equal quantity p + qx + 7\*2 + sx’ +, See., is also divisible by p, and cannot, therefore, be a prime number. Hence, then, it appears, that, in any algebraical formula, such a value may be given to the indeterminate quantity, as will render it di- visible by some other number; and, therefore, no algebraical formula can be found that contains prime numbers only. — a. e. d. Scholkim. But although no algebraical for- mula can be found, that contains prime numbers only, there are several remarkable ones that con- tain a great many; thus x^ + x + 41, by making successively x — 0, 1, 2, 3, 4, &c., will give a series 41, 43, 47, 53, 6l, 7I5 &c., the first forty terms of which are prime numbers. The above formula is mentioned by Euler in the Memoirs of Berlin (1772, page 36).Prime Numbers. We may likewise add the two following; viz. x° + x+ If, and 2a;4 + 29, of which the former has seventeen of its first terms primes, and the latter twenty-nine. Fermat asserted, that the formula 2”\+1 was always a prime, while m was taken any term in the series 1,2, 4, 8, l6, &c.; but Euler found, that 234 + 1 =641 x 6700417 was not a prime. These examples are sufficient fpr showing, how easily we are led into error from induction, and how little it is to be depended upon, in mathematical investi- gations. prop. v. 33. The number of prime numbers is infinite. For if not, let the number of them be represented by », and the greatest of all those primes by p, then it is evident, that the continued product of all the prime numbers, not exceeding p, as •2.3. ·, 5 . 7 . 11--p, will be divisible by each of those numbers, and, therefore, if 1 be added to it, it will be divisible by no one of them; and, consequently, if the formula (2,3.5.7.11------------p) + 1 be divisible by any prime number, it must be by one greater than p; and if it be not divisible by any prime whatever, it is itself a prime number, which is necessarily greater than p; therefore, in either case, we must have a prime number greater than p; and, consequently, p is not the greatest; neither is n the greatest number of them, and the same is true of all finite numbers p, and «;Go Prime Numbers\ therefore, the number of prime numbers is in- finite*. — a. e. d. PROP. VI. 34. If a be any number whatever, and b, b', b", &c., all those numbers that are less than 2a, and also prime to it; then will all prime numbers be contained in one or other of the forms 4an±b, 4an±b', 4an± b", See. For every number, when divided by 4a, will leave for a remainder one of the terms in the series 0, 1, 2, 3, 4, &c., 4α—1; or, which is the same, + 0, +1, + 2, Hh 3, + 4, &c., 2a. But it is evident, that, when any one of these re- mainders has a common divisor with 4a, the num- ber contained under that form is not a prime num- ber, it having the same common divisor as the remainder and modulus; rejecting, therefore, all those remainders that have divisors common with 4a, and representing the others by fe, b\ b"> &c., it * It has also been demonstrated by Legendre (art. 404, Essai sur la Tkeorie des Nombres), that every arithmetical progression, of which the first term and common difference are prime to each other, contains an infinite number of prime numbers. And, also, N that if n represent anv number, then will —---— : be the * ~ k.logti- 1.08366 ’ number of prime numbers that are less than n, very nearly. We should have added here the demonstration of this very curious theorem, but that it depends upon a fluxional consideration, which could not be well introduced into an elementary work of this kind.Prime Numbers. 61 is manifest, that all prime numbers will be con- tained in the forms 4an± b, 4an± b', 4an ± b", &c., at least all those that exceed the prime factors of a. — a. e. d. Cor. 1. Hence we may deduce, generally^the forms obtained at art. 30 and 31; viz. if η— 1, all prime numbers tfc 4n± 1; If n — 2, all prime numbers ^^3^3! JLf n = 3, all prime numbers -f ί’ I m-4 1 JtlL T O * f efcl6n±l; If n— 4, all prime numbers s ^ J^ 5! b tfcl6»± 7- The number of forms, therefore, to any par- ticular modulus, depends upon the number of in- tegers, that are less than half that modulus, and also prime to it; and we have shown (art. 24) how this number may always be ascertained. Sup- pose, for example, that 2n was the given modulus; make N X κ = 0, is. „ 10000 x 8 , , , . for every 10000,---^-----= 1333 ; and out of this number, for the first 10000, there are 1230, which are really primes; whence the probability of a num- ber being a prime that falls under one of the above . . 1230 forms is ——, ir it be under 10000. 13o3 Probability If the number be between 10000 and 20000, If the number be between 20000 and 30000, 1033 1333 983 1333 If the number be between 30000 and 40000, ——- 5 1333 &e. &c. This calculation is made upon a supposition, that the number of primes, in the above eight forms, are equal to each other, which is not strictly true; such an hypothesis, however, may be assumed, in a rough estimation of this kind, without affecting, in any great degree, the truth of the result.64 Prime Numbers. PROP. VII. 35. Three prime numbers cannot be in arith- metical progression, unless the Common difference of them be divisible by 2 x 3 =6; except 3 be the first of the prime numbers, in which case there may be three prime numbers in arithmetical pro- gression, whose common difference is not divisible by 6, but there can be only three. For all prime numbers greater than 3 are of one of the forms 6n + 1, or 6n + 5. Also, if the com- mon difference be not divisible by 6, it must be of one of the forms 6m + 1, 6m + 2, 6m + 3, 6m + 4, or 6m + 5; or it may be otherwise represented by 6m+ r, r being any one of the numbers 1, 2, 3, 4, or 5. And, therefore, three numbers in arithmetical progression will be, either 1st, 6n + 1, 6(n + m) + r +1, 6(n + 2m) + 2r+ 1; or 2d, 6n + 5, 6{n + m) + r + 5, 6{n + 2m) + 2r + 5. And it is to be proved, that some one of these terms, in both series, is divisible by 2, 3, or 6; and, consequently, that they are not all primes. First, let r— 1, 2, 3, 4, or 5 ; then we have, in the'.first series, f r+l=2, 3, 4, 5, or 6; \ 2r + l =3, 5, 7, 9, or 11. And here it is evident, that either r +1, or the corresponding term 2r + 1, is divisible by 2, 3, or 6; and, consequently, one of the three terms in the first series is also divisible by the same; and, there- fore, they are not all three primes. And we arePrime Numbers. 65 led to the same result in the second series 5 for, by making r=l, 2, 3 4, 5, f /· + 5 = G, 7, 8; 9, or 10; j_ 2r + 5 = 7, 9, 11, 1>1, or 15. Where, also, one or other of the two corre- sponding terms has a common measure with 6; and, therefore, these three terms are not all primes. Consequently there cannot be three prime numbers in arithmetical progression, unless their common diiference be divisible by 6; if we except the ease where the first term of the progression is 3. And in this case there can be only three, for otherwise, by taking away the first, there would still remain three prime numbers in arithmetical progression, of which the common difference is not divisible by 6, which is contrary to what has been demonstrated above. — a. E. d. prop. vui. 36. There cannot be five prime numbers in arithmetical progression, unless their Common difference be divisible by 2x3 x 5 = 30; except when the first term of the progression is 5, in which case there may be five prime numbers in arithmetical progression, whose common difference; is not divisible by 30, but there can be no more than five. For all prime numbers greater than 5 to modulus 30, are of one of the following forms: 30n + 1, 30n + 7, 30n+ 115 30ή +13, 30η +17, 30η + 19, 30η+23, 30η + 29. And since, by the foregoing proposition, three VOL. 1. E66 Prime Numbers. prime numbers cannot be in arithmetical pro- gression, unless their common difference be divisible by 6, it follows a fortiori, that 5 cannot be so un- less the common difference be also divisible by 6; therefore, the common difference, in this case, as compared with modulus 30, must be of one of the forms 30m + 6, 30m + 12, 30m + 18, or 30m + 24, all other forms being rejected as not being divisible by 6, which we have seen is a necessary condition. Assuming, therefore, 30»·+r for a general ex- pression for prime numbers, and 30m + 6p a ge- neral expression for the common difference, our five terms of the progression will be 30» + r, 30(» + m) + r + 6p, 30(« + 2 m) + r + 12p, 30(n + 3m) + r + 1 Sp, and 30 (n + 4m) + r + 2 4p. Now it readily appears, that whatever value is given to r, of the above; viz. 1, 7» 11» 13, 17» 19» 23, or 29; and to p of those which it repre- sents; viz. 1, 2, 3, or 4; one or other of the ex- pressions r + 6p, r + 12p, r + 18p, or r + 24p, is di- visible by one of the numbers, 2, 3, or 5. Thus, Ifr=l, andp= l, then r + 24p-^*5. lfr=l, and^?=2, then r + \2p^ 5. Ifr=l, andρ = 3, then r+ 18/)-*- 5. Ifr=l, andj»=4, then r+ 6p·+. 5, And so on of any other values of p and r. * This character signifies divisible by, and is only employed to save the repetition of those words.Prime Numbers'. 6f Whence it follows, that these five numbers Can- hot be all prime numbers; that is, five prime num- bers cannot be in arithmetical progression, unless their Common difference be divisible by 30, if we except the case hi which the first terni of the pro- gression is 5 ; which evidently is excepted in the demonstration, as our forms are for primes greater than 5. The two primes, 2 and 3, are also excepted ; and, with regard to the first, it is evident it cannot form the first term of such a progression, because it is aii even number; but 3 may be taken for a first term, and, by giving to r this value, the same im- possibility will appear; There Cannot, therefore, be five prime numbers in arithmetical progression, unless their Common difference be divisible by 2x3x5, excepting only the case Where the first term is δ, add in this case there Can be only five; for if there were six, by taking away the first, there Would still remain five prime numbers itt arithmetical pro- gression, whose common difference would not be divisible by 30, which is contrary to what has been shown above. — a. e. d. Cor; In the Same niahner it may be demon- strated that sCveh prime numbers Cannot be in arithmetical progression, unless their common dif- ference be divisible by 2 x 3 x 5 x 7 == 210; except the first of those prime numbers be 7> in which case there may be sevett prime numbers in arith- metical progression, of which the common dif- ference is not divisible by 210, but there cannot be more than seven. And, generally, there cannot be n prime numbers in arithmetical progression, unless their common difference be divisible bjr F 26*8 Prime Numbers. 2 x 3 x 5 x / x 11, See., n; except the case in which n is the first teVm of the progression, in which case there may be n such numbers, but not more. PROP. IX. 37· The sum of any number of prime num- bers in arithmetical progression is a composite number. This is evidently true, if the number of terms in the progression be an even number, because then their sum will be even, and, therefore, composite. We have, therefore, only to consider the case, in which the number of terms in the progression is odd. Let, then, p be the first prime number, and d the common difference of the progression; then, if we consider at first only three terms, they will be p+(p + d) + (p + 2d) = 3p + 3d, which is evidently divisible by 3, and, therefore, a composite number. If we take five terms, they will be p + i^p + d) + (p + 2d) + {p"i~ 3d) + (j9 4- 4d) = bp + lOrZ, which is evidently divisible by 5; and, generally, We may assume p+(p + d) + (p + 2d) +, See., (p + 2nd) for any progression, the sum of which is . . (1 + 2n)dx 2n . . . . . (2n+ l)p + ^-------------= (2 n+ 1 )p+ (2 n+ 1 )nd; it will be divisible by 2n+ 1, and is, therefore, a composite number. — a. E. D.Prime Numbers. 69 PROP. X. 38. If a and b be any two numbers prime to each other, and each of the terms of the series b, 2b, 3b, 4b,-----(a—l)b, be divided by a, they will each leave a different positive remainder. For if any two of these terms, when divided by a, leave the same remainder, let them be repre- sented by xb and yb, and their common remainder by r; so that xb = na + r, and yb — ma + r; then it is evident, that xb—yb — na — ma will be divisible by a. But this is impossible, for xb—yb = bx(x—y); in which product the factor b is prime to a, and (x—y) < a; because both x and y are less than a, by the hypothesis; con- sequently, their difference must be so; but if, of two factors, one be prime and the other less than a third number, the product is not divisible by this number (cor. 6, art. 11); therefore, bx (x — y) is not di- visible by a; and, therefore, xb = na + r, and yb = ma + r, are impossible; that is, no two of those terms can leave the same remainder, when both are divided by a. — a. e. d. Cor. 1. Since, then, the remainders arising from the division of each of the terms in the series b, 2b, 3b, 4b,-----(a—l)b, by a, are different from each other and <7—1 in number, also all of them necessarily less than a; it follows, that these remainders include all num- bers from 1 to a — 1.f O Prime Numbers. Gor. 2. Hence, again, it appears, that some pne of the above terms will leave a remainder 1 \ and that, therefore, if b and a he any two numbers prime to each other, a nuinber x when atfe4»+1; and | ^1.2. 3. 4--- when a^An — 1. Whence it follows, that every prime number of the form An +1 is a divisor of the sum of two squares. Again, the latter forjn may be resolved into the two factox's I |^1 .2.3.4 I ^ 1 . 2 . 3 . 4 which product, being divisible by a, it follows,74 Prime Numbers, that a is a divisor of one or other of these factors* when it is a prime number of the form 4n — 1. Cor. 2. From the first product, which we have demonstrated to he divisible by a, viz. 1 . 2 . 3 . 4, &c., (a— l) +1 ■--------------------------------= e, an integer, we may derive a great many others ; as l8. 28.3.4.5, &c., (α-3)(α-ΐ) + 1 •-----------——-—--------------==e, an integer; Is. 28.3*.4.5, &c., (α-4)(α-ΐ) + 1 »—------~--------—.---------------= e, an integer; and so on, till we arrive at the same form as that in cor. 1. Scholium. The theorem above demonstrated was invented by sir John Wilson, as we are in- formed by Waring, in his Meditationes Alge- braical, page 380; but, notwithstanding the sim- ple principles on which its demonstration is founded, it escaped the observation of these two celebrated mathematicians; the latter of whom speaks of it, at the place above quoted, as an extremely difficult proposition to demonstrate, on account of our having no formula for expressing prime numbers. Lagrange was the first who demonstrated this theorem, in the New Memoirs of the Academy of Berlin, 1771 (which demonstration is, as might be expected from the celebrity of its author, very in- genious) ; and, afterwards, Euler gave a different de- monstration of the same proposition, in his Opusc. Anak/t. tom. i. page 32.9, which is upon a similarPrime Numbers. 75 principle as the foregoing; and, finally, Gauss, in his Disquisitiones Arithmetical, extended the theo- rem by demonstrating, that “ The product of all those numbers less than, and prime to, a given num- ber a± 1, is divisible by the ambiguous sign being +, when a is of the form pm, or 2pm, p being any prime number greater than 2; and, also, when « = 4; but positive in all other cases (Recherches Arithmetiques, page 57). The theorem of sir John Wilson furnishes us with an infallible rule, in abstracto, for ascertaining whether a given number be a prime or not; for it evidently belongs exclusively to those numbers, as it fails in all other cases, but is of no use in a practical point of view, on account of the great magnitude of the product even for a few terms. * PROP, XII. 40. If a and b be any two numbers prime to each other, the equation ax— by — ±c is always possible; and an infinite number of dif- ferent values may be given to x and y, that answers the condition of the ecpiation, in integer numbers. For, by (cor. 3, art. 38) the ecpiation, ax — by — + 1 is always possible, while a and b are prime to each other; and, consecpiently, acx — bey = ±c, or ax' — by' == ± e; by milking cx=x', and cy=y': and we have evidently the same result if we write*6 Prime Numbers. a(x' + mb) for ax', and b(y'±ma) for by'-, for these still give a(x' + mb) — b(y'± ma) = ± c. Or, again, making x' + mb = x, and y' + ma =y, our equation becomes ax—by—±c; which is, therefore, always possible. And it is evident, that by means of the am- biguous sign ±, and the indeterminate quantity m, the formula? x'±mb = x, y' ± ma —y, will furnish an infinite number of values of x and Tf, that answers the condition of the equation ax — by = + c, in integers, a and b being prime to each other. — a. e. d. It is also obvious, that m may be so assumed, that x shall be less than b, or y < a. Cor. Hence, in any of our future investigations, if we have two quantities, t and u, prime to each other, we may always substitute tx — iiy = c, c being any number whatever, when the state of ' the question requires such a substitution, without con- sidering the particular values of x and y, it being sufficient for our purpose, in many cases, to know that the equation is possible. But if t and u have any common measure, then such a substitution cannot be made, unless c have the same common measure.Prime Numbers. 77 PROP. XIII. <41. The equation ax + by = c is always possible, if a and b be prime to each other, and c > (ab — a — b). For let c = (ab — a — b)+r, then the equation be- comes ax + by = (ab — a — b)+r; the possibility of which depends upon ab — a — b — by + r x- a being an integer. Now this equation is the same as (y -f 1 )b — r x- ■b- 1 and, therefore, it depends upon the possibility of (y+l)b-r --------— = a? being an integer; or, which is still the same, by calling y -f 1 =y', upon the possibility of the equation y'b — ax'=z r\ which we have seen may always be established, so that y' (ab — a — b). —* eu e. d. Cor. The two foregoing propositions are very useful in judging of the possibility, or impossiblity, of indeterminate equations of this kind; and, con- sequently, also, in proposing them* so that they may be within possible limits.79 CHAP. IV. ,Oh the possible and impossible Forms of Square Numbers, and their Application to Nu- merical Propositions. PROP. i. 42. Every square number is of one of the forms 4n, or An + 1. For every number, being either even or odd, is of one of the forms 2n, or 2n + I; and, conse- quently, every square number is of one of the forms 4η1, or (An1 + An + 1); but 4«6 efc 4m, and {4m2 + 4m + 1) =4(m2 + m) + 1 i&4n- f· 1,. a. e. d. Cor. 1. Every square of the form An is ne- cessarily even ; and every square of the form An 4-1 is evidently odd; therefore, every even square is of the form An, and every odd square of the form An + 1. Cor. 2. By the foregoing proposition it appears, that every odd square is of the form 4(m2 + η) +1; and hence it follows, that it is also of the form 8η + 1: for if n be odd, it is odd, and if n be even, id is even also; therefore, in both cases, id A- n is even; and, consequently, 4(ns -f η) + 1 efe8η -f1; that is, every odd square is of the form 8η + 1. If,80 Forms of Square Numbers. therefore, a number be of the form An + 1, but not of the form 8n + 1, that number is not a square. Cor. 3. No numbers of the forms An + 2, or An + 3, can be square numbers. Nor can any numbers of the forms 8»+ 2, 8 m+ 3, 8m + 5, 8m+ 6, or 8m+ 7* be square numbers. Cor 4. The sum of two odd squares cannot form a square, for (An + l) + (An'A-1)^An + 2, which cannot be a square (cor. 3). Cor. 5. An odd square, subtracted from an even square, cannot leave a square remainder. For An— (An' + l) = 4(n — η') — 1 tfrAn + 3, which cannot be a square. Therefore, if the dif- ference of an even and odd square be a square, the odd square must be the greatest. Cor. 6. If the sum of an even and odd square be a square, the even square must be divisible by 16, or be of the form 4qri\ For all odd squares are of the form 8n +1 (cor. 2); and, therefore, if the even square had only the form An", nn being odd, the sum of the two would be 4m'2 + 8% + 1 = 4(m'2 + 2 η) + 1; and since nn is odd, (n'q + 2m) is odd also; and, therefore, 4 (η'2 + 2n) + 1 is not of the form 8m + 1; and, consequently, it is not a square (cor. 2). prop. ir. 43. Every square number is of one of the forms 5m, or 5m + 1. For all numbers, compared by modulus 5, are of one of the forms 5m, 5«+ 1, 5m± 2; that is, every number is either divisible by 5, or will leave for aForms of Square Numbers. 81 remainder 1, 2, 3, or 4; or, whifch is the same, + 1, or +2: and, consequently, all square num- bers are of one of the following forms: Numbers. Squares. Farms. bn, 2bn* 35 bn bn± 1, 2bHi± 10« + 1 fafc bn + 1 bn±2, 2bn‘±20n + 4 fate 5ra + 4fafc5«—1. Consequently, all square numbers are of one of the forms bn, or bn± 1. — a. e. d. Cor. 1. If a square number be divisible by 5, it is also divisible by 25; and, if a number be di- visible by 5, and not by 25, it is not a square* Cor. 2. No number of the form bn + 2, or bn + 3, is a square number. Cor. 3. If the sum of two equates be a square, one of the three is divisible by 5, and, consequently, also by 25. For all the possible combinations of the three forms bn, 5n+l, and 5η — 1, are as follows: ft (bn 4“ 1) "f· {bn? 4-1) ^bn 4- 2, {bn— 1) + (bn'— ljfate5w— 2faft5/2»+3, '< bn + bn' '35 bn, bn + (5w'+ l)fa35n +1, bn + (bn'— 1)355»— 1, (bn+ l) + (bn'— 1)355ή* Now of these six forms, the latter four have on# of the squares divisible by 5, and, therefore, also by 25 (cor. l). And the two first are each impossible forms for square numbers; that is, neither of these two combinations can produce squares: therefore, if the sum of two squares be a square, one of the three squares is divisible by 25. VOL. I. G82 Forms of Square Numbers. Cor. 4. By means of the two foregoing pro-1 positions, and their corollaries, it appears, that no number contained under a repetend digit can be a square number. For every number expressed by a repetend digit is equal to the same number of repetend units, multiplied by the particular digit under the repetend of which the number is contained. But every repetend of units is of the form 100n+ Ilste4«-+3tfc5»+ 1; and it is only necessary to show', that no number of the form 4n + 3, or bn + 1, multiplied by any one of the nine digits, can be a square. Now the following products, (4?i + 3) x I, (4n + 3)x4, (4n + 3) x 9, cannot produce squares, because one of the factors is a square, and the other not; and, consequently, the product cannot be a square (art. 15). Again, (4n + 3) x 2 efe4w' + 2, (4n + 3) x 5 tte4ii/ -f 3, (An + 3) X 6t&4n' + 2; which are all impossible forms for squares (cor. 3, art. 42). And since a repetend unit is likew ise of the form bn + 1, we have (bn+ l) x 3z&bn' + 3, (bn + l) x “ % bn' + 2, (bn+ l) x 8zp.5n' + 3, * each of which is an impossible form (cor. 2, art. 43) jForms of Square Numbers* 83 and, consequently, no repetend digit can be a square number. — ? 10»+ I 10n± 4 10»+ 5 11 jll» 11 n + I 1 1»-j- 9, 1U+ 3 11»+ 4 1 ln+ S 12 12 n 12 n+ 1 12 ??+ 4 12»+ 9 33 ’ 13n 13»+ I 13??± 3 13»+ 4 14 114n 14» -|- 1 14 » -[- 8 1 4-tz —[— 2 14??+ 9 14»+ 4 14n+l 1 14??+ 7 15 |15n 15»+ 1 15»+10 15n-j- 4 15»+ 6 I5w+ 9 36 16??; 16»+ l 16 ??4 16»+ 9. 17 17 n I7»± 1 17??+ 2 17»± 4 17 w+ 8, 18 13 8?? 1 8n+ 1 lSn+13 18?? *4- 4 18??+ 16 18»+ 7 18??+10 10 19,»+ 1 19?!+ 7 19»+17 19??+ 4 19??+ 9 19»+ 5 19n + H 19 ?? + 6 19??+16 20 1 i2o/i 20»+ 1 20»+16 20??+ 4 20» + 5 ,, 20n+ 987 Forms of Square Numbers. 46. Scholium 2. Hence, by way of exclusion, arises the following table, which exhibits all those forms, referred to the same moduli, that can never be- come squares, and by means of which we may frequent- ly ascertain whether a given number be a square or not, without absolutely performing the extraction. Table of impossible Forms for Square Numbers for every Modulus, from 2 to 20. Impossible Formula. 3 3»+ 2 4 4»+ 2 4 »-}- 3 5 5»+ 2 5»+ 3 6 6n+ 2 6n+ 5 7 7 »+ 3 7» + 5 7»+ 6 8 8»+ 2 8»+ 3 8»+ 7 9 9»+ 2 9» + 3 9»+ 5 9»+ 8 10 10»± 2 , 10n± 3 11 11»+ 2 llw+ 6 11»+ 7 11»+ 8 11»+10 12 ^ 1 2» + 2 12»+ 3 12»+10 12n± 5 12»+ 6 12«+ 8 13 13»± 2 13»± 5 13»± 6 14 jl4»+ 3 14»+ 5 14«+13 14»+ 6 14·»+10 14»+I2 15 jl5n+ 2 15«+ 3 15)2+11 I5n+ 5 15ii+\2 I5n+ 7 15»+13 15«+ 8 15»+14 16 ^ 16»+ 2 16/2+ 3 10»+ 8 I6n± 5 16?i+12 16»± 6 16»+* 15 16»+ 7 17 17n± 3 17»± 5 17n± 6 17»± 7 18 jl8»+ 2 18»± 3 18»+ 9 IS τι 4- 5 . 18?ι+Π 1S» + 6 I 14 18»+ 8 18»+17 19 |19»+ 2 19)2+ 3 19n+13 19»+ 8 19«-f-1 -4 19 » +■10 19»+ 15 19«+12 19»+18 io |20»± 2 20)2+ 3 20)2+10 20n± 6 20»+·-! 1 20»+ 7 20»+15 20»+ 8 20»+1988 Forms of Square Numbers. Lemma. 47. In order to simplify and abridge the de- monstration in the following propositions, it will be proper to make a few general observations on equations of the form af + bu1 = tv*. And, first, we may always consider a and b as quantities that have no square factor, or divisor; for, if a = α'<β2, and b = b'f, our equation becomes a'qff + M'ti1 — ivl; or, making q>t=f, and Qu = u', we have a'tn + b'un = if; and, consequently, if the above equation obtain when the quantities a and b, or either of them, have a square divisor, it may always be put in another form, a't'~ + b'lf = if, in which the similar quantities a' and b' have not a square divisor; and, therefore, in what follows, with regard to the possibility or impossibility of equations of the form af + bu1, we may always consider a and b as not having a square divisor. Again, if the equation af + bif = if be possible, when f, if, and if, have a common square divisor Φ5, it is also possible when divided by it; thus, if β<ρ"/'2 + b$iun = φ*ινη be possible, so also is atn±bif = wn, which is a similar equation to the first, and in* which f‘, if, and w'~, have now no common square divisor. And it is evident, that no two of these squares can , have a common divisor, unless the third square has the same. For, if it be possible, let f= tn<^, and if — if$9; then, atn<$1± hif<$=-if, where the first side of the equation is divisible by ± 3qu2 = tv2, or <)pn + 6n + 3p + 2 + 3qv2 — ur, or 3(3pn-f 2n + p± qu2) + 2 = w2; that is, v/ip.Sn'+ 2,Forms of Square Numbers. 91 which is impossible (art. 46); therefore, the equation is always impossible, under the above limitation of q. — a. Er x>. Remark. If 5 had not been taken under the above restriction, our demonstration would necessarily havefailed; because, in that ease, if we put q — 3q', we should have had 3quz= q'. 9«2; or, making 9&2 — u'2, 3qw* = q'u'2; wdiich would evidently have altered the form of the equation. But, under the above re- striction, we are led to several impossible cases, by taking p — o, or an integer number, and q any num- ber not divisible by 3 : thus p= 0, then 2 f± 3u* — w2; p — 1, then 5f±3u*=w°i J9=2, then 8f±3u2—w2; &c. &c. &c. are all impossible equations, wdiich may be carried on at pleasure. In the above, we have taken q= 1; but if q — 2 and p = 0, then 2t'2 ± 6uq = ur; p=l, then 5t2±6u2 = w2‘, p=2, then 8 f ± 6u~=tv2; &c. &c. &c. are also impossible equations; and thus we may pi’oceed to find impossible equations, to any length, pt pleasure. prop. v. 49. The equation (5p± 2)f^5qu° = w2 is im- possible, when q is prime to 5. For t must be of one of the three forms, 5 m, 5m + 1, 5m— 1; which furnish the three following equations:92 Forms of Square Numbers. 1st, (bp±2) x bn^bqif — w1·, 2d, (bp± 2) x (bn+ l)cfcbqu*=wt; 3d, (bp ± 2) x (bn — 1) φ bqu2 == w*. In the first equation, in which we suppose <2tte5n, we have evidently uf^bn' also; that is, f and wi are both divisible by 5, whereas, by the preceding lemma, they are prime to each other, which is absurd; therefore, the equation is im- possible, when f*&bn. In the second equation, in which we suppose /*sft5n + 1, we have (bp± 2) x (bn + Ι)φ55'ίίϊ = ίϋ2, or 2bpn± 1 On4" bp± 2^bqut = ivi, or b(bpn± 2n+pz^qu2) ± 2—uf·, that is, w2mbn± 2, which is impossible (cor. 2, nrt. 43). In the third equation, we suppose t2mbn — 1, whiph gives (bp± 2) x (bn— l)dpqu'i = uf, or 2 bpn ± 10» — bp + 2 Φ qif = w°, or b(bpn± 2n—p^qu2) +2 = uf·, that is, w2^ubn + 2, which is also impossible; and, consequently, as t* must be of one of those forms, it follows, that the equation (bp± 2)f^bqu1 — uf is always impossible, when q is prime to 5. Cor. By means of this proposition, we arrive at the following set of impossible forms to modulus 5, which may be carried to any length. 2t3^bu‘i = ivi, - 2tdp\0u- — ur, 3f ^bu2 = u>2 ■ w ff^bu2-- 8#2φ5Μ2 = ?.υ2, kp. &c, &c. 3#2φ10 u==,w'} 7/2φ1θΜ2 = ητ, 8ί2φ10«2 — w2, &c. &c. &c*Forms of Square Numbers. 9* PROP. VI. 50. Every equation that falls under any of the following forms is impossible; viz. (7p + 3)f + fqu1 = iv2, (7jv 4- 5)f + 7 qw‘=w*, {7p + 6)f ± 7qu°-=w% it being always understood that q is not divisible by 7. For every square number is of one of the forms 7n, 7«+l5 7n + 2i or 7n + 4 (art. 45). But t cannot be of the form 7n> in any of the equations, because then w° would be of the same form; that is, both f and wl would be divisible by 7, which is contrary to the supposition, since they are prime to each other; therefore, f must be of one of the other forms, if there be any case in which these equations are possible. Now in the first equation, if we suppose /5sfc7w+ we have (7P + 3) x (7n+ \)±7qut — ivl, or 4Qpn + 21w, + 7j» + 3± 7qu* — iv1, or 7\7pn + 3n+p± qtP) + 3 = w~, or tv* ζ&7 n'+ 3 > an impossible form for squares (art. 46). Again, suppose 7n + 2> then (7p + 3) x (7n + 2) + 7qus = or 49/m+ 21n+ I4p + 6±7qu* = wi, or 7{7pn + 3n + 2p± qu ) + 6 — w2, or w-m7n + 6; an impossible form, by the same article.94 Forms of Square Numbers. If f^7n + 4, we have (7|> + 3) x (7« + 4) ± 7qu~=uf, or 4$pn + 21m + 28/» + 12 + 7qu*~w*, or 7(7/m + 3m + 4p + qi?) + 12=uf, or wiiti^n+ 5; an impossible form. Therefore, the first equation is impossible under every form of f, q being prime to 7· In the second equation, {7p + 5)t°+7qus=w*, by assuming f successively of the three forms 7m+1, 7m+2, 7m+ 3, we are led to the following results. ίί*7«+ 1, then { + * (7.<+l) + 7?»*=«>*» 7„ + i, then ( (7i + 5) x (7« + 2)±7s»*=»·* fa=7n+4, then { «'ί, + 5> x (7«+4)_+75»·=»'* all of which are impossible forms: and, conse- quently, the second equation, (7P + ϊ>)ΐι± 7qu- = uf, is always impossible, q being prime to 7· In the third equation, [7p + 6)t‘l ± 7qu~—w9f by assuming, as before, f of the forms 7m+1, 7m+ 2, 7m+ 3, we have ■ .Forms of Square Numbers. 95 + l, then ·< f {7p + 6) x (7«+ l) ± 7qu°~ L 7 n - 6; + 2, then to modulus 7· 7p + 6, J And hence we are led to an inference, that the same is true for any other prime modulus; that is, the9<> Forms of Square Number#. Equations (11 jo + 2)f±llqu* = w% (11 p + 6)f*±liqu*=uf, (1 Ip + 7)t*± llqif — u?, (11|>+ 8)f±\\qui = uf, (1 \p + 10) t% + 11 qiC- — to®, are all impossible, while q is taken prime to 11, Also, (l3p± 2)fd^\3qui = w'1, (13p± b fdplSqu^u?, (I3p + 6)fdfci3qu%=w*, when q is taken prime to the modulus 13; And (i7p± 3)f*i7qu*=vf, (I7p± 5)ίϊφΐ75ιΜ2=κ)2, (I7p± 6)ί2φΐ7qtii=ioi, (l7p± 7)fdFiqqui=zv-, when q is taken prime to the modulus 17; Likewise, (I9p+ 2)f±l$qu*=w\ (I9p+ 3)^± XSqic-uf-, (19p + 8)f- ± 19qif = w\ (19P+ 10) f ± 19qu* = IV-, (19p + 12)t:i± I9quq = zv~, (I9p+I3)f±19qu* = zv'1, (19p + 14)f ± 19qu* = a;2, (19P+ \b)f±\9qif=zv*, (19P + 18)f- ± 19qui = zo%, when q is prime to 19; are all impossible forms of equations in rational numbers. These latter forms are only deduced from ob- servation, upon the supposition that the product ofForms of Square Numbers. a possible and impossible form is also of an im- possible form; which property is, however, ri- gorously demonstrated in the two following pro- positions. With regard to such moduli as are not prime numbers, they are evidently reducible to others that are prime, by means of the indeterminate letter q. ritop. vii. 51. Let a be a prime number, and φ any number prime to a; then, if the series of square numbers, 2ψ, 3ψ, 4ψ, &c., · - - Ψ*, be divided by d, they will each leave a different positive remainder. For if it be possible* that any two of these squares, when divided by a, can leave the same re- mainder, let them be represented by /«2 up + r,, up + &c., represent all the possible forms of square numbers, to modulus a (art. 44); and, since the number of100 Forms of Square Numbers. these remainders r„ n, r3, &e., never exceed? a ■“ i it follows, that there are the same'number of 2 impossible forms ; which may be represented by aq + sl} aq + Sc2, aq+s3, &c.; and it is to be demonstrated, that any one of these impossible forms, being multiplied by any of the above possible forms, will produce an impossible form. For let ηι^φΐϊ&αρ + rm represent any possible form, and aq + sn any one of the impossible forms; then, if (ap + rm) x (aq + su) produce a possible form, m2φ2 x (aq + sn) = aqm2φ2 + snm?φ2 will be the same; or, i„m2φ2, because the first part aqm1 φ2 is divisible by a; but if this be a possible form, that is, if, when divided by a, the remainder be found in the. series of possible remainders, r„ r3, r„ &c., let it be represented by rv, then it is evident, that’ the square ο2φ2, whence this remainder is derived,·, being of the form ap + rc, and φ2 being supposed also of the form ap' + r\, we should have » ί„»ϊ2φ* - ί>®φ2 = (αρ' + rt.) - (ap + rv) » (ap' — ap) divisible by and, consequently, r (sjiftp2 — ϊτφ2) = divisible by a; but a; but (sjtf — mV) = ms(sa — xa), and m is less than a, and, therefore, prime to it; con- sequently, if m°(s„ — 7Γ') be divisible by a, (sn — yr) must be so likewise; that is, w® divided by a must have a remainder sn, or π*ζραρ + s„ ·, but this is an impossible form of squares by the hypothesis, there- fore, (sn — 7Γ9) is not divisible by a; that is, the product (ap + rm) x (aq + sa) cannot, when divided by a, leave for a remainder any number in the series of possible remainders, r„ r,, r,, r4, &c.; and, therefore, the remainder of this product must fall in the other series Sly but 6 is not found there, whence the equation is impossible. Ex. 2. Find whether the equation fF + 1 if = 23zq be possible or impossible. lls*?7w + 4, and 23t*57w + 2· And 4 and 2 being both found to belong to mo- dulus 7, the equation may be possible. Again, ^ttilln + 7) and 23tfclira + 1. Now one of these remainders, 1, belongs to mo- dulus 11, but 7 does not, therefore the equation is impossible. Ex. 3. Find whether the equation 14a? + Gif = 1 fz0' be possible or impossible. 6efci4n + 6, 17 =*14» + 3. And neither 6 nor 3 belongs to modulus 14, there- fore the equation may be possible. Again, 14ste6n + 2, iy^6n + 5. And neither 2 nor 5 belongs to modulus 6, the equation, therefore, may still be possible. Also, 14tfc7?i + 14, and 17—6j*?17»+11.· And neither 11 nor 14 belongs to modulus 17, therefore the equation is possible. In fact, 14 . lla + 6.12=17.10b These examples will be quite sufficient for ex-ΐοβ Forms of Square Nmibers. plaining our operation; it may not, however* be superfluous to add* that* when an equation appears under the form ax2 — by1 = c%2, it is immediately transformed to the sort of equation we have been in- vestigating* by writing it c%2 + by* = ax2. The cases* in which one or two of the coefficients become unity* are evidently involved in the general form above given* and* therefore* need no examples*. prop* x. 54. The equation %2—y2~az2 is always possible in integers. For* if we resolve x2—y2 into its factors χΛ-y, and x — y (which are the only two literal factors that the formula admits of)* and also az2 into any two factors amt12, and mu2, we have* by comparison* v x+y = amt, 1 Qr f x+y = nm2, x—y = nut, j x—y = amt, which, by multiplication, becomes x^—y~ = am'tht, or x1 - y~ — azr, by making z—mtu. Now these equations give, amt + mui , amt —mm 1st, x=-----------, andy —------------·, 2 2 nut + amt mit — amt 2d, x = ■ ■- _ , and y =---—— . On making m — 2, in order to clear the ex- pressions of fractions, they become, 1 st, x = at + it, and y = at — it; 2d, x = it + at, and y — it — at ·. therefore, the equation is always possible in in- tegers.— «.. E. D. * See part ii. chap, iii, prop. 5.ιο7 Forms of Square Numbers. We may also take ra= 1, or" any odd number, only observing, that if a be odd, we must have t and u both odd; for, otherwise, x and y would not be integers. (And if a be even, then u must be even likewise. Cor. 1. If a be a prime number, the solution above given is the only one the equation admits of in integers, for x + y and x—y are the only literal factors of of — y°; and amt and mu2 are the only factors of az2, with regard to form; and, conse- quently, one of the two equalities must obtain; but the quantities t and u bping indeterminate, they will furnish an infinite number of numerical solu- tions. But if a be a composite number, then the equation may have, beside the two solutions given above, as many different literal solutions as there are different ways of producing a by two factors; thus, if a = he, we may have 1st, · f x+y=pamf, ( x — y = mu1, 2(1, < f x+y = bm f, ■ ·? ^ x —y ~ cmtt\ (X + ,„ = »«·, , [x—y=amfr; f x+y — cmif,' \ x — y = bmf. Cor. 2. The equation x2—y2 = az2 includes the two forms xl — az2 = tf, and x~ + az2 if; for, by transposition, the first of these becomes x* — if = az2> and the latter if — x2 =?= az2, which are evidently both pf the same form. Therefore, if it be required to make x1 az2 --if a square, we may haAre x = af — if, or -- u1 — af, and % == 2tu; whence x2 -f az2 = (af + uf ; or we may have x=‘———, and z = tu, Avhich give 2108 Forms of Square Numbers. x2 -t· cϊ— ’e/* + M*V And' to make x° — az2 — if a square, we assume x = af + ?r, and z — 2tu, which give x~ — az~ — (at1 — u2)2, or = (u2 — af f; or we may take x- af + u2 and x == tu. may Cor. 3. But if α= 1, and the equation becomes a’5 + Z'.—y2, then we may have indifferently x= f — u\ andx = 2tu, or x — 2tu, andx = f — u2, unless there be any thing in the nature of the equation that limits these fonns: as, for example, if it be necessary that one of the quantities, x or x, be even; then it i;s obvious, that tbe even quantity must have the form 2tu. With regard to the equation ar — z2=y2, it gives either x = t2 + u2, and x = 2tu, or z — f — u~, both of which values of z answer the required conditions of the equation, Ex. Find the values pf x, y, and x, in the equa- tion Λ' — if — .30x°\ Here the following substitutions may be made, f x+y= mf, or f x+y = 30mf, \ x—y = 3Qmu2, \ x—y= mu2. f x + y = 3mf, f x+y — lOmt2, \x—y=10mu2, \x—y— 3 mu1. f x + y = 2mf, oj. f x + y=lbmf, x—y— 15»m2, f x—y — 2mu2, f x + y — 5mf, ( x + y = 6mf, \x—y= 6mu, 01 \ x—y— bmu2. And making, in each of these, m = 2, in order toForhis of Square Numbers. ICKf hvoid fractions, we have the following general in- tegral values of x and y: f x = f + 30m5, f «= 30 f 4" u\ •i y= f 30 m2, or < ly= 30 f — ui f X — 3f + 10m3, f x= 1012 + 3 m5, 1 y- 3f 10m2, or < l#= 10 f 3 in { x = y- 2f + 15M®, 1 5 m2, or < f # = u= 15 f \bf 4- 2 m2, 2 m5; r X — 5 f* 6i«2, or < fx = 6f 4- 5 m2, 1 y- 5f 6V, iy= 6f — 5 m2; In which formulae, t and u may be any integer numbers whatever. PROP. XI. 55. The two indeterminate equations, x2 4- f = z*9 and x*—yz = z*r, cannot both obtain, with the same values of x and y* For, in the first place, x and y may be con-' sidered prime to each other (art. 47), and, there- fore, x and y both odd, or one even and one odd; and we see, immediately, that it is y that must he even: for if + 1, and y^4n+ 1, then x* ^-yl^4n 4- 2, which cannot be a square; and if χ^Φ,Αη, and y^^AnA- 1, then x*'— y*^AnA- 3, which is also an impossible form; therefore x is odd and y even. Hence, then (cor. 3, art. 54), we nnist have, f x = r" — ly^2rs ‘2d, \ y~ 2tu* Which furnish the following equations: { r3 — s3 =* t3 + u3, rs' = tu.•iio Forms of Sqiiare Nmnbers. Now, in these equations, r is prime to s, andi t prime to u; for otherwise, x and y would have a common measure, which is contrary to the supposition; and, farther, as x~r the above equation becomes aV2 __ JV® = ctW -f r'V2* And here, since r is prime to s, and t to u; r', s', and fe, are all prime among themselves, as is evident ; for if we suppose any two of the quantities to have a common measure, as, for example, a and δ, then, since a and b enter, either separately or connectedly, into three of the above quantities, the fourth, r's% must have the same common measure, that is, t = abj and u = r's', would have a common measure, whereas we have seen that they are prime to each other; and, consequently, r', s', a, and έ, are all prime to one another. Now, by transposition, this equation becomesForms of Square Numbers. Ill were both possible, with the same values of x and y> it would also be possible to find similar equations, Xn+yn=zZn, Xn _.y2 — Wn y ivhich would also be possible, and in which y' < y. And, in the same manner, if these last were possible, we might still find others, { { x"*+y^ = z"*, x'n —y/n = w"*j where y” < y, and so on of others, ad infinitiltn. But it is impossible for a series of positive integers, r, y > v r\ &c., to go on decreasing to infinity, without becoming £ero; in which case our equations are And, consequently, the two proposed equations can never obtain, \vith the same values of x and y, ex- cept when y — o; that is, the double equality x9, -f- y ~ == [ X*—yq=r>lO*, is impossible. — o,. e. d. Cor. 1. Hence, also, it appears, that the tw<* equations, f x*+y2 = 2z\ \ x2 -~y* = 2wq, are impossible, with the same values of <% and y^ for these may be reduced toForms of Square Numbers* 113 f ^ = s;2 + z'V ■- I y« = *»_«;·; tod the two last being impossible, the former ar« impossible also. Cor. 2. The twO equations ( 2a? + y% — z0-, \ 2x?-y* = zo% tire both impossible, with the same values of x and y. For we may consider x and y as prime to each Other; and, therefore, both odd, or one even and one odd; but they cannot be both odd, for then 2x?+yi = 2{4n + l) + (4w/+ l) tt:4» +3> which cannot be a square. Neither can x be even and y odd, for then 2x-—y° = 2(4») — (4«' + l) y?4»i. + 3, which is an impossible form. And if y were even and x odd, then 2a? + f = 2 (4n + 1) + 4n' tfc 4n + 2, which is also impossible; and, therefore, the two given equations cannot both obtain. Cor. 3. And this again shows the impossibility of the two equations f a? + 2y*=±2s2, \,x*-2y* = 2w*% for, by doubling these, we have f 2x^+{2yY = (2z)\ l2tf-(3y)* = (2a>)y which we have seen are impossible, .·, VOL. i. i114 Forms of Square Numbers. PROP. XII. 56. The two indeterminate equations. { X2 + 2if = z2, X2 ■ 2f = W% are impossible, with the same values of x and y. As, in the foregoing proposition, we may con- sider x and y as numbers prime to each other, also, as in that, x must he odd and y even; and, there- fore, we must have (cor. 3, art. 54), 1st, { x r 2'^5 } or { * \y= 2 rs, J \y = 2d> Therefore> 2s* — r°, 2 rs. { 2s2 = t2 + 2n*, or 2s* -- r® = f + 2m*, rs = tu; and it is to be demonstrated, that these two equali- ties cannot obtain at the same time. Now, for the same reason as in the foregoing proposition, r is prime to s, and t to m; also, as in that, we may make r = ar", s = hs', t — ab, and M=rV; which four quantities are all prime to each other, for the same reason as in the foregoing pro- position; and these values, being substituted for r, s, t, and u, give, 1st, r'2d- -2^Ψ = αΨ + 2Λ'\ 2d, 2s'2b2-rnd =αΨ + 2r'V2: which, by transposition, &c., become ^\d-2sn) = b\d + 2sn), b\2sn-d) =r'*(2s'* + a*); and, by division,Forms of Square Numbers. llS a2 + 2/2__r'2 n 2sn + at_bi 1St’ d-2d*~Y'’ 2d’ 2s'*-a*~r*' Now, since .r is prime to d, the numerators of these two first fractions are prime to their re- spective denominators, or they have only the common measure 2; for if a2 + 2.s/2, and a® — 2/% have any common measure, their sum 2a2 will have the same ; but 2d, and a2 + 2/, can have no other common measure than 2, and this can only be when a is even; for; if ft2 be odd, d 4- 2/ is odd, and is, therefore, not divisible by 2; and, if a be even, 5 must be odd, because they are prime to each other: also in this case we may make a = 2a', whence our two expressions become Ha'", and 4a'2 + 2./’; aftd, after dividing by 2, we have 4«/2, and 2an + sH, one of which is even and the other odd, and they are, therefore, in this state, prime to each other; because / is prime to a'. There are, therefore, two cases to consider separately: first, when the nume- rators and denominators are prime to each Other; and, secondly, when they have the common mea- sure 2» In the first case, we must have, 1st, f a2 + 2sn = rn, X d~2 2d, f 2s'* + a* = //, \ 2sn — d — r,i. The second supposition gives, 1st, f d + 2s'" — 2r'~, t d-2s'i=2d·, 2d, f 2sn + dt = 2b% f 2s'3 — d == 2rni Now the second and third of these forms are im~ iπ6 Forms of Square Numbers. possible (corollaries 2 and 3 of the foregoing pro- position) ; and the fourth, being doubled, is similar to the first, being (2sJ+2d = {2b)\ (2s'fi - 2a2 = (2r')\ And, therefore, if the original equation, ( x% + 2y* = s2, \ x2 — 2y2 = w2, be possible, it is also possible to find a similar equation, a2 + 2sn = rn, Ί ( x'2 + 2y’2 = z'% a2 — 2s'2 = b\ j °Γ t x,2-2yn^w'2, in which s ory'f — 2z2, l 2x9- y= wq. \ xr — 2f = 2wq. 5. f x* + 2if = **, 6. f 2xq + f=2z3, Λ o x~ — 2f = w'1. \ 2x*- yq-2w\ All of which are impossible forms when taken in pairs, and similar impossible forms in pairs might be deduced from like investigations; such are the following, the demonstrations of which may be made to serve for practical exercises for the Student. H 2. f X5- ο q • ?/' = *> _ x* + 2y1 = 1 X2- ■ 2i/' = W*. 3. { \x*+ y* = z*, 4. / ■ ιί1 — v2 ■ y —z , x2 + Sy1 = «Λ l AT- . =: U)Q\ 5. { ' #* +2ya = se, 6. f /- 2f = %\ t X2- 3yq=iv\ ys = sV 8. f X2 + f = z\ of + 2lf — IV\ 1 X2- 2yq = re2. a*2-b 10. ff- y = *2, x% +Ay* = w\ t Xs- 4yq = w\118 Forms of Square Numbers. x°- + y~ = 2®, 12.' f **- f = z\ 0? — 3yl = wl. \ xl + 3,«/2=vo1 xf + 3_f/ = 2®, 14. f x1 — 3y- = 2®. x2 + 4y1 == u?, \ x* — Ay*—it? And, generally, the pair of equations, f af± cfs=z2, 1 X? ± f =5 *0% lire impossible, if the two equations, f wr± erf — (c — 1 )p% [m!+ n3 = (c—l)q% x be impossible; and, conversely, if these last two be possible, so also are the former ; the possibility or impossibility of which two last equations may be ascertained by inspection, from the table at page 104. prop, xm. 57. The difference of two biquadrates cannot be equal to a square, or the equation x*—y*==z*· is impossible. For xi—yi = (a2 + f) ( &C-; each of which expressions, reduced to an improper fraction, gives the sides of a rational right angled /ν,β triangle. And if in the fraction 2 rs we make 5=1, and r—2n + 2, our expression becomes 4ril + 8n + 3 4n + 3 = « + ' 4n + 4 4» + 4’ and here, making n= 1, 2, 3, 4, &e., we have this other series, n + - 4n + 3 1^» 2^ 11 15 19 4n + 4 8" 12’ 16* 20 which has the same property as the former 23 o 5—, &c., 24 5123 CHAP. V. On the possible and impossible Forms of Cubes and Higher Powers, PROP. i. 6o, All cube numbers are of one of the forms 4n, or An ± 1. For every number is of one of the forms An, 4«.+ 1, or An± 2. And the cubes of these formulae are (An)3 —64n3 afc4rc, (An ± 1 y = 64n3 ± 48»* + 12n ± 1 ^ An ± 1, (An + 2)5 == 64n3 + g6«4 ■+· 48» ± 8 tte An, Therefore, all cubes are of one of the forms An, or 4w + 1, a. e. d. Cor. 1, No number of the form An+ 2 is a cube. Cor. 2, As in these forms n must be either even or odd, that is, of one of the forms 2n', or 2n' + 1, the above formulae may be again subdivided into the following: \ If n^P.2n*, If n^2n' Λ-1, ( An νί, \ An± 1 tfe8«,± 1. f An z&Snf + 4, \ An± 1 ys8« ± 3. But, since 8» + 4 is divisible by 4, and not by 8,124 Forms of Cubes, and Higher Powers. this form cannot contain a cube; and, therefore, all cubes to modulus 8 are of one of the forms 8η, 8«± 1, or 8w±3. Cor. 3. No numbers of the form 8» + 2, 8n + 4, or 8w + 6, are cubes. PROP. II. 61. All cube numbers are of one of the forms 7n, or 7# ± 1. For every number is of one of the forms 7n, 7n+ 1, 7^ + 2, 7«±3. And the cubes of these formulae take the fol- lowing forms; viz. (7n Y - -- -- -- -- - ^7nt {7n± l)3 = 7V±3. 7V + 3. 7»± l3s7#± i* (7»± 2)3 = 73W3 ±3.2.7V4· 3.24 7«± 8 Sfe7w± (7?i ± 3 )3 = 7 V ±3.3,7 V + 3.3*. 7» ± 27 eft 7»i.± 1. Therefore, all cube numbers are of the forms 7n, or 7n± 1. a. E. D. Cor. 1. No numbers of the forms 7n + 2, 7n + 3, 7n + 4, 7n + 5, can be cubes. Cor. 2. If a cube number be divisible by 7> it is also divisible by 73· And, conversely, if a number be divisible by 7> and not also divisible by 7\ that number is not a cube. Cor. 3. As n, in the above forms, must be either even or odd, we may subdivide these into the following: f 7n zp.\An', \ 7n± l-tftl4«'± 1. n^p.2iiForms of Cubes, and Higher Powers. 123 ’ , , . f 7n ttel4M' + 7j «*2M+1, {f„±la=u„'±6. Therefore, all cubes to modulus 14 are of one of the forms, 14m, 14n + 1, 14m+ 7, 14m+ 6. And, conversely, no numbers of the form 1 4m + 2, 14 + 3, 14m + 4, 14m+5, 14m+9> 14m+10, 14n+ll, 14m+12, can be cube numbers. PROP. III. 62. All cube numbers are of one of the forms 9», or Qn ± 1. For all numbers to modulus 9 must fall under one or other of the following forms, viz. 9m, 9n+ 1, 9m±2, 9m+ 3, 9n + 4; the cubes of which give (9». )* - - - - '.............*&9n, (9m± i)s = 9sms + 3 . 9V + 3. 9m+ i^9«±i, (9 m + 2)3 = 93m3 + 3.2.9V + 3.2s. 9m + 8 ttt 9m +1, (9m + 3)3 = 93m3 ±3.3.9V + 3.3*. 9m + 27 efc9M, (9« + 4)3 = 93M3 + 3.4.9V + 3.4‘2.9m + 64^9m + 1. * Therefore, all cubes are of one of the forms 9m, or 9m + 1. a. e. d. Cor. 1. No numbers of the form Qn + 2, 9m + 3, 9m+ 4, <)n+5, 9^ + 6, 9n + 7 can be cubes. Cor. 2. By applying here the same reasoning as in the corollary above, we shall find, that all cube numbers to modulus 18 are of one of the forms 18m, 18m+1, 18m+ 8, 18m + 9; and, therefore, conversely, no number in any of the forms 18m + 2, 18m + 3, 18m + 4, l 8m + 5, 18m + 6, 1Sm + 7,126 Forms of Cubes, and Higher Powers. 18»+ 11, 18» + 12, 18»+I3, 18»+ 14, 18»+ 1$, 18»+ l6, can be a cube. PROP. IV. 6.3. All cube numbers are of tbe same form to any modulus a, as the cubes Ο3, l3, 23, .3’, &c., {a — l)3. For every number n may be represented by thU formula an + r, in which r < a (art. 11). But {an + r)3 — a3n‘ + 3«°»V + 3anr°- + r’= a(arn3 + 3a»V + 3nr1) + r3, and is, therefore, of the same, when compared by modulus a, as the cube r3; because all the other part of the formula is divisible by a; but since r < it, it must be one of the terms in the series 1, 2, 3, 4, &c., (a—1); and, consequently, all cubes are of the same form to any particular modulus a, as the cubes 0s, l3, 23, 3s, &c., (a— l)3. a. e. d. Cor. 1. Hence, in order to ascertain the forms of cube numbers to any given modulus a, we need only find those of all the cubes less than a; that i% of the series Q3, l3, 23, 33, &c., (a — l)3: and hence a table of those forms might be readily constructed; but as, in many cases, the number of forms would be equal to the number expressing the modulus, no advantage could be derived from the classification, because no numbers are in these cases excluded; thus, to modulus 10 we should have the ten forms 10», 10»+ 1, 10»+ 2, 10»+ 3,Forms of Cubes, and Higher Powers. 127 lOn + 4, 10»+ 5, 10n + 6, 10n + 7, 10ra + 8, 10n + 9; so that no number is excluded with this modulus; and hence it appears, that cube num- bers may terminate with any of the digits, whereas in squares we have seen (cor. 5, art. 43), that they always terminate in Ο, 1, 4, 5, 6, or 9. prop. v. 64. All cube numbers, with regard to modulus 6, are of the same forms as their roots. For all numbers are of one of the forms 6n, 6n + 1, 6n + 2, 6n + 3, 6n + 4, 6n + 5; and the cubes of these formulae will evidently take the following forms: (6n + 0)y (6n+1)3, (6n-f-2)3, (6n + 3)3, ^ (6n + 4)3, (6n + 5)3,_ the same form as "03tte6n + 0 ls:*:6n+ 1 29tfc6n + 2 s 39tfc6n + 3 43y?6n + 4 .53tfe6n + 5 which are manifestly the same forms as the cubes that they represent. Cor. Hence the difference between any integral cube and its root, is always divisible by 6. PROP. VI. 65. The equation (4p + 2)f± 4qus = w3 is always impossible in integers, while q is prime to 4. For (cor. art. 47) the three cubes f, u3, and w3, may be considered as prime to each other. Also all cubes are of one of the forms 4n, or 4n ± 1 (art. 60); and, since 4qu3 is always of the form 4n,128 Forms of Cubes, and Higher Powers. whatever may be the form of u3, we shall have, in the first place, by making fz&4n, (4p + 2)4» + 4qu3 = w3 &ζ4η'; that is, f and tv3 both of the form 4n, which is absurd, because they are prime to each other by hypothesis. Again, if ^^4»+ 1, we have (4p + 2)(4» + l) + 4qu3 = w3, or \6p + 8n+4p+2 + 4qu3 = w3, or 4(4p +2n± p± qu3) ±2 —w3; that is, w’st;4» + 2, which is an impossible form for cubes (cor. Ϊ, art. 6o); and, consequently, the equation (4p + 2)f±4qw> = w3, is impossible, while q is prime to 4. -— a. E. D. The condition of q being prime to 4 is evidently necessary; for if q had the form 2q', then the equa- tion would become (4p + 2)f ±q'.(2u)3 = w3, and the possibility or impossibility would depend upon the form of q', and would, therefore, require a dif- ferent mode of demonstration; hence, in this, and also in the following propositions, q must always be taken prime to the respective modulus with which it enters. Cor. 1. By means of the general formula above given, we derive the following particular cases, which are all impossible in integers, q being taken prime to 4. „ ' 2t3 ± 4u3 = tv3, 6f + 4u3 = w3, 10 t3±4u3 = w3, &c. &c. &c. 2 f± 12 U3 = W3, 6f± 12u3 — w3, \Ot3±\2^-W3, &c. &c. &c.ForMs of Cubes, and Higher Poitiers. 129 And it is obvious how these may be extended to any length at pleasure; and others may be found by taking q = 5, 7> 9, &c< prop. vii. 66. The two general equations; f {7p±2)f± fqus = ws, \ (ip ± 3)1? + fqus = U?, are impossible in integers, while q is prime to 7· For we may, as before, consider t, u3, and w3, as prime to each other; And since it has been shown (art. 6i), that all Cube numbers are of one of the forms 7n> or +1; and, because 'Jqu3 is always of the form 7% we need only give to f the two forms *ln, or 7w± 1, to ascertain the possibility Of impossibility of the above equations.· But in the case of t3 i* In, it is evident, that vf would then hate the Same form 7n; so that i? and zd3, being both of the form 7ri, would not be prime to each other, which is Contrary to the hypothesis; and, therefore, if the equations be possible, it must be when f^7n± 1: let us, therefore, in- vestigate the equations upon this supposition. Suppose, then, fife 7 n± 15 whence the first equa- tion becomes (7^±2)(7«±i)±7<]u3 =m>5, or 49/m ± 14 n ± 7P±2± 7qu* = n>5, or 7(7pn±2n±p±qu>)±2 =uf, or ws7ni 2; which we have seen (cor. 1, art; 6l) is an impossible form for cubes; and, consequently, the first equa- tion is impossible. VOL. 1. κ* 130 Forms qfOubei, cmd Higher Powers·. In the second equation, by giving F the same form, we have (7p±3)F±7qu3 = tv*, or (7P ± 3) (7n ± 1) ± 7qu3 = w3, or 4$pn±2ln±7p±3±7qu3=-w3) or 7(7pn± 3n±p± qu3) ± 3 —w3, or w3z&7n + 3, which is an impossible form (cor. 1, art. 6l); and, therefore, both the original equations are impos- sible. — a. e. D. Cor. 1. By means of these general formulae are readily deduced the following particular cases: 2 F ± 7 U3 = W3, 3f ± 7u3 = w3, 4F± 7u3 — tv3, 5 F ± 7u3= w3, i)F ± 7u3 = w3, loF ±7u3 = to3, &c. &c. &c. 2 F + 14m3 = w3, 3 F± 1 4u3~w3, 4F ±14u3 = ws, 5f± 1 4u3 = w3, <)F + 1 4 m3 = w\ lOi3 + 14 U3=-Vfy &c. &.c. &c. Which are impossible forms for cubes, and they may be farther extended by giving other values to p and q, observing to take q prime to 7- PROP. VIII. 67. The three general equations, f (9p±2)F±9qu3 = w3, < (9P±3)F±9qus-ws, i. (9p± 4)F± 9qu3 = w3, are all impossible in integers, q being taken prime to the modulus 9. For here again we may suppose F, u3, and w3, as being prime to each other; also, Qqu3 is always of the form 9«, whatever be the form of u3: it is, there-Forms of Cubes, and Higher Powers, ί 31 fore, only necessary to investigate the possibility or impossibility of the three given equations, under the different forms of f; viz. when ft&gn, and F ± 1. Now, with i-egard to the first, viz. fmgn, we see immediately, that in all the equations, uf would have the same form 9??,; and, ‘therefore, £3 apd uf would thus have a common divisor, Which is con- trary to the hypothesis, as all the three cubes are prime to each other; 'and, consequently,, if the equations be possible, it must be when ± 1, Now, this form being substituted for f, we have, First equation, l9P ±2)F isqti* = te3, or ,, {9p +2){9n± l) ±9qu3 =w% or 81 pn± i8»± 9p± 2 ± 9qu3 — uf,;or 9(9Pn ±2n±p± qu3) ± 2 ' — tip, ■ or w3mgn± 2, which is an Λ impossible form for cubes (cor. 1, art. f>2); and, therefore, the first equation is .im- possible. The second equation gives (9p±3)f ±9qus — ic3, or (9p±3)(9^±l)±Qqu3 = ws, or 8 lpn± 2f 11 ± 9p + 3 ± Qqu3 = w3, or 9(9pn±3n±p ±qu3) ±3 = w3, or ιν3ζ&9η±3, Which is likewise an impossible form for cubes (cor. 1, art. 62); and, therefore, the second equa- tion is impossible. In the third equation, we have^. κ 2132 Forms of Cubes, and Higher Powers. (9P± 4)^±99m’ —iP, or (9p ± 4) (9« ± 1) ± 9QU$ = ws, or 81 pn±36n±Qp±4±9qitl=io>, or 9(9pn±4n±p±qus)±4 =ws, or tv,its9n±4, which is also an impossible form for cubes (cor. 1, art. 6 2); and, consequently, the third equation, as well as the first and second, is impossible.—a. e. d. Cor. The above three general equations furnish the following particular eases of impossible form* of cubes: 2/1 ± 9 m*=u>9, 3i3±9u,=wt, 4t* ±Qua = w’, bf + ==?£>’, 6^±9«’=m>5, 7 i*±9tt’=w% 11 &c. &c. &c. 2/5 Hr 18us = Wr, 3f±l 8u3~ioa, 4f± 18 us = w3, 5f* + 1 8m’ = uf, 6<* ± 18tt?=i»*, 7<’± 18tt’ = tt>s, ΙΙί’ + Ιβ»^^, &c. &c. &c. Which, like the other tables of impossible forms, may be carried to any length at pleasure, by giving different values to p and q; observing always,, that q is prime to modulus 9* PROP. IX. 68. If there be any case in which the equation x>—f = zs be possible, the following conditions must obtain; viz. C x—yvzr*, < x — «ttes5, or two of these quantities will be of the form her*Forms of Cubes, and Higher Powers. 133 ■given, and the other of the form 9φ5, which resolve into the four following cases; viz. C x—y up. r3, f x^y^9f, 1st, ■ ( X — ZHi s?, 2d, < : x — z vr s3, ty + ztte f. + f, f χ-y** f, Γx-yzp Vs, 3d, ■ < x-z^Qs3, 4th, - c a? —2 tft s', by + Zito f. ly + zi* 9P. And it is to be demonstrated, that one of these four conditions must obtain, if the equation x? — ys = 2s be possible; where r5, s’, and f, may be any numbers whatever, indicating only that x—y, x — z, y+z, are complete cubes, or that they are of the forms there given. In the first place, we may consider x, y, and z, as being prime to each other (cor. art. 47); and since x>y, put x—y + d; then, because x is prime to y, d is prime both to x and y, for if y and d had a common measure, x would have the same, because x=y + d; and if x and d had a common measure, y would have the same, because y=x—d; and, therefore, since x and y have no common measure, d is prime both to x and y. Nqw, substituting y + d instead of x, in the given equation, it becomes {y + df—y3 =**’, or 3fd +3yd + d ~£s, or d{3y* + 3yd + d*)p*z3. And here, since d is prime to y, it follows, that the first side of this equation is divisible by d once, and after that, neither by d nor by any factor of d, unless 3 be one of its factors, in which case it is divisible by 3d once, and after that, neither by d134 Forms of Cubes, arid Highkr Powers. nor by any factor of d. For of the thr£e terms (3y* + 3yd+d2), which form the quotient, two of them have d enter into their composition^ and are, therefore divisible by d; but in the ether term, y* is prime to d, atid, consequently, the whole quantity taken collectively is also prime to d, unless 3 be one of the factors of d> in which case, as we have said above, the first side will have 3d for a divisor, but after that, the quotient .will be prime to d; and whatever is true of the first side of the equation is evidently so of the other side £3, because they are equal quantities; and, consequently, z* is divisible by donee, and after that neither by d nor by any factor of dP unless 3 be one of its factors ; in which ease, it is divisible by 3d once, and sifter that neither by c? nor by any factor of d: and, therefore, d in the first case, and 3d in the second, must be complete cubes (cor. I, art. l6); that is, we must have either d^p.r3, or d^Qr3, in order that 3d^3V: or, since d=x—y, it follows that x—y ^r3, or Now if the equation x3 -r:y3 = z3 be possible, so likewise is x3 — z3—y3, and it is evident, that, were we to consider the equation under this form, the result would be exactly similar to that obtained above; viz. x — z must be of one of the forms s\ or 9s3. Again, the same equation, by transposition, be- comes ?/ -f %3 And making y + z — m, or t) — m — z, we have {tn — %)3 H-V==+("3 ~rT'" And subtracting the first equation from the sun* of the two latter, to which it is equal, we have ( t9 — 3 f(s* + r3) + 3f {2(f — r3)2 — (s? + r3)2} l (.Vs + r3)3 = 0; or ( t9-3f(s> + r?) + 3f(^ + fy-(f \hriy^ | 24#W, because . 2(s3 — r3)2 + 8sV = 2(.s3 + rJ)2, whence { f — (s" + r3) }3 =?. 24 fs3r3. And here the impossibility of the equation i& manifest, under the present supposition; because we have got an integral cube, equal to three times, another integral cube, which is absurd. - But by substituting Qr3 for r3, <)sJ for s', and Qf for f, the impossibility is not so immediately obvious; for, in these cases, by putting f)r’ for r3, we have {f — (a·3 + Qr"‘) ]9 = 2l6iVr3 = (6for)3. Again, writing fM'3 for we obtain ! f - (9s3 + r3} J ■3 = 216i\s r = (6fer)*.‘ And gf for f gives (9 f - (f + r3) }* = 2i6Wr* = (6tof.Forms of 43ubes\ tend Higher Powers. 139 And it fceffe remains to be shown, that these equalities cannot subsist. Now these equations, by extraction, become, .·> 1st, f- tf-'9r*=*6ter9 ' gd, #3 —9/ — r3 = 6fov. 3d, 9f- s*-\r3 = 6tsr. Whence, again, by division, we have 1st, 3d, e t _.9^_ 6 sr ' tr ~ st ■ t\. _9£_ : r* a sr tr ’ M ~ 0* 1 ^ 1 ..s' r5 6 sr tr ~~st = \j 0 And one of these equations must be possible, if the equation whence they were derived be so. But, since r, '$> and t, are prime to each other, each of the above fractions is in its simplest form; and they each contain a factor in their de- nominator, that is not common with the other de- nominators ; and, therefore, these fractions cannot any how combined be equal to an integer (cor. 2, art. 13). Having therefore shown, that if the equation r3 — tf — z3 were possible, one of the above ex- pressions must be equal to the integer 6; and having also demonstrated, that these fractions can- not be equal to any integer whatever, or that the above equalities are impossible; it therefore ne- cessarily follows, that the equation whence they were derived is so likewise; that is, the equation s3 is impossible: blit the impossibility of the equation tf3—y3== s3 involves in it the impos-140 Forms of Cubes, and Higher Powers. sibility of the general equation a?±yi=z,i and, consequently, the equation x,±ys=:z> is impossible in integral numbers. — a. B. D. Cor. Since xs ± y3 = z3 is impossible, so likewise Is -ψ±-~ = -pr, for this may be reduced to , , ' ' . z3p3 a3 where the latter cube must be an integer, which we have seen is impossible; therefore, the equation cannot obtain, either in integers or fractions. PROP. XI. 70. The third differences of consecutive cube numbers are constant, and equal to 1 . 2.3 = 6. For let {x-lf, x3, {x+ l)3, (* + 2)’, represent any four consecutive cubes, then {x — l)3 = xf- -3#2 + 3x — 1, X? = X3, (#+ l)3 = a^ + 3x* + 3x+l, (# + 2)J = a;5 + 6a;2 + 12a? + 8, Γ3χ*-3χ+ 1, 1st differences, \ 3d?2 4- 3d? + 1, V 3x* 4- 9«r 4- 7 * 2d differences, f 6x} \ 6x + 6. 3d difference, = 6=1 .2.3, Cor. 1. In the same manner it may be shown, that the third differences of cubes, the roots of which are in arithmetical progression, are equal toForms of Cubes, and Higher Po wers. 141 1 .- 2.3.. if , .where d is the common difference of the roots. Cor. 2. The second differences of consecutive squares are equal to 2; or to 2d2, if their roots form an arithihetical series, whose common dif- ference is d; which is readily demonstrated on the same principles as those employed above. PROP. XII. 71. Every complete biquadrate, or 4th power, is of one of the forms bn, or 5» + 1. This is evident, for every square number is of one of the forms bn, or 5n± 1; and a 4th power being the square of a square, we have (5n )2 = 52«* ttibn, (bn± l)4 = 52»4± 10« + 1 tt?5n + 1; therefore, every 4th power is of one of the forms 5«, or5w+l.— a. E. d. Cor. 1. If a 4th power be divisible by 5, it is also divisible by 54. And, conversely, if a number be divisible by 5, and not by 54, that number is not a 4th power. Cor. 2. No number of the form bn + 2, or bn + 3, or bn + 4, is a biquadrate. Cor. 3. Every 4th power being of one of the forms bn, or bn + 1, we have, by supposing n even and odd, the four following fdnns to modulus 10; viz. n^2n' «tfc2ny+ 1 { { bn tfclO n', b7i T 1 i* I On' + 1; bn t&lOn' + b, bn+ imlOn' + 6; and hence every 4th power terminates with one of142 Forms of Cubes, and Higher Powers. the digits, 0, 1, 5, or 6.' And,, conversely, no number terminating in 2, 3, 4, 7? 8, or 9, is at biquadrate. Cor, 4. Since it is demonstrated (cor. 2, art, 42), that all even squares are of the form 4», and all odd squares of the form 8ft-H J, we have for ,tlie squares of these (Art )3 = 1 Gir efc 16n', (8ft 4- 1 )3 = 64ft24- l6ft4- l^l6«'+ 1. And, consequently, every complete 4th power is of one of the forms l6‘«, or 16ft 4-1; that is, every even 4th power is of the form l6ft, and every odd 4 th power of the form 16ft 4-lw PROP. xiii. 72. All 4th powers are of the Same form with regard to any number a as a modulus, as the 4th powers Ο4,; l4, 24, 34, &c., (ia)4, ; ' ' when a is even; and as when a is odd. For every number whatever may be represented by the formula an±r, where r never exceeds 4ft (art. 10). But {an + r)4 = a4rt* ± Aa’rfr + 6aW + Aanr3 + r4, and all the terms, but the last, of this expression, being divisible by a, the whole quantity is evidently of the same form, with regard to a as a modulus, as the last term r4; but r never exceeds 4a, therefore,Farms of Cubes, and Higher Powers. 143 every 4th power to modulus a is of the same form as the 4th powers Ο4, l4, 24, 34, &c., (4«)4, when a is even; and as Ο4, l4, 24, 34, &c., ’ w^en a *s 0Ror. xv. 74. The three general equations C (bp + 2) t4 5^M4 = w*$ < (5/> + o) t' ψ 5 qu4 = w4, t (5p + 4)f + bqu4 =w*i hre impossiblej q being taken prime to 5. For, in the first place, we may always consider t, u\ arid nf, as prittie to each other (cor., art. 47). And since all 4th powers are of one of the forms bn, or bn + 1, we shall have, by giving to t4 these foririSj the following equations j If t' an 5« j f (op+ 2) x 5ndip bqh4 — 1C4as5n', 1. < (bp + 3) x bn^5qii4 = w4^tibn'j (bp + 4) x bn + bqu4 = w4as bn't And if i4as5» + li {(bp+ 2) x (bn+ \)dpbqu4—w4^bn±2, (bp + Ά) x (5«+ l)=fcbqu4 = tv**ti5n±3, (bp + 4) x (bn + 1) + bqu4=M>4as5«.+ 4. L146 Forms of Cubes, and Higher Powers. Now, in the first set of these equations, in which it is supposed that f4sfc5n, we have evidently, also, ζϋ*ϊ&.5η', whereas it has been seen, that t\ id, and zo*, are prime to each other; therefore, the equations are impossible, when tf^.bv. Also, in the second set of equations, in which £4 ste 5» + 1, we have {w* ikbn± 2, η,4 'Φ. bn + 3, IV* aft 5 n + 4'; which are all impossible forms for 4th powers (cor. 2, art, 70· Therefore the equations C (bp + 2)t*^b qu4 = w*, < (bp + 3) i4 dp bqu4 = itf, f (bp + 4)t* + bqu* as»w*, are impossible, either in integers or fractions,, a. E. if* Prop. xvi. 75» The general indeterminate equations (l6p + r )tf4 τ 2u* = iv*, (1 βρ -tr')t + 3n* =»u·’4, (lbp + r")f + Aid = w4, &e. &c. &e» are impossible, r being any number > 1 and less than 14; that is, r< 14, r’ < 13, r" < 12, &c. For t*, u*, and m4, being prime to each other, and all 4th powers, being of one of the forms ΐβκ or l6n+ I (cor. 4, art. 70, it follows, that either t* and- u* are each of the form l6»+ 1, or one of them is of this form and the other of the form 160 ; which suppositions furnish the following cases:Forms of Cubes, and Higher Powers. 14 f 1st, #4ifel"6«+ 1, and M4tfcl6n'+ 1. Then f (\6p-+r ) x (l6^+l) + 2(i6w/ + i) = !j;4te { 16»" + »·+ 2, - f (i6p + r') x (i6p + 1) +3(l6ra'+ i)±=n>4sfc· \ 16n" + r' + 3, f (l6p + r') x (l6p + 1) + 4(l6w' + 1) == t£J4y=i \ l6n" + r" + 4. &Ci &C. &Ci 2d, t^p.16n, and li 16n' + i. Then C{\6p + r ) x l6n +2{l6n'+ΐ) — ίϋ4αζΐ6η"+ dj, 2. < (ifip + r') x l6n + 3(l6rc'+l) —U/4skl6rc,' + 3, (_ (lfip + r") x l6n+ 4(16rt' + l) — w4^p;l6n" + 4, 3d, f sfc 16n + 1, aiid u4sfc 16n. Then Γ(16p + r ) x (i6n.+ i) + \6n' ^w4^iQn" 4-r ; ■ 3, < (16p + r') x (16n + ί) + 16V = w4 tfc 16n" + r' j (16p + r") x (16n + 1) + 16n' ~ to4 m 16n" + r". Now, in the first set of these equations, we have 16n*' + r + 2, 16n” + r' + 3, l6« + r" + 4; and since f< 14, r' < 13, r” < 12, it is evident, that each of these forms = 1 (m + ; some quantity greater than 1 j and less than lb;-and, therefore, they are all im- possible forms, by the converse of cor. 4, art. 71· And the second set of these equations are evi- dently impossible, as are also the third; because r, r', r", &c., are each >1,' but < l6. And, con- sequently, all the equations in these forms are im- possible.— a. e. D. Scholium. The above set of forms furhistiefc an in- finite number of impossible formulae for biquadrateS, Or 4th powers, and various others might have! been found; but as all thdse given for squares ate equally applicable to 4th powers, it would be useless 1-2148 Forms of Cubes, and Higher Powers. to multiply them by other particular cases, as the methods which have been explained, the reader will easily apply to any particular case that may occur; and as he is thus furnished with so many cases of the impossibility of equations of the forms ax? ± by~ = &2, ax? + by3 — z3, and ax' ± by' == z', which might have been carried to a much greater extent; it will always be proper, when any equa- tions of these forms are proposed, to examine, first, whether they be possible or impossible; as, in the latter case, much unnecessary labour will he avoided. But it may be necessary to caution the young practitioner, that though an equation may fall under a possible form to one mo- dulus, it may be impossible under another; and, therefore, that it is not so easy to show that an equation is possible, when it really is so, as to show the impossibility in those cases that are impossible. In short, there are no means of showing that an equation, wrhich exceeds the 2d degree, is possible, but by solving it; but the impossibility may be fre- quently demonstrated by the methods above taught. PROP. XVII. 76. No triangular number, except unity, is a biquadrate. For, if possible, let x(x +1) 4 . • ------— =/, or a?(ar+ 1) = %/ ; now, since the two factors sc and x+ 1 differ from each other only by unity, they are necessarily prime to each other; but, if 2yy be resolved into- twoForms of Cubes, and Higher Powers. 149 factors prime to each other, they must be 2m4 x n4; for it cannot be otherwise resolved into factors, that are prime to each other, and this leads to the fol- lowing equations: f x = 2m4, , f x — n (a?+l=«4. \x+l=2 mi The first gives n4 — 2m4 = 1, and the second 2 m4 — η4 — 1. The latter of these equations, by transposition., becomes 1 + n4 — 2m4, which is impossible (cor. 2, art. 57); and, from the first, we derive ms + n4 — (m4 + 1 )2, or (m3)4 + n4 == (m* + 1 )9, which equation is also impossible (art. 58); there- fore, no triangular number, except 1, is a bi- quadrate, PROP. XVIII, 77· The 4th differences of consecutive 4th powers are constant, and equal to 1 x 2 x 3 x 4 = 24. For let x — 2, x— 1, x, x + 1, x + 2, represent the roots of any five consecutive 4th powers; then (x + 2)4^x4 + 8a3 + 24a? + 32a? + l6, (a? + l),=a?4 + 4.i;1+ 6a?2 + Ax + 1, (ar + 0)45=a?4, (a:—l)4=5=a?4 —4a?* + 6a:2— 4x+ 1, (x — 2)4 = x4 — 8a3 + 24a? — 32a? + 16. 1st diff,- 4a3 + 18a?2 -f 28a; +15, 4a?3 + 6a?2 + 4x+ 1, 4a;3— 6a;2 + 4a?— 1, 4af — 18a·2 + 28a? — 15ΛI SO Forms of Cubes, and Higher Powers* 4tildiff. =24 = 1 . 2 . 3 . 4. And if the common difference of their roots be then the 4th differences will be 1 .2.3.4 . d*. PROP. XXX. 78. Every 5th power is terminated with the same digit as its root. Or all 5th powers are of the same form, with regard to modulus 10, as the roots of those powers. For all numbers to modulus IQ are of one of the following forms: (10m )5 051 oV oil On" (lOn+l)505l0n'+1 O5l0n"+1, (1 On + 2)5 051 On' + 25 051 On" + 2, (1 Om + 3 )5 051 On' + 35 051 On" + 3, (1 OM + 4)5 051 On' + 4s Oi 1On" + 4, (10M + 5)505 10m" + 5'051 On" + 5, (ion + 6)4θ5 ion' + 65θ5 iom" + 6, (1 On + 7 )4051 On' + 75 051 On" + 7, (10m+ 8)’Oil On' + 85 05 10m" + 8, (io?x + 9)5 05 ion' + 95 Oiion" + 9. Where the latter formulae are evidently the same as the first; and, consequently, the powers have the same forms to modulus 10 as the roots of those powers, or they are terminated with the same di- gits. — a. e. D, a. e. p.Forms of Cubes, and Higher Powers. 151 Cor. It has been demonstrated (art. 64), that all cubes have the same forms as their roots to modulus 6; and, in the above proposition, that all 5th powers have the same forms as their roots to modulus 10; and the same is universally true for prime powers; namely, that they are of the same form as their roots to modulus double the exponent of the power; viz. all 7th powers are of the same form as their roots to modulus 14, and 11th powers of the same form as their roots to modulus 32: and sq on for any other prime powers, PROP, XX, 79· The 5th differences of consecutive 5 th pow- ers are constant, and equal to 1 . 2,3 , 4.5 =120. For let x-r2, ®—1, x, ®+l, ®+2, x-\-3, re- present the x'oots of any six consecutive 5 th powers, then (® + 3)5 = x5 4- 15®4’ + 90®5 4"« 270®° + 405® + 243, (® 4- 2)5 ~xb + 10®4 + 40®3 4- 80®2 + 80® + 32, (® + i)5 ~X5· 4- 5®4 + 10®3 4- 10®“ 4- 5® + (x 4- oy ==.r5 (X — vy — 5®4 4- 10®9 -· 10®2 4 5®— 1» (® — 2y = xJ — 10®4 4- 40®3 — 80,r 4- 80® — 32. 1st diii. -< 5 a?4 + 50,.;° + iqo.r + 325,r + 211, 5®4 + 30®3 + /Οι?2 + 75® + 31, 5x4 + 10®1 +10®*+ 5® + 1, 5 a?4 — 10®3 + 10a?2- 5® + 1, 5,v4 — 3Q.F.+ fOx* — 75x+ 3U V153 Forms of Cities, and Higher Powers, 3d diff. f l 20x3 + 120x" + 250λ? + 180, 20od + 60#® + 70^+ 30, 20,r3 + Q#* + 10# + o, 20#’ — 6‘0#3 + fOx — 30, {6o#® + 180# + 150, 60#2 + 6 or + 30, 60#2 — 60# + 30, 4th diff. 5th diff f 120# + 120, 120# + 0, = 120=1 ,2.3 4.5. a. e. d, And if the roots of any set of 5 th powers form an arithmetical progression, the common difference of which is, d, then will their 5th differences be equal to 1 . 2.3,4,5 , d\ Scholium,· It is also undoubtedly true, that the nth differences of consecutive nth powers are con- stant, and equal to 1 , 2,3 , 4.5.6, &c. n; but it is difficult to demonstrate this on pure ele- mentary principles. The demonstration appears to rest on the following theorem; viz. the product 1 , 2.3 . 4,5, — — nn—n{n—l)" + n(n— 1) 1 . 2 (ra — 2)* n — n(n— 1)(« — 2) ’ 1 ,2,3 (n — 3)” + &c. which is readily deduced from the Differential Cal- culus; but a demonstration, founded on those prin- ciples, could not, with propriety, be introduced into an elementary work of this kind,153 CHAP. VI. On the Properties of Powers in General. PROP. i. 80. The difference of any two equal powers, of different numbers, is divisible by the difference of their roots; that is, xn—^ = u{x—y)*. For make x—y — d, or x = d+y; then x"—yn be- comes (d + y)n—yn- and we have to demonstrate, that this expression is always divisible by d, or x-y. Now, by the developement of (d+y)n, and writing for the coefficients of the respective terms, 1, n, m, p, &c., η, 1, we have (d+y)n-yn = dn + ndn ~ 'y + mdn' y + pd" ~3y3 +-ndy” ~1 = d(dn~1 + ndK~°y + mdn~3if+pdn~iy% +-nyn~l), which latter fonn is evidently divisible by d; and, consequently, the equal quantity (d + y)n—y", or xn—yn * The abridged expression m(x—y) indicates a multiple of the quantity within the parentheses, and may be read either xn-~yn equai a multiple of x—y\ or xn—yn divisible by x—y-154 Powers in General. is divisible also by d, or by x—y, since x—y — d; that is, Xn — yn = M(x — y). G. E. D. Cor. 1. When x is prime to y, then d=x—y is prime to both x and y (cor. 1, art. 7); and, com sequently, the above quotient dn~x + ndn~'y + md’^ 'y1 + pdn~*y% + - - - nyn~x — d(d'*-2 + nd"~hj + mdn~iyl + pdr‘~ by* +----) + nyn~x is prime to d, unless the power n be equal to d, or some multiple of d; for all the terms, except the last, are divisible by d; but the last ny’l~l is prime to d, unless n be a multiple of it; because we have seen that d is prime to y, and, therefore, the quo- tient is not divisible by d, except in the latter case; and hence we conclude, that x"—y'i is always di- visible by x—y once, but after that, the quotient is not again divisible by x—y, unless n=x—y, or n= some multiple of X—y . Cor. 2. If n be a prime number, then the co- efficients of the expanded binomial d + y may be represented by 1, n, na, nb, &c., nb, na, η, 1 (cor. 1, art, 12), in which case {d + y)n—yn becomes f dn + ndn~xy + nad’l~"~y~ + nbdn~3y\ ■% - - nad'y"~Cl + t ndyK-{ = f d(dn~' + ndr‘~9y + nadri~']y’1 + nbdn~*y*, - - - nadyv~~ l + which, being divided by d, gives for a quotient f d(d*~* + nd*~9y + md*~4y> + nbdn~by}, - - - nay"~% 1 + »/"*,153 Powers in General. which is evidently not again divisible by d, unless n^=d; for since n is a prime, it cannot be a multiple of d. But this quotient may be divisible by n, if n be a factor of d; for if we make d = nd', the above becomes f nd’\ (nd'Y"^ + n(nd{Y~3y + na(nd')n~iyi + - - -t ' \ way®} +■ ny“~l, which is evidently divisible by n, giving for a quotient f d'{ (ηά')η~Λ + n(nd')n~3y + na(nd')"~4y3 + — ·* l my*-*} 4-y1; but this quotient is not again divisible by n; for, since n is a factor of d, and d is prime to y, y" “1 is prime to n; and, since all the first part of this quotient is divisible by n, but the other part, yn~ \ is prime to n; therefore, the whole quotient is also prime to n. And hence we conclude, that the difference of two powers, xw—yn (when x and y are prime to each other, and n is a prime number) can only be divided once by x—y\ and after that, nei- ther by x—y nor by any factor of x—y, unless n — x—y, or some factor of x—y, in which case, xa—yn is divisible once by n(x—y), but after that, neither by n nor by (x—y), nor by any factor of x-y. Cor. 3. The quotient really arising from the division xn—yn by x—y is —-- = xn~' + xn~qy + xn~tyi + xn~4y$ +----xy*~* x—y which quotient, therefore, from what has been shown above, is always prime to x—y (x and y being156 Potvers in General. supposed prime to each other); except when n is equal to x—y, or when it is some multiple or factor οϊχ—y. PROP. II. 81. The difference of two equal powers is always divisible by the sum of their roots, when the ex- ponent of the power is an even number; that is, xn—yn = u{x + y), when n is even. For make x+y=s, or x=s—y, then xn—yn be- comes (s—y)n—yn, which we have to prove is always divisible by x+y or s. Now by the developement of (x + y)v, and waiting fpr the coefficients of this expanded binomial, 1, nj w, p, &c., m} η, 1, it becomes is~yY-yn- f sn — nsn~'y+ tnsn'2y2 — psn~3y3 + --- + ms!yn~’1 — \ nsy’1'1; because, since n is even, the last term of (s—y)n, namely yn, will have the sign +, and will, there- fore, be cancelled by — yn. Now this quantity may be put under the form f s(sn~1 — nsn~-y + msn~3y2 —psn~*xf +-+ msyn~* 1 -nyn-'), which is evidently divisible by s; and, consequently, the equal quantity (s—y)n—yn, or xn — yn is also divisible by s, or by x+y, that is, xn-yn = u(x + y), when n is an even number. — a. e. r>. Cor. It may also be demonstrated, by the same reasoning as that employed above (cor. 1, art. 80),Powers in General. 157 that, if x and y be prime to each other, then x* -~yn can only be divided once by x+y> unless n be equal to x +y, or some multiple of x 4- y. PROP. III. 82. The sum of two equal odd powers is always divisible by the sum of their roots; that is, xn + y’‘ = M(x + y), when the exponent n is an odd number. For make x +y=s, or x=s — y, then xn + yn be- comes (s —y)“+yn; which we have to demonstrate is always divisible by s, or oc + y. Now by the developement of (s—y)*, and writing 1, n, m, p, &c., m, η, 1, for the coefficients of the expanded binomial (s—y)n, we have (s-y)n+yv = ( ,vn — ns”' 'y + ms*' Sf — ps" ~ “’-if +-+ msSf" * — 1 nsyn'\ for since n is odd, the last term of the expanded binomial (s~y)n, or yr\ Λνϋΐ have the sign —, and will, therefore, be cancelled by 4- yn. And this expression may be put under the form f s(sn “1 — nsn~ hj + msu “ 3yl — ps11" h/3 4--4- msy11"s l -ny'1-1), which is evidently divisible by s; and, therefore, the equal quantity (s—yY+y*, or χΆ + yn, is also divisible by s, or by x + y, that is, xn +y* — M{x + y), when n is an odd number. — a. E. d. Cor. 1. It may also be demonstrated, by the same158 Powers in General. reasoning as that employed at cor, i * ait- 80, that if x and y be prime to each other, then xn +yn can only be divided by x + y once,- rinless ft == (# -f cl), of some multiple of (x + d). Cdr. 2. And if n be a prime number, and x and y prime to each other, then xn + yn can only be di- vided by x + if once; and after that neither by x+y nor by any factor of x+y, unless n be one of its factors, in which case it may be divisible by n{x-\-y) once, but after that neither by n nor by («r + 7/), nor by any factor of (x+y); as is evident from the same reasoning as that employed at cor. 2$ art. 80. Scholium. By means of the three foregoing pro- positions, and their corollaries, we may draw the following general conclusion with regard to the divisors of the formula xn±yn; viz. 1. If n be even, or of the form 2n'9 then (xn—yn)i or (xQn' —y^1') — m(x +y); and m(x —y). But if x he prime to y then will x*n'—yqn' be di- visible only once, by each of those cpiantities, unless 2n' ~x+y, or <£ —i/, or some multiple of one of those quantities. 2. If ή be odd, or of the form 2ri -f 1, then (af — y11), or (a?s”#+1 —y*'1'*1) — m(x— y) ; and (xn +f)> or (ο^71'+ί +/η'+1) ~m(x +y). But if in these formulae x and ?/ be prime to each other, then each of those quantities are only di- visible by their respective divisors once, unless 2η' + 1 ζζχ—yt or some multiple of x—y> in the first case; or 2n' -f 1 ~x-\-y, or some multiple of x +y, in the second cate. And if in these two last forms.,Powers in Genera L 159 φ arid y be prime to each other, and n be a prime number, then, in the first form, xn—yn is divisible by x—y once, and after that neither by x—y nor by any factor of (x—y), unless n be one of its factors; in which case xn—yn is divisible n(x—y) once, and after that neither by n nor by (x—y), nor by any factor of (x—y). And in the second form, (xn+yK) is divisible, by x+y once, and after that neither by (x + y) nor by any factor of x +y, unless n be one of its factors ; in which case it is divisible by n(x + y) once, and after that neither by n nor by (x +y), nor by any factor of x +y. By means of the above propositions, We are also enabled to ascertain the divisors of the sum or dif- ference of unequal powers of the same root; viz. (xm — xn) == m(;c — 1), and m(# +1), when m — n is even, or of the form 2n', for' xm-af-af‘x(xn-**-l)i and since m — n^2n', therefore, 1) = l2") =M(a\— 1), andMfr+i); and, consequently, xn x (xm^n— l) = (xm — xn) — m(x — l), and m(x + l). Again, if n — ni be odd, or of the form 2uf 4-1, then (xm — xn) = u(x—l), and (xm + xn) = m(x + l). For (*“-Λ·")=Λ'η X ί), and (xm + Xn) = Xn X (xm ~ * + 1) ; also, since m — n^p 2n' + 1, therefore,l6o Powers in General. (xm~* - l) = 08n+1 - l2n+1) = m(x-1), aiid (xm-n + 1) - (x9n+I + 1tn+') = m(x + 1); and, consequently, xn x (x*1-* + 1) = (xm + χη) == m(x + 1). Lemmct. 83. In demonstrating the impossibility of the eqμation xn ± j/” = zn, it will be sufficient to consider n as a prime number. For suppose n be not a prime, but equal to the product of two or more prime factoi’s, as n =pq, then the equation becomes χ"±y*. :**«=* (x?)?± (yry-fia (ζρ)% being a similar equation, in which the power q is a prime number; and, therefoi'e, if the equation be possible when n is a composite number, it is alsoi possible for a prime power; and, conversely, if the equation be impossible when the power is a piime, it is also impossible for every Composite power; we shall, therefore, in what follows, consider n as a prime number. Again, we may always suppose x, j/, and %, as prime to each other; for it is evident, in the first place, that two df these numbers cannot contain a common divisor, unless the third contains the same. Suppose, for example, that xn and yn contained any common divisor, as V“ + ΦΥ” — φ V“, or, dividing by the greatest common divisor, x'd±y'n-z>n: ifj therefore, the equation xn±y* — zn be possible,* when x, y, arid z, have a common divisor, it is also possible after being divided by that common divisor, and ill which latter equation the three re- sulting quantities, xf, if, and z\ are prime to each other; and, conversely, if the latter be impossible, the former is impossible also ; We shall, therefore, only consider the cases in which x, y, arid z, are prime amongst themselves. It will also be sufficient to consider the ambiguous sign + under either of its forms + or — ; for if the equation xn+yn — zn be possible, so also is the equatiori a” —y* — x"; arid if the equation be im- possible Under the latter form, it is likewise im-* possible under the former. We shall therefore limit our demonstration to the equation xn—y’a==zn, in which n is a prme number, and x, y, and %, numbers prime to each other 5 the impossibility of which, from what is Said above, involves with it the impossibility of the general equation xn + yn — zn, when x, y, arid z, are any numbers whatever, and it any number ex- cept 2, or some power of 2. Now, with regard to ti — 2, we know*, that the equation is not impossible, but the case of it equal 4 has been demonstrated to r \i6s Powers in General. be impossible (art. 73); and this latter case involves- that of every higher power of 2, thus = Zs = (λ;5)4 ± (y*)4 = (»T, which being impossible in the latter form, is ne- cessarily so in the former; and, in the same manner, the impossibility of the equation for any higher power of 2 may be shown to be involved in that of 11 — 4: it is evident, therefore, that our equation, together with that of n = 4, involves every possible value of n greater than 2. PROP. IV. 84. If the equation xn—y* = zn be possible [n being a prime number, and x, y, and z, prime to each other), then one of the four following con- ditions must obtain; viz., C x—y t&rn, f x—yd?nn~xrn, 1st, · < x—zms”, 1 X 2d, - c x — zd?s*, ί x—yd? r", C x—y^rn, 3d, < x — zd?nn~'sn, 4th, < i X — Zd?,f, \3/ + Z*Rtn. Where r, s, and t, may represent any numbers whatever, indicating only, that (x—y), (x — z), (y + z), &c., are complete nth powers, or that they are of the form rn, sn, t‘. This follows from what has been demonstrated cor. 2, art. 80; viz. that xn—yn is divisible by x—y once, and. after that neither by x—y nor by any factor of x—y, unless n be one of its factors, in which case xn—yn is divisible by n{x—y) once, and after that neither by n nor by x—y, nor by any.163 Powers in General. factor of x—y; and the same must necessarily be true of the equal quantity a”; vis. that it is divisible by x—y once, or by n(x—y) once, when n is a factor of x —y, but after that it is neither divisible by n nor by x—y, nor by any factor of x—y, and, therefore, (cor. 1, art. i6) x—y, in tlft first case, and n(x—y) in the second, must be complete nth powers; that is* x—yt&rn, or n[x —y) t&rn; hut the latter of these forms, since n is a prim* number* must be n(x—y)i&??r'n, or x—y^?in~lr'n ·, and, consequently, if the equation oc“-yn = zn be possible, we iiiust have x —ynzrn, of «*" V*. But the equation x" —yn = zn maybe put under the form x* — za =yn; and, consequently, we have also the same result as to the difference x — z; viz. x — z^sn, or fin-'sn. And again, by writing the equation thus, yn + z” = xn, we shall* by means of cor. 2, art. 82, and the same reasoning as that employed above, find, that y +■ stfcT* or n"'T. Hence, then, if the equation X‘l-y" = Zn be possible, the following conditions must ob- tain; viz. The difference of the roots x —y^rn, or rin~1rn. The difference of the roots x— zmsn, or ?in~1sn. The sum of the roots y + or n"~Hn. m 2l£>4 Powers in General. But since (x—y), (x — z), and (y + z), are re- spectively divisors qf the three nth powers, zn, y% and xH, and since these three quantities are prune to each other, their divisors must also be prime to each other; and, consequently, only one of these can b^pf the latter form above given, as they would otherwise have a common divisor n. There- fore, if the equation be possible, we shall hate either f x—y^r", < X— Zi&Sn, by + stfcf', or two of these quantities will be of this form, and the third of the form ?ιη~'φ’\ which evidently re- solves into the four following cases, one of which must necessarily obtain. if the equation xn—yn = be possible; viz. C x—yzRr1, 2d, K-rK)}u=tnn + nt' r’iy + nhr-3’'(.sn-rn) -n(sn- ' &c. • ru) +natm-tn(sn- * See note, page 137 ·i6G Powers in General. (s'* — r”) + natnn~~v(s” f {Γ + (sn + r")}" = tnn + nt”-" (■? + r”) + natm-qn(sn + { rnY + nbtm~3"(sn + f)*— &c.* And here, since the sum of the two first expressions is equal to the third, it is evident that the latter, subtracted from the sum of the two former, is equal to zero. But in adding the two first together, the 2d, 4th, &c., terms cancel; and, consequently, in subtracting the latter from that sum, the 2d, 4th, &c., terms will remain the same, except that the signs will be changed from + to —. And as to the 1st, 3d, &c., terms of the first two equations, and the same terms of the third, we shall have, by observing that (s* - r'f = ($* + rnY - 4svr", (s' - rn'f = (sn + rny - 8(^"r“ + n x 8sY"(s2" + r2!1), 7th terms, &c. And, consequently, subtracting from those sums the 1st, 3d, &c., terms of the third line, namely, * By writing n, na, nb, nc, &x,, for the coefiicients of the bino- mial, cor. 1, art. 12. 1st term t’“\ 3d term + 5th term nct’nn~4n(sn + rv)4; the remainders of these particular terms tr rll be,Powers in General. ιβγ 1st rem. — t'm, 3d rem. = natm,'i"(sn + r’*)2 — 2natm~^ x 4s“r’1, ( 5th rem. = nctm~in{sn + — 2nctnK~in x 8δντ*(δ** -t- 1. o. 7th rem. &c. &c. In short, the whole of the remainder which is equal to zero, will be expressed by j {f - (sn + rn) }·- {2mtm-3\4snrn) - (2net™-"'. 8vV) 1 («** + r*·)- &c. And here it is only necessary to observe, that all the terms on the latter side of this expression are divisible by t"s”r%, so that, for perspicuity sake, we may write it thus, {t" - (s'1 + r") }* ~ fjVA = 0; and, consequently, {tn-(f + rn)}n== a·, and here, since the first side is a complete nth power, the latter side, which is equal to it, must be so likewise; and, consequently, a must be a, complete nth power, or a = a'" ; that is, {#" - (sn + rn)}" = fsVV”; and, therefore, tn-sn~rv = tsrA': or, dividing by trs, we have />!-! -n-1 .,n-l :—_______________-y sr tr st 5 which must necessarily be an integer. But these three fractions are in their lowest terms, be- cause r, s, and t, are prime to. each other, amil6s Powers in General. each of the denominators contains a factor that is not common to the other two; they cannot, there- fore, he equal to an integer (cor. 2, art. 13); and, consequently, the equation is impossible under the first condition. And in order to arrive at the re- sults of the other thi'ee conditions, we have only to substitute nn~'rn for rn; h’"V for sn ; and n’l"'P for f1, whence we draw the four following concilia sions; 1st, 2d, 3d, 4 th, r~‘ rs - l tv st l * * < it s” ■1 if- lpn"1 . — \β/ vs tr st A * tn-1 γ,η-l vs tr st — A * .1 y.i f-' . ///* ?\v tr st memr A - according as we assume the 1st, 2d, 3d, or 4th, cosk dition. Inwhich expressions weought to have one of the quantities a', a", a"', a"", an· integer number, if the given equation were possible ; but since in each of these expressions we have three fractions, in their lowest terms, and the denominator of each contains a factor not common to the other two, therefore (cor. 2, art. 13) they cannot produce an integer number. Having shown, therefore, that, if the equation xn—yn=zzn vrere possible, one of the quantities a', a", a"‘, or a"", would be an integer; and having also demonstrated that no one of these quantities can be an integer; it follow’s, that the equationPowers in General, 169 Whence they were derived is impossible; that is, the equation x?—yn — zn is impossible, when n is a prime number. We have also demonstrated, art. 83, that the impossibility of the equation xn—yn = zn, when n js a prime, involves with it the impossibility of every equation of the form wn±yn=zn, in which n is any number whatever except 2, or some power of 2; and we have likewise shown that the impossibility of the equation, when n is any power of 2, is involved in that of x*—y* = z*, which particular ease has been demonstrated to be impossible (art, 73); and, consequently, the equation Xn±yn=%* is always impossible, when n is any integer number whatever greater than 2. — a. E, d. Cor, Since the equation xn±yn = zn fJQn Ί/η is impossible, so also is —- = —, for this is the r m y q znm"pn , , - , same as x" + yn = : and, therefore, the equa- tion is likewise impossible in fractions. PROP. VI. 86. If m be a prime number and x any number not divisible by to, then will the remainder arising from the division of x by to be the same as that from the division of xm by to.Powers in General. 170 For make x=x' + 1, then we have = [xf + 1 )ro = x'm + nix'm~1 + max'm's -1-mx + 1 / And since all the terms of this expanded bino- mial, except the first and last, are divisible by m (cor. 1, art. 12), it follows that the remainder from the division (af + I )m by m is the same as that from the division x/m + l by m; which, by rejecting the multiples of m, may be expressed thus: xm = (F + 1 )m = + I. Making now v' = v" + 1, we shall have, on the same principles, xm = (a' + 1 )* = x'm + 1 = (x" + 1)" + 1 = x"m + 2, Again, let x" = x"' + 1, and we obtain xm = x'm+l=x"^+2-x"/m + 3. And thus, by continual substitutions, we have xw= x- + 1 = x"m + 2 = x",m + 3 = &c.; or, f X* =(x - 1)m + 1 = (* - 2)+ 2 = {x - 3)w + a &c. t (x-x)m + X, the last of which terms is equal to x; whence it follows, that the remainder arising from the di- vision of x by m is the same as that from the di- vision of x* by m. — a. e. d. PROP. VII. 87- If in be a prime number, and x any num- ber not divisible by m, then will the formula xn~’ — 1 be divisible by in, or, which is the same, (v”*'1 — l)=M(m). For, by the foregoing proposition, the remainder . x . . xm ®f— is the same as the remainder of —; and, eon- m m ■' ■Powers in General. 171 sequently, the difference xm — x is divisible by m. But xm — x = x(xm~l — l), and since this product is divisible by m, and the factor a? is prime to m, it must therefore be the other factor (xm~' — l), that is divisible by m (cor. 5, art. 11). — a. e. d. Cor. 1. Since xm~1 — 1 is always divisible by m, if x be prime to m, and m itself a prime, there aye, necessarily, m — 1 values ofx less than m, that satisfy the equation a?™'1-! -——— = e, an integer; that is, x may be any number in the series 1, 2, 3, 4, 5, &c., m— 1, because all of these numbers are necessarily prime to m; and, since m—l is an even number, we shall have also m—l values of x, comprised between the limits — im and ~m; that is, x may be any num- ber in the series m—l ±1, ±2, ±3, ±4 τ t r + — 2/ so thaty in both cases,, we have m— 1 values x is divisible by m, and m is a prime imniber, one of these factors must be divisible by m; that is, m - 1 a·3 Ηζηια± 1; and, consequently, every power, the double of whose exponent plus 1 is a prime number, as («?), is of one of the forms am, or am + 1; and hence again, we derive the forms of many other higher powers ; thus, x 3 at: 7«5 or 7n ± 1; x at·, 1 lw, or 1 \n± 1; a;6st; 13», or 13»+1; x ssfclor 17» + 1; x ?n?A<)n, or l$n± 1; a?11 at: 23», or 23»+I; &c. &C. &e. And hence we have the following forms of allPowers in General· 173 powers from 2 to 12, the 7th powers only excepted, which cannot be introduced into these forms, be- cause neither 7 + 1? nor 2 · 7 + 1? is a prime num- ber. Table of the possible Forms of Powers, from 2 to 12. ,r4sR 3n, or 3w + lift bn, or 5m+1; #3sr - - - - sr 7», or 7»± 1; .i?4sr bn, or 5m + ltfc - - - - - X ssr - - - - tfcllM, orllM+1; xeitt 7«? or pi+lmlSn, orl3M+l; a:7eR - - - - sr ' - - - - - a?*sR - - - - s)s17m, 01'17m±1; #9sr - - - - srIQm, orl9M±l; a;10sftllM, or 11m+1 sr - - - - #usr - - - - sr23m, ογ23μ+1; λ;1* sr l 3m, or 13m + 1 sr - - - - - Scholium. By means of the foregoing table of formulae, we may frequently satisfy ourselves of the possibility or impossibility of equations of the form axa±byn = dzn. And also whether any given number a is a comr plete power or not, without the trouble of extracting its root: it is to be observed, however, that a given number may be of a possible form, though it be not a complete power; but if it be of an impossible form, then we are certain, without any farther trouble, that it is not a complete power.if 4 Powers in Generali PROP. VIII. 8S. If m be a prime number, and p be made to represent any polynomial of tlie nth degree, as p sf + ax11'1 + bx71^2 + cn~3 +------ then^ I say, there cannot be more than n values of „r, between the limits and — x?n, that reh-/ der this polynomial divisible by m. For let ft be the first value of x, that renders P divisible by irt, so that am — kn + ak7l~l + khn~~ + eft""3 -{- - q; then, by subtraction, we have f p — am = [xn — ft”) + a(xn~l — kn~1) + h{xn~2 — ft"~ l &c. But the latter side of this equation, being divided by ft (art. 80), we shall have for a quotient a polynomial of the degree η — 1; which, being re-* presented by p', gives P — Am=^ (x— ft)p', or p±=(tf — ft)p' + Am. Let now ft' be a second value of x3 that renders p divisible by m, then it follows, that {x — ft)p' + Am is also divisible by m■; and, consequently, (#—ft)p' divisible by m, but the factor x — ft, which now becomes (ft' —ft), cannot be divisible by m, because both ft' and ft are less than ±m-; therefore, p cannot be divisible a second time by m, unless p' be divi- sible by nu The polynomial P is, therefore, only once more divisible by m than the polynomial p'; and, in the same manner, it may be shown, that p', of the de- gree η — 1, is only once more divisible by than?Powers in General. 1.71 p", of the n — 2 degree, &c.; and hence It follows, that, p being a polynomial of the n degree, tilery can be only n different values of x, comprised be- tween the limits +-‘m and — \m, that render it divisible by m. — a. e. d. Cor. We have seen (cor. 1, art. 87), that if m be a prime number, the formula xm~l— 1 has m—l values of x, between the limits + ^m and — \m, that render it divisible by m. Now, this being put un- / m-1 \ / m-l \ der the form I j ) x ( jF — 1 )? h follows,that each of these factors has —*— values of x9 between 2 the limits -f 4-wz and — that render them di- visible by m. For neither of them can have more than------- such values* by the foregoing propo- 2 sition; and, since their product has m—l, it is obvious, that they have each the same number of values of x between the above limits, and that this * . m-l number is ------. 2ίγ6 chap. vii. On the Products and Transformations of ceftdm Algebraical Formulae. PROP. i. 89. The product of the sum and difference of any two quantities is equal to the difference of their squares. For, (x+^)(*—3/} = λ4—a. έ. d, PROP. II. 90. The product of a sum of two squares, by double a square, is also the sum of two squares; dr (λ" + f) x2s! tte a/2 4- yn. For, (vf+f) x 23*=(x+yY · s* 4- (x—yY ·«% which is evidently +y'*. Cor. Hence if a number be the sum of two squares, its double is also the sum of two squares^ Also if a number n be the sum of two squares, 2"tf is so likewise. Thus, for example, 5 = 2*4-1*; a x 2 = 10=3*4-1* | 10x2 = 20 = 4*4-2*; 20 x 2 = 40=*6*4- 2* &c. PROP. III. 91. The product arising from the sum of two squares by the sum of two squares, is also the sum of two squares; orProducts of Algebraical Formulae. Iff (x% + if){xn+yn)ip.0fn For, (*■+/)(;+ ' J /v υ ' (_ or (xx —yy)~+(xy +xy), as will be evideiit from the development of these expressions; and, consequently, (x2 + yi)(x'i+y'^^kx^2 + y"~i a. e. d. Cor. Hence the product may be divided into two squares two different ways. And if this pro- duct be again multiplied by another, that is the sum of two Squares; the resulting product may be di- vided into two squares four different ways; and; generally, if a number N bte the product of n factors; each of which is the sum of two squares, then will Si be the sum of two squares, and may be resolved into two squares 2n different ways. For example, 5 = 24 + 1® 13 = 32 + 2* Then the product 65 = 8a + l4, dr 72 + 4e; Again, - - 17 = 42 +12 1105 = 322 + 9s = 243 + 232. 332 + 42 = 3T+ 12* And this resolution of the given product into square parts; is readily effected by the foregoing theo- rem ; for f (8*·+ l)(4* + Is) = (4.8 + 1)2 + (8.1 - 4 . l)2 = \ (4.8- 1)2 + (8.1+4. l)2, and / (72 + 42)(42 + l) = (4.7 +1.4)2+(4.4-7.l)s = \ (4 i 7 - 1.4)2 + (7.1 + 4.4)2. And in the same mariner may any other product, arising from factors of this form; be resolved into its square parts-178 Products of Algebraical Formula;. PROP. iv. 92. The product of the sum of three squares, by the sum of two squares, is the sum of four squares; or (a2 + tf + *2)(a'2 + y»)&w'n 4 a;"2 +ym 4 a:"2. For, (a9 +y" + s'2) (a'2 + yr!) = (xx'+yy'f + [xy*—yx'f 4· a,9s2 4 J/,3s9, as will appear immediately, from the development of these formulae; and, consequently, (a9 + f + %2) (xn + yn) *wm 4 a"2 +ym 4 *"*. α. E. d. For example, - 14 = 3* 4 22 4 15 5 = 2a+l rpi , f 70=(3.2 4 2.l)94(2.2-3.l)e Then the product j + = 8*+i* +22+l2; and a similar decomposition may be effected on any other similar product. PROP. v. 93. The.product arising from the sum of four squares, by the sum of two squares, is the sum of four squares; or (ic5 + a9 4 y2 4 z~) (x/e 4 y'2) + a'2 4 yn 4 s/2. For, (m2 4 a2) (a/2 4 y') ^ wn 4 a'2, and (/425)(a'24ya)^/24^, by art. 89; and, consequently, (ur + a2 4 f 4 z2)(a/2 + yn) ^ion 4 a'24/24 s'*. ft. E. D.Products of Algebraical Formulae. 1^9 PROi*. VI. 94. The product of the sum of four squares, by the sum of four squares, is also of the same form; or r (m*+a?+y*+**)(«>'*+**+y*+ I to'n^x"'2 + y"* + z"i.) For, (tt>* + Λ* + «/ + X*)(vf* + Λ·'* + y* + *"*) 5 (««o' + auf + ?/y + 2.sf)* + (ioaf - .no' + ys' - zy')* + (toy' — xz' —yw' + zx'Y + (ivz' + ay' —yaf — zw')q, as will appear immediately fromi the development of the above formulae; and, consequently, the pro- duct in question ^[tv"2 + x"* + y"* + z"'2).— o.. £. D. Cor. 1; As in this product^ there arts only com- plete squares enter, we may change at pleasure the signs of the simple quantities; and, Conse- quently, there will result Several different formulae equal to the same product, and each equal to the sum of four squares; and in so many different ways may any number that arises from the product of factors of the above form, be resolved into the sum of four squares. Cor. 2. This proposition may be rendered mores general by the following annunciation: _ The product of the two formulae, ’ (w1 — bx° — cy2 + bcz}) (tv'* — baf° — cy,% + bczn) tfc (w": - bx"* - cy"* + bcz"2). For, («?* — bx* — cy2 + bcz~) (tv'* — bxn — cy'* + bcz'*) ** {(ivw' + bxaf + cyy' ± bczz')* ~ b(waf + iv'x + cyz' ± cy'z)* — c(wy' — bxz' ± yw' + bzx')* + bc(xy'—wz'± zw' + yaf)*, N 2180 Products of Algebraical FormutcO. as will be evident from the development; and, con- sequently, the product in question is of the same form as each of its factors. PROP. VII, 95. The product of the two formulae (a3 — ay2) and (xn — ayn), is of the same form as each of them 5 that is, (x2 — ay1) (xn— ay'2) ^ x'n — ay"2* For, (x°--ay2)(x'2-ay'2) and, consequently, (xx' + ayy'f — a(xy' +yx'f, or (xx'—ayy'f—a(xy'—xy'fi (x2 - ay2)(x'2 ~ ay'2)&x"2 - ay"2. a. e. 0, Cor. The product of any number of factors* each of the form (y2 — ay2), is always of the same form. PROP. VIII. 96. The two formulae (a.·2 +y2 + z2) and (x2+y2 + 2z2), are so related to each other, that the double of the one produces the other; that is, (x2 + y2 + z2) x 2 &x'2 + y'2 + 2z'2, and (x2 +y2 + 2z2) x 2 st: x'2+y'~ + z'2. For, 2 (a-2 +y2 + z2) — 2x2 + 2y2 + 2*2 == (x+yf + (x—y)2 + 2z2^xn+y'2 + 2 zn. And 2(x2 +y2 + 2 z2) = 2x2 + 2y2 + 4z2 =* (λ+yf + (λ—yf + 4z? zpix'2 +y'2 + z'2. a. e. v.181 Products of Algebraical Formulae. For example, 14 ==3® + 2® + l® Mult, by - 2 The product And - - Mult, by - ?28 ==(3 + 2)*+ (3-2)* + 2,1* ;5*+l- + 2, 15 =s3a+ 2S+ 2,1* 2 The product { = <3++?>*+ <3 ~ 2>*+ 2* And the same of all other numbers of these forms. PROP, IX, 97· The formula x%— 2?/ may be always trans- formed to another of the form 2x'°~—yJi, and this last may be converted into the former; that is, f of — 2yi 2xn — i/'2, \ 2x~ — y° sfc x,& — 2y'~. For, x* — 2y^=^2(x±y)i — (% + 2yY^2xn—yn, and 2x~—y‘! ~{x± 2y'f — 2 (x ± y)s tfcx"* — 2yn; as is evident from, the development of these formula:; and, consequently, a number that is of one of these forms is also of the other. — a. e. d. For example, 14 =5? 2.32 — 22 = 42 — 2 , l2. Also, - - 28 == 62 — 2.2® = 2, 42 — 2*· And the same of any other numbers of either of these forms, prop, x, 98, The formula x~ — by1 may be always trans- formed to another of the form bxn—y'~, and this last may be converted into the former; that is*leg Products of Algebraical FormuL·. f x1 — by1 i* 5x'~ — yn, \ 5p, or > r, this formula may always bp transformed to a similar one, p'yn + 2 q'y'%' + rV2 =r= Φ, which shall be equal to the same quantity φ, and in which 2q' shall not exceed either // or r'. Let us suppose, first, 2q>p\ and in the case in which also 2q > r, let p be the least of these two numbers p and r, abstracting from their signs. Now make y = ?/ — mz, m being an indeterminate coefficient; and substituting for this value of y in the given equation, we have p(y'— ms)*+ 2qz(y'— φζ) + ΐ'Ζ2 = φ, or fyn — 2(pm — q)y'z + (pm2 — 2 qm + r)z* = φ. And here we may always take the indeterminate quantity m, so that ± (pm — q)

p, and 2q' r', we may proceed in a similar manner tp obtain a new transformation, in which the mean coefficient (which we may represent by q") shall be less than q', and so on again for others, in which the mean coefficient 2q'" is less than 2q"% But the series of integers %, q'% f\ &c„ cannot go on continually decreasing, without be- coming finally less than the extreme coefficients ; and, therefore, by continuing these transformations, we must necessarily at last arrive at that, which ad-‘ rnits not of any farther reduction; and which will be consequently such, that the mean coefficient is less than either of the extremes, pr at least not greater than the least of them; for with anyDivisors of Algebraical ForniuL·. 18$ formula, in which this is not the case, a farther reduction may be made. Therefore* every formula* pf + 2qyz + rx\ in which the itleafl coefficient 2q exceeds either* or both, of the extreme coefficients, may he trans- formed to another, in which the mean coefficient 2q shall be less than either of the extreme co- efficients, or at least not greater than the least of them. — a. e. d. Cor. In the successive transformation of the formula, py*+2q y % + r za, to pyn + 2(/ y'z + rV, to p'y'2+2q"y'z' + r'%n, £c., tve have always pr — q* =pr' — qn =p'r' — q"~, &t\, each of these quantities having the same sign ; for W6 have seen this equality take place in the trans- formation that we have effected, and it is evident, that the same would still have place in any farther reduction, the operations being all effected in the same manner. The following example may be of some use in illustrating the foregoing proposition. Let there be proposed the formula SSy* + 172yz + 21 Qz® = <£, in which the m+an coefficient 172 exceeds the first 35 j and let it vbe required to transform this to another equal' anv'l similar one, in which the mean coefficient sh all be less than either off the extremes.190 Divisors of Algebraical Formula.· First, put y =y'—tnz, which value of y, being substituted in the given formula^ gives 35y,2 — (fOm — l’/2)y'z + (35m2— 172m + 210)z\ And now, in order that fom — 172 < 35, take m = 2, which reduces the above to 3 byn + 32y'z + 6z°- = φ, in which the mean coefficient 32, though < 35, is still > 6; and, therefore, we must proceed to an- other similar reduction* Let, then, z —V — my', and the second trans- formed formula will become 6z/s — (12m — 32)yV+ (6nf — 32m + 35)y^. And here, taking m = 3 in order that 12m —32 < 6,· we obtain 6z'2-4zy~7y'2 = And again, since — = r; tlierefore,- pr —y~ — d; And we have seen (art. 102) how every indeter- minate formula, pif + 2qyu, + rip, toay be transformed to another similar and equal formula, , p'f+aq'yz + r's?, iri which the mean coefficient 2q' < p’ aiid < r (the sign < not excluding equality), and in which pr — (f is always equal to the same constant quan- tity a, And, consequently, every divisor of the formula t + aiP has its divisors contained in the formula py1 + 2 qyz + rz\Divisors of Algebraical Formula. 193 and in which 2q does not exceed p or r, and also such that jor— q*==a.— a. e. d. Remark 1. Since 2q < p, and 2q < r, independently of the signs of these quantities, we have 4q' 3(f , dr q < \f -: this is evident, because when & is positive, p and r are both positive. Cor. 2. Every divisor of the formula tCi — au2 may be represented by the formula pf + 2 qyz — rz%, iri which —pr —ψ= — tij or pr + f — a, because, When a is negative, pr is necessarily so likewise; and, consequently, one of these quantities, p or r, is + and the other —, arid it is indifferent to which we give the sign - a > 5q*, or q < ■; and here, since pr < 4qi, we have \4 Remark 2i We may have Cases in which p = r = 2q·, as, for example, when p — 2, q — l, and r —2; for theti 2q does not exceed either ϋ104 Divisors of Algebraical Fortnidet· p or r, neither are p, q, and r, divisible by the same number, which condition is, therefore, strictly Within the limits of the proposition; and hence it follows, that we must not consider the sign < in the two expressions q < \J 7^ and 9 < s/f i to ex" elude equality. Prop. iv. 105. Every divisor of the formula f + ut, t anil , u being prime to each other, is always of the same form y1 + s'2. Or the sum of two squares, which are prime to each other, can only be divided by numbers that are also the sums of two squares. For by cor. 1 of the foregoing proposition, every divisor of the formula t~ + au7 is included in the formula pf + 2qyz + rz*, and in which q < \J—, andpr — 9®== a. Now in the present case «= I, therefore* q < \for q ==0, there being no integer < \/~$ and, since pr — y2 = 1, we have pr — 1, and there- fore p — l, and r— 1; and, consequently, the above formula, which includes all the divisors of #* + if, becomes tf + z* i that is, every divisor of the formula il + tt2 is of the form f + z% or every divisor of the sum of two squares, prime to each other, is also the sum of two squares. ·— a. e. d.Dimsoirs of Algebraical Formula. 193 Thus, for example, 65 — 8® + l2 can only be divided by 13 arid 5, both 6f which are the sums of two squares. Also 50 — 72+l have for divisors 2 — l2 4- Γ2, 5 = 22+l2, 10 — 32 4- 12, 25=4® 4-3®. Again, 221 = 102 + Ϊ 1~ is only divisible by 13 and 17, Which are both the sums of two squares; arid the Same for all other numbers included in the formula il 4- if, t arid u being prime to each other; PROP; V; 106, Every divisor of the formula t2 + 2 if, t arid u being prime td each other, is of the same form y1 4- 2s2. Or the divisors of the surii of a Square, and double a square, are also the sum of a square, and double a Square. For every divisox; of this forrilrila f + aif is con- tained in the formula py~ 4- 2 qyz 4- rz$ in which q < \ Jand pr — cf — a (cor. i, art. 104), O But iri this case a —2, therefore q <\f ~, or q — 0; also, since pr — tf — 2, we have pi'= 2, whence p — ‘2, and r=l, or p — 1, and f = 2; therefore, the above formula becomes ( 2f 4- s2, in the first case, and \ y2 + 2%-, in the second, which are two identical forms, by charigirig y into Z, and z irito y; consequently, every divisor of the formula f + 2if is also of the same form as itself. Cor. 1. With regard td the divisor 2, it cari only be of the fdrm y14- 2S2, when y — 0 and 2 == 1; to that, in this case, we have 02 + 2.12. o 21$6 Divisors of Algebraical Formula. As an example to this proposition, we may take 99 = 1 + 2.which inn only be divided by 3 = 1-+ 2.1s, 9 — la + 2.2s, ll=3® + 2.1®, 33 = 5® + 2.2s; and it is the same with every number that is con- tained under the above form. prop. V!. lof. Every divisor of the formula t* — 2w5, t and u being prime to each other, is of the same form y1 — 2m®. Or the difference of a square, and double a square, can only be divided by those numbers that are equal to the difference of a square and double a square. For since every divisor of the formula f — air is contained in the formula pif + 2 qy% — i'z2, in which pr + q a, and also q < (cor. 2, art. 104), it follows that q — 0, whence also pr~2; and, therefore, p = 2 and r—l, or p = 1 and r = 2; and, consequently, the above formula becomes either 2orf-2z\ which two forms are precisely the same, because 2f — s® = (2y + s)2 — 2(y + «)*; therefore, every divisor of the formula f — 2u2 is also of the same form. Or the difference of a square, and double a square, can only be dividedDivisors of Algebraical Formula. 197 by numbers that are also the difference of a square and double a square. For example, 98 —10" — 2.12 can only he divided by 2 = 2a-2.1®, 7=3®—2.I2, 14 = 4--2. I', 49 = 9"-2.4®; and the same of all other numbers in this formula. PROP. VII. 108. Every odd divisor of the formula f1 + 3u°' is also of the same form f + 3z~. For since all its divisors are contained in the formula pif + 2qy% + rs2, in which pr — q" = a, or pr^-cf — 3; and also /3 q — or < \f — (Remarlt 2, art. 104), so that q = l, or q = 0; therefore* in the first case* since %q is not greater than p or r* and pr — qi = 3> we must have p = 2* and r = 2* which renders the above formula 2y2 + 2 qyz + 2s2; but as this is evidently an even divisor* it does not belong to the case at present under consideration* which only relates to the odd divisors of the given formula. In our case* therefore, q = 0; and* consequently* pr — q2 — 3, or pr == 3; therefore* p==3 and r = 1 * or p~ 1 and r = 3; whence the above formula is re- duced to 3y°~ + s2* or y1 + 3z2y198 Divisors of Algebraical Formulas. two expressions which are identical as to their form; and, therefore, every odd divisor of the formula f + 3ti2 is also of the form f + 3z2. Remark. With regard to the divisor 3, it is obvious that we must have ?/ = 0, and z = 1; or. 3=02 + 3.12; but for all other divisors, this ex- ception has not place. For example, 52 + 3 . 62==133 = 7 · 19 ; and 7 = 2* +3 . I2, also 19 =? 42 + 3 , 1*, both of the same form, PROP. VIII, 109. Every odd divisor of the formula f — 5u~ is also of the same form f — 5z2. For all its divisors are contained in the formula py° + 2 qyz — rz2, in which —pr — q2 — — a, or pr + q* = 5, and q = or < \f—\ and, consequently, q~ \ orG; hut the first case gives only even divisors, the same as in the foregoing proposition; and the latter case of q — 0 redaces the above formula fo 5f — z2, or yf — 5z2, which are identical forms; because by2 — z2 — {by ± 2z)^ — 5(2y ± z)2; and, consequently, every odd divisor of the formula t2 — bu2 is itself of the same form, As an example in this case, we may assume 95 — 10® — 5 . I 2, which is only divisible by 5 and 19. Now 5==52 — 5.22, and 19 = 72 — 5-2s; and; the same of all other numbers in the above form. Scholium. From the foregoing propositions itDivisors of Algebraical For mula\ 199 appears, that all numbers which are comprised in the following formulae, viz. f + u*, tf + Qu*3 #* —2w®, t -I- 3zr, and f — 5w*, t and u being prime to each other, can only have divisors that are of the same form. It is only ne- cessary to except those divisors of the two latter forms, a ' _ i f + 3ir, and t — 5u\ that are double of an odd number; the reason for which exception is explained in arts. 108 and 109, It frequently happens, that a number falls under two or more of the above forms, in which case its divisors are also of the same double or treble forms. And in some cases we have numbers that belong to each of the forms above given. Thus §41 == 152 H- 4* =5= 13" + 2.62 = 213 - 2,10°" == f + 3 31δ-5,12\200 CHAP. IX. On the Quadratic Forms of Prime Numbers, with) Rules for determining them in certain Cases, Lemma. 110. Since all square numbers are of one of the forms An, or 8n +1, we establish at once the three following theorems: 1. Every odd number represented by the formula ty* + 2*ste4» + 1. 2. Every odd number x-epresented by the formula f + 3x2 8η +1, or 8n + 3. 3. Every odd number represented by the formula y% — 222tte8«+ l, or 8??- + 7· And from these three arise, by way of exclusion, three others; viz. 4. No number of the form An — 1 can be re- presented by the formula y1 + 2®. 5. No number of the form 8n + 5, or 8n + f, can be represented by the formula f + 22®. 6. No number of the form 8n + 3, or 8n + 5, can be represented by the formula f — 22®. prop. x. 111. Every prime number of the form An + 1 is the sum of two squares, or is contained in the formula f + 2®.Quadratic Forms of Prime Numbers. 201 For let m represent a prime number of this form, pr m— An + 1; then (art. 87) {xm~1 — I) = m(m), pr (x4,‘ — 1) = m(m). But xin — 1 = (x°n + 1)(*2Λ — l), and each of these fac- tors has 2n values of x contained between the limits + -\m and — 4m, that render them divisible by m (cor. 1, art. 88), whence the factor x^+.l is di- visible by m; but #2n + 1 is the sum of two squares, and therefore its divisor m is also the sum of two squares; because every divisor of the formula ri + w is itself of the same form (art. 105). — a. e, d. Cor. 1. As the form An + 1 includes the two, 8n + 1 and 8n + 5; therefore, every prime number contained in these two latter forms is also the sum of two squares. Thus, 5, 13, 17, 29, 37, and 41, are prime numbers of the form An + 1, and each of these is the sum of two squares; for 5 = 22 + l2, 13 = 32 4- 2*, 17 = 4® + 1,29 = 52 + 22,37 = 62 -t-1*, and 41 = 52 + 42; and so on for all other prime numbers of this form. Cor. 2. We have seen (art. 91) that every num- ber, which is produced from the multiplication of factors that are the sums of two scpiares, is itself pf the same form, and ipay be resojved into two squares different ways, according to the number of its factors; and hence we may find a number, that is resolvible into two squares as many ways as we please, by multiplying together different prime numbers of the fprm 4« + 1. PROP. 11. 112. Every prime number 8/1 + 1 is of the three forms f + z% y~ + 2z2, and f — 2z3.SO 2 Quadratic Forms of Prime Numbers. For let m be a prime number of the form 8« + 1, or m — 8» + 1; then, as this form is included in that of 422+1, we know, from the foregoing proposition, that m + s2; and it therefore only remains to demonstrate the two latter cases. Now since (xm~1 — 1) — u{m), or (of*— 1) = m(#i) (art. 87), we may put this under the form (λ,4:1+ 1 )(af"— l) ; and each of these factors will have An values of x < Am, that render them divisible by m (cor. 1, art. 88), so that there are so many different values of x, that render the binomial xin + 1 divisible by m; but this may be put under the form (xin — l)2 + 2a;2”, and m being a divisor of this formula, it is itself of the same form y3, + 2s2 (art, 1θ6), We may also put the same quantity #4”+ 1, un- der the form (x2” +l)2 —2«2“; and m being also a diviso,r °f this formula, is itself of the same form y~2*2 (art, 107). Hence, every prime number of the form 8w + 1 is of the three forms f + z2, f + 2z2, and f — 2z2. a. e. d, Thus 41 = 5s + 42 = 3s + 2 .4*= 72 - 2.22, And 73 = 82 + 32=l2+2,63 = 9'-2.22. PROP. III. 113. Every prime number 8» + 3 is of the form y* + 2z2, For let m be a prime number of this form, or m=8n+ 1; then we have (by art. 87) i) = m(»i), or (a;8"+2—l) — M(m); and there are 8n + 2 values of x contained in theQuadratic Forms of Prime Numbers. 203 series I, 2, 3, 4, &c., 8m + 2, that render this formula divisible hy m (cor. 1, art. 87) ; and, con- sequently, (28,,+2— 1) = m (m). But 28"+2-l = (24n+' + l)(24n+I-l), and, there- fore, one of these factors is divisible by m; and it cannot be the latter, because this may be written 2.24” — }, which is of the form 2 f — tv, or f — 2iv; and, therefore, if m was a divisor of this, it would be itself of the same form (art. 107), or mmy" — 2z~; but this formula cannot represent any number of the form 8m+ 3 (art. 110), whence, since m cannot be a divisor of this factor, it must therefore he a divisor of the other factor 24”+l + 1. But 24B+l + 1 = ? . 24n + 1 =fc2f + μ2; and, consequently, its divisor m is of the same form (art. 106); that is, msr v/2 + 2z% — a. e. d. For example, 11, 19, and 43, are prime numbers of the above forrq; and 11 =3® + 2.12,19= 1* + 2.3®, and 43 = 52 + 2.32; and the same of others. PROP. IV. 114, Every prime number 8n + 7 is of the form y- — 2v\ For let m be a prime number of this form, or m — 8n + 7; then we have (by art. 87) (xm~l — l) = m(m), or (a;8"4·0 — l) = m(wj) ; and there are 8m + 6 values of x, contained in the series 1, 2, 3, 4, &c., 8m+ 6, that render this formula divisible by in (cor. 1, art. 87); and, con- sequently, (28"+β—1) = μ(μϊ), But 28,‘+0-l = (24'1+3 + l)(24"+5-l^ and there-S04 Quadratic Forms of Prime Numbers. fore one of these factors is divisible by m; and, consequently, m will also be a divisor of one of them when doubled; that is, it is a divisor of one of the two quantities 2(24”+3+ 1), or 2(24"+3— 1), which two expressions thus become 24“’ 'r 2 . Is, and 24”' - 2 . V, and m is necessarily a divisor of one of them. But it cannot be a divisor of the first, because this being of the form f + 2m2, if m was a divisor of it, we should have m^pf + 2a2 (art. lOfi); but mdp8n + 7, and no odd number of the form yL + 2a2 is of the form 8 n + 7 (art, 11Q): since, therefore, m is not a divisor of this factor, it must necessarily be a divisor of the other factor 24"' — 2.12, which is of the 'form f — 2u~; and, consequently, its divisor mis also of the same form (art. 107); that is, mefcy2—2a2. ft# E· D. For example, 31 — 72 — 2.3% and 47 = 7a — 2.1*.; and the same of all other prime numbers in this form. Scholium. From the last four propositions, we may draw the following theorems: 1. All prime numbers of the forms 8»+ 1, and 8n + 5, are, exclusively of all others, contained in the formula if + a2. 2. All prime numbers of the form 8»+l, and 8» + 3, are, exclusively of all others, contained in the formula if + 2 a2. 3. All prime numbers of the form 8« + 1, and 8n + 7, are, exclusively of all others, contained in the formula if - 2a2,Quadratic Forrhs of Priffle Nufnbers. 205 4. All prime numbers of the form 8n + 1 are, at the same time, of the three forms • f + f + 2z% and y* ~~ 2%** PROP. V. 115. To ascertain whether a given number of the form 4η + 1 be a prime number. Since every prime number p of the form An -f 1 is the sum of two squares, or p = #*+pa, it is obvious, that in order to determine whether a given number of this form be a prime, we have only to ascertain whether'.it can be resolved into two squares; and, if it can, in how many ways this resolution may be effected; then, if it happen that the given number may be decomposed into two squares, in one way only, the number is a prime, but otherwise it is composite; and the object of the present propo- sition is to teach the easiest method of performing this decomposition. Now, because p = ar+pa, and since these squares cannot be equal, it necessarily follows, that one of them is greater and the other less than; if, therefore, every square >±p and

5 is of one of the forms 10»+1, 3; 7i or 9; or, which is the same, it is terminated by one of those digits. Again, all squares are of one of the forms 10», 10» + 1, 4, 6, or 9 > hr they are terminated in one of the digits, Ο, 1, 4, 5, 6, or 9 ? and therefore no num- ber terminating in 2y 3^ 7* or 8* can be a square number; therefore., 1 * When the last digit of the proposed number is 1, We may omit all squares terminating in or Q, because these will give remainders terminating in or 2, and^ therefore, such remainders can- not be squares. 2. When the last digit of the given number is 3, we may omit all squares that terminate in O, 1, 5, or 6; because these would give remainders ter- minating in 3, 2, 8, and 7 ; which, therefore, can- not be squares. 3. If the last digit be f, we may omit all squares terminating in 0, 4, 5, or 9, for the same reason as above. 4. If the last digit be 9> we may omit all squaresf terminating in I or 6s By these remarks the number of operations in subtraction wil 1 be reduced, generally, about one half, and will be considerably less than the num- ber of operations in division by the common rule. Ex. 1. Let it be proposed to ascertain whether the number 10133 be a prime. Since this number terminates in 3* the onlyQuadratic Forms of Prime Numbers. 20f Squares between 5066 and 10133, that do not ter- minate in 0, 1, 5, or 6, are the following; vizi Given AT°. Squares. Remainders. 10133 ~ 5329 =ώ 4804 10133 - 5929 =s 4204 10133 - 608 4 = 4049 10133 — 6724 = 8409 10133 - 688g 3244 10133 - 7569 = 2564 10133 - 7744 2389 10133 8464 1669 10133 - S649 =■ 1484 10133 - 9409 = 724 10133 - 9604 =5 529 Here the last remainder is 529 = 23;, and it is the only Square; therefore, the given number 10133 is a prime. Thus eleven operations in subtraction are fnade to answer the purpose of tWenty-four divisions, and even this supposes all prime numbers under 100 to be known ; for otherwise the number of divisions Would be much more considerable. Ex. 2: Is 7129 a prime number? Ex. 3. Find whether 47933 be a prime number. Ex. 4. Find whether 47881 be a prime number. PROP. VI. I16. To ascertain whether a given number of the form 8n + 3 be a prime number. Every prime number p of the form 8n + 3 is also of the form a? + 2if, or y = / + and here x and y must be both odd squares, for otherwise208 Quadratic Fortiis of Prime Numberί; x1 + 2yl cotdd not have the form 8» + 3; also y1 is necessarily less than \p\ we must, therefore, sub- tract from p the double of every odd square < '/pi and if amongst the remainders there be found one Square, and no more, the given number is a prime,' but otherwise it is not. These operations may be considerably abridged from the following considerations: We have seen, that all prime numbers terminate1 in one of the digits 1, 3, 7> or 9> and the doubles of square numbers terminate in one of the digits 0, 2, or 8; therefore, 1. If the given number terminate in 1, we may omit all those squares, the doubles of which ter- minate in 8; because these would have remainders terminating in 3, which cannot be squares. 2. If the last digit of the given number be 3 or 7, we may omit all squares the doubles of which terminate in 0, because the remainders of these will terminate in 3 or 7, and, therefore, are not squares. 3. If the last digit of the given number be 9, we may omit all squares, the doubles of which ter- minate in 2; because these will leave remainders terminating in 7, which cannot be squares. 4. It may be farther remarked, that every odd square has the last digit but one even, and, there- fore, in general, all those double squares may be omitted, that leave an odd digit in the last place but one of the remainder. Ex. Let it be proposed to ascertain whether the number 11051, which is of the form 8?i + 3, be a prime number.Quadratic Forms of Prime Numbers. 20§ Here, the last digit being 1, we may omit all those squares terminating in 9, because the doubles of these terminate in 8, and, therefore, the re- mainders in 3. Hence the operation, Given Ar°. Double Squares. Remainders, 11051 - 10082 == 969 11051 - 9522 — 1529 11051 - 8450 = 1—ί II r-H' O CD 11051 - 7442 = 3609 11051 — 6962 == 4089 11051 6050 5001 11051 - 5202 == 5849 11051 - 4802 = 6249 11051 - 4050 =£ 7001 11051 - 3362 = 76 89 11051 - 3042 8ΟΟ9 11051 - 2450 8601 11051 - 1922 == 9129 11051 1682 == 9369 11051 - 1250 — 9801 = 99* ring thus found two square remainders iiiay conclude with certainty, that the given num- ber is not a prime, and discontinue the operation. Remark. Our first rule extends to all numbers of the form An + 1, which includes the two forms 8n+l and 8w+5, and the above applies to all numbers of the form 8n -f 3; but those that fall tinder the form 8n + 7 are still excluded, nor can they be submitted to a similar test; for these num- bers being of the form x2 — 2f (art. 114), there are ho limits to the values of x and y, nor to the num*210 Quadratic Forms of Prime Namier ber of ways in which a given number may be re- solved into this form; for (.r2 — 2f) x (tfA2 — 2yn) may be resolved two wavs into the same form (art. 95): and* since we may find xn — 2f~ — 1, by taking x = 3 and y— 2, it follows, that this pro- duct is still that is/a number of the form — 2f may be resolved into this form in as many ways as we please, whether it be a prime number or not, which is not the case with the two forms xa~ + yl and or + 2yl. PROP. VII. 11/. If ct be any prime number, and the series of squares 1\ 2\ S’, 4% &c., be divided by a, they will each leave ai different positive remainder. This is in fact only a particular case of the ge- neral proposition demonstrated (art. 51); for, by making φ = 1, the series of squares, Φ\ 2ψ, 3ψ, 4ψ, &C., becomes I4, 24, 3% 4' &c., (ffj, each of which, when divided by a, will leave a different remainder, as is demonstrated in that article. Cor. 1. And the same is evidently true of theQuadratic Forms of Prime Numbers. 211 negative remainders, which arise from taking the quotients iri excess (cor. 1, art. 51). Cor. 2. Hence, also, we may see in what cases the positive and negative remainders are equal to each other; for then it is evident, that a will be a divisor of the sum of two squares, and we shall have r2 + / ; , = 1, then, because a' is a divisor of the formula lo2 + x2+y2 + z2, it is also a divisor of the formula (iv - aa')2 + (x — βα')2 + (y — yaf)2 +(z — oaf, where each of the roots is less than \a' (cor., art. 118); assuming, therefore, . (iv — aa')2 + (λ? — βα')2 + (y — ya')2 + (% — ha')2 = a"a\ we shall have, for the same reason as above, a"a' < a'2, or a" < a'. Now, by means of the formula (art. 94), if we multiply together the values of aa', and a"a'9 we shall find a product that is the sum of four squares, and*of which each is divisible by a'2; and having performed this division, wre obtain a" a = (a — aw — βχ — yij — ο*)2 + (αχ — fiw + yz — hy)2 ' -f (ay — yw -I- tx — βζ)2 + (az — hw + βη/ — yx)2;Quadratic Forms of Prime Numbers. 215 .or, for the sake of abridging this expression, μ/* + λ?,2 + yn + %n-a"a\ and here we have a" < a'. If now a"=l, the above becomes wn jr xn + y'2 4- £/2= a, and the proposition will be demonstrated; but if a", though < a', be >1, we may proceed, in the same manner, to find amew product, wn + + y"2 + %,n = a'" a. and in which a!" < a!'; and by continuing thus the decreasing series of integers a, a', a", a/y/, a"", &c., we must necessarily, finally, arrive at a term a{m) equal to unity, and then we shall have a equal to the sum of four squares. — α. E, d. prop. x. 120. Every integral number whatever is either a square, or the sum of two, three, or four squares. This follows immediately from the foregoing proposition, and the formula (art. 94) ; for every number is either a prime, or produced by the mul- tiplication of prime factors ; and since every prime number is of the form {uf + x* +y* + and the product of two or more such formulae being still of the same form (art. 94), it necessarily follows, that every integral number whatever is of the form But it is to be observed, that no limitation in the course of the demonstration of the foregoing pro-.B16 Quadratic Forms of Prime Ν-umbers· position was made, that could prevent any one or more of these squares from becoming zero; there- fore, every integral number whatever is either a square, or the sum of two, three, or four squares. α. e. d. Cor. Ail that has been proved in the foregoing proposition for integral numbers, is equally true of fractions; for every fraction may be expressed by an equivalent one having a square denomiqator$ therefore, every fraction is of the form W* 4- x* 4- if + uf x1 if 2? s. - — - 4- ■ - + L—z 4- ■ g i nr m: nr nr m this curious property, therefore, extends to every rational number whatever. Scholium. The theorem that we have demon- strated, in the two foregoing propositions, forms a part of a general property of polygonal numbers, discovered by Fermat; which is this, “ Every num- ber is either a triangular number, or the sum of two or three triangular numbers. A square, or the sum of two, three, or four squares. A pentagonal, or the sum of two, three, four, or five pentagonals. And so on for hexagonals, &c. Or the same may be more generally expressed thus: If m represent the denomination of any order of polygonals, then is every number n the sum of m polygonals of that order; it being understood that any of these po- lygonals may become zero. Let, therefore, n be any given number, and a1, ?/, indeterminate quantities; then the different parts of the general theorem may be detailed in the fol- lowing order:Quadratic Forms of Prime Numbers. 217 ο . o , 0 . < x +x y+y z‘+2' + + — 1st, N = * 2 2 2 2d, n —+ + + . 3uq - u 3ιν* — w 3cr — a? , 3if-y , 32Γ-* 3d, NF---------:+—-------+■--------+-1----~+-------: y ' 2 2 2 2 2 ' 4th, &c. &c, &c. &c. The second form which relates to the squares has. been demonstrated in the foregoing pror position, and Legendre has also demonstrated the first case, for triangular numbers; but all the other cases, past the second, still remain without demonstration, notwithstanding the re- searches and investigations of many of the ablest mathematicians of the present time, and of others now no more: amongst the former we may mention Lagrange, Legendre, and Gauss ; and of the latter, Euler, Waring, and Fermat himself; the latter of whom, however, as appears from one of liis notes on Diophantus, was in possession of the demon- stration, although it was never published, which circumstance renders the theorem still more in- teresting to mathematicians, and the demonstration of it the more desirable. We have demonstrated the second case, but this carries us no farther, whereas, if we had demon- strated the first, the second would flow from it as a corollary; and it may not he uninteresting to show in what manner these different parts of the same theorem are connected w ith each other. First, let us suppose the possibility of the equation xQ + x if 4- if + %218 Quadratic Forms of Prime Numbers. to have been demonstrated, from which may be drawn this, 8n + 3 = (2x + l)2 + (2y + l)2 + (22+ l)2, or 8n + 3 =x'2 + i/2 + zn, or 8n + 4=xn + y'% + 2/2 + 1; and since these four squares are all odd, the num- bers x' +y', x'—y', 2'+ 1, and — 1, are all even; and hence we have, in integers, 4n + 2 = /x' + y'\ /xl-y' V /z'+ IV. A'-lV irf-) +(—) +(—)* or, for the sake of abridging, 4n + 2=+ x,n+y'n + %'/2; of which squares two are even and two odd, for otherwise their sum could not have the form 4n + 2; we may, therefore, write 4n + 2 = 4r- + 4s2 + (2 i +1)2 + (2+ l)2; from which we deduce 2n+ 1 = (r + s)2 + (r —s)2 + (t + v+ l)2+ (t — vf; that is, every odd number is the sum of four squares, and the double of a number that is the sum of four squares is itself the sum of four squares, for ( 2(m0' + n" +p~ + (f) — \ (m + w)2 + (m — nf + (p + qf + (p — qf; and, therefore, every number is the sum of four squares. If, therefore, the case which relates to triangular numbers was demonstrated, that which relates to squares would be readily deduced from it; but theQuadratic Forms of Prime Numbers. 219 converse has not place; that is, we cannot deduce the first case from the second. The third case gives Άιι1 — u 3w° — iv Άχ1 — x 3i/° — n Sz^ — z N = —------1--------l· —----h —------ H-----, or 2 2 2 2 2 24n + 5 = {6u - 1 γ + (6w -1 )a + (6# -1 γ + {6y-1 γ+{6ζ_ 1 )3, So that the enunciation of this particular part re- turns to this, Every number of the form 24n + 5 is the sum of five squares, of which each of the roots is of the form 6n— 1. The fourth case returns to this, Every number of the form 8n + 6 may be de- composed into six squares, of which the roots are of the form An — 1. And, in general, the proposition is always re- ducible to the decomposition of a number into squares, and all the partial propositions that we have considered are included in the general form, $aN + (a + 2) (a — 2)3 = (2a#-a+2)3 + (2ay — a + 2)3 + (2az — a + 2)3 ψ &c. the number of squares on the latter side of the equation being (a + 2),220 CHAP. X. On the different Scales of Notation, and their Application to the Solution of Arithmetical Problems. PROP. I. 121. Every number N may be reduced to the form n = arn + brn~' + er”'2 + &c. pA + qr + w, where r may be any number whatever, and a, b, c, &c., integers less than r. For let n be divided by the greatest power of r contained in it, as A, and let the quotient be a, and remainder ν', so that n == or” + Divide again ν' by the next lower power of r, as rn_l, and let the quotient be δ, which will be an integer or zero, according as. n'> or < A"', and the remainder n", whence n = aA + bA~1 + n". Dividing again n" by r”"2, and supposing the quotient c, and remainder ν'", we have n = aA + bA"1 + cA ‘2 + ν'". And by thus continually dividing the remainder by the next lower power of r, Ave shall be evidently brought finally to the form n — aA + hrn~' + cA~° - - - pA + qr -1- w ·,221 Different Scales of Notation. in which expression, as a, b} c, &c., are the quo- tients arising from the division of a number by the highest power of r contained in that number, it necessarily follows, that each of those coefficients, $ , a, b, c, &c., is less than r. — a. e. d. * ***% ■Cor. If r— 10, then a, b, c, &c., are the digits^C^S^ifoi by which any number is expressed in our common method of notation; thus, 76034=7.104 + 6.103 + 0.10* + 3.10 f 4, 18461 = 1 . 104 + 8 .103 + 4.10* + 6.10+1, which form is always understood in enumerating the value of any number proposed; that is*, we give j to every digit a Ior.nl, as well a.a its nri.nrinal nr natural value: thus, in the number 76034. the second digit from the right is 3. but we consider it as representing 30. on account of its local situation, being in the second place from the right: iutlw game manner, the 6 represents 6000. and the 7. 70000, so that the value of each digit is estimated according to its local situation and its original value, the former indicating the power of 10, and the latter the number of those powers that are in- tended to be expressed. Cor. 2. It is evident, from the foregoing pro- position, that a number may be in the same man- ner represented by any other value of the radix r, and hence arise the. difterent scales of notation, which receive the following particular denomi- nations according to the value of the radix r. If r= 2, it is termed the Binary scale. r = 3, - * - - Ternary. r~ 4, - - - - Quaternary.222 Different Scales of Notation. If r= 5, it is termed the Quinary. r = 6, - - - - Senary. r—10, - - - - Denary, or common scale.· r = 12, - - - - Duodenary. And since, hv the foregoing proposition, a, b, ci &.C., are always less than r, the radix of any system into which they enter: therefore it follows that for every scale we must have as many cha- racters, including the cipher, as are equal to the number expressing the radix of the system.. Thus, the only characters are, for the Binary scale, - - 0, 1. 0, I, 2, σ, 1, 2, 3. 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Ternary, - - - Quaternary, - - Senary, - - ■ Denary, or com- ^ raon scale, And hence it follows, that in the duodenary scale, we must have two additional characters for representing 10 and 11, and as these characters may be assumed at pleasure, we shall, in what follows, express 10 by the symbol φ, and 11 by tt, whence the digits of the duodenary scale will be O, 1, 2, 3, 4, 5, 6, 7, 8, 9, φ, 7Γ, A - PROP. II. 122. Having given the equation N — arn + brn~1 + cr11'9 - — pr~ + qr + w, in which n and r are given numbers, to find tire unknown coefficients, a, b, c, &c., and the ex- ponent n. Or, which is the same, to transform aDifferent Scales of Notation. · 223 Humber from the denary to any other scale of no- tation. It is evident that this may be done by propo- sition 1; namely, by dividing n successively by the highest power of r that is contained in it; but it is more readily performed by dividing n successively by r; thus, if n = arn + hr" "' +t crn~" — - pr1 + qr +to be divided by r, the quotient will be arn-'+ &νπ~2 + crn~3 - - - pr + q, and the remainder w. This last quotient being again divided by r gives for a new quotient arn~~ + fer’*"3 + cr"'4-p, and a remainder q. And this quotient, divided by r, gives a quotient arn~3 + brn~* + crn~5, and a remainder p. Whence it is evident, that the successive re- mainders will be the coefficients w, q, p, &c., or the digits that express any number in the scale of which r is the radix. Ex. 1. Having given the equation ΐ7’486=α.6’! + έ.6”'Ι + ο.6“-2------w, tq find a, b, c, &c. Or, which is the same, let it be proposed to convert 17486 from the common to the senary scale.224 Different Scales of Notation; Here, by the foregoing proposition, 6)17486 6)2914 rem. —2 = w 6)485 - - - 4 = q 6)80 - - - 5 =p 6)13 - - - 2 =C 6)2 - - - 1 = 6 0 - — 2 =a *Therefore, 17486, in the denary scale, is ex- pressed by 212542 in the senary. Ex. 2. Transform 1810 into both the binary and ternary scales. 2)1810 3)1810 2)905 rem. = = 0 3)603 rem. = = I 2)452 - - - I 3)201 0 2)226 0 3)67 - - - 0 2)113 - - - 0 3)22 - - - - I 2)56 1 3)7 - - - 1 2)28 - - - 0 3)2 1 2)14 - - - σ 0 --- 2 2)7 - - 0 2)3 i 2)1 - - - 1 0 - - - 1 Different Scales of Notation. 225 Therefore, 1810 = Γ1100010010, in the binary Scale; and 1810 = 2111001, in the ternary scale. Ex. 3. Transform the two numbers, 844371 and 215855, from the denary to the duodenary scale. 12)844371 12)70364 rein. = 12)5863 12)488 12)40 12) o - - - 12)215855 = 3 12)17987 rem. =11=^ 8 12)1498 - - - 11 = 7Γ 7 12)124 - - - 10 = φ 8 12)10 - - - 4 = 4 4 0 - - - 10 = φ 0 - Hence 844371=348783 1 . a , , , and 215855 = φ4φττ7Γ, J J And thus a number is readily transformed from the denary to any other system of which the radix is given, and hence we find 1000 is expressed in the following maimer according to the value of the radix r. r = 2, 1000 = 11.11101000; r = 3, 1000 = 1101001; r = 4, 1000 = 33220; r = 5, 1000 = 13000; r = 6, 1000 = 4344; r = 7, 1000 = 2626; r = % 1000 = 1750; •r = 9, 1000 = 1331; r = 10, iooo== 1000; ? = 11, 1000- 82φ; r = 12, 1000 = 6w4. a226 Different Scales of Notation. Hence it is evident, as it is indeed from the na- ture of the subject under investigation, that the greater the radix is, the less will be the number of digits necessary for expressing any given number ·, but the operations of multiplication, division, &c., will be the more complex; and, therefore, in judging of the advantages and disadvantages of different systems, we ought to keep both these circumstances in view, as also a third, which is the number of prime divisors of the radix; and, on a just estimate of the whole, the radix 12 will be found preferable to any of the other systems: but on this subject we shall add a few remarks at the conclusion of this chapter. PROP. III» 123. To transform a number from any other scale of notation to the denary, or common scale. This proposition is the converse of the foregoing one, and it is readily effected by the reverse opera- tion. For let arn -i- + crn~* — - pr* + qr + w represent a number ip any known scale of notation, whose radix is r; then, since a, b, c, &e., are also known, we have only to collect the successive values of the different terms, and their sum will be the number transformed, as required. Ex. 1. Transform 7184 from the duodenary to the common scale of notation. First, 7184 = 7.123 -f 1 .12° + 8.12 + 4»Different Scales of Notation. 22f Therefore, we have, 7.12" = 12096 T . 12* = 144 8.12= 96 4 = 4 Duodenary 7184 =12340 Denary scale; Ex. 2. Transform 1534 from the senary to the denary Scale. 1534 = 1.63 + 5 . 6* + 3 ; β + 4 l.63 = 2i6 5 6*= 180 3.6 = 18 4=4 Senary 1534 =418 in the common scale. Cow By means of the two foregoing propositions \ r. a number may be transformed from one scale of / notation to another, neither of which is the denary, f jjn, by first transforming it from the given scale to the V common scale, and then into the particular one J required; bROP. tv. 124; In every scale of notation, whose radix is r, the sum of all the digits expressing any number, When divided by r— 1, will leave the same re- mainder as the whole number divided by r— 1; that is, if N = nr” +Zir”-1 + cr”'B - - - pr + (jr + w, then will n -s- (r — 1), leave the same remainder, as (α + 6 + c - - - p + q + w)*{r~l). a 2228 Different Scales of Notation. For make r— l^r', or r = r'4 1, .then (r— l) = (r'4 . will leave a remainder 1, because every term of the expanded binomial (r'+l)n is divisible by except the last; which is 1; and* consequently* (r'41 )v -f- r', or rn-^(r— l), will leave a remainder 1, and this property is entirely independent of the value of n; and hence it follows, that every,power of r divided bv r— 1 will leave a remainder 1, or the powers f\ rM~l, rn“a, &c., are all of the form ?n(r— 1)41; that is, 7y^m(r~ 1)4 1, whatever integer value is given to n\ And hence it follows, that if the work be right. For let a and b represent any two factors, and make ■α = 9η + α?, -b^Qm + b'. Then ah = 9(9/27/1 + mat + nb') + a'l·'-. and, therefore, ab~ 9 leaves the same remainder as a'b' divided by 9: but the remainder of a 9 is * the same as the digits of a by Q, and the remainder f of b -*- 9 is the same as the digits of b-+- 9? a*id the J same of the product ab; and hence the reason of the rule. The same is obviously true for any other system of notation, by taking the number next less than the radix for the divisor. Thus, for example, we have seen that 215855 = φ4φττ7Γ in the dim-230 Different Scales of Notation. denary scale, and 215855-4-11 leaves a re* jnainder 2: but φ + 4 + φ + 7Γ + χ=10 + 4 + 10 + 11 + 11 =46, which, divided by 11, gives also a remainder 2. Suppose it was required to multiply φ4φππ· by φφ4, the operation and proof would stand thus: Operation. φΑφππ rem. 2 <ρψ4 rem. 2 3577?r8 8811t2 881 1tt2 95088918 rem. 4 It is unnecessary to observe, that in this operation, as in all others in which the radix is r, we must in multiplying, dividing, &c., divide by the radix; that is, by 12 in the above example, and set down the overplus, instead of dividing by 10 and setting down the overplus, as is done in the common scale. Proof by 11. PROP. V. 125. In any scale of notation whose radix is r, the diiference of the remainders of the sum of the 1st, 3d, 5th, &c., digits by r + 1, and the sum of the 2d, 4th, 6th, &c., digits divided also by r-f 1, is equal to the remainder of the whole number divided byr+1. Let N=ar" + brn~l + cr"'2 - - - pr^ + qr+w, then, I say, the remainder of {w +p + b &c.) -4- {r + i),231 Different Scales of Notation. minus the remainder of (q + c + a See.) -s- (r + 1), is equal to the remainder of N h- (r + 1). For make r + 1 = r', or r = r" — 1, then it is evident that (r'-l)” will leave a remainder +1, or — 1, according as n is even or odd; for all the terms in the expanded binomial (r' — l)“ are divisible by r'} except the last, which is +1 or — 1, according as n is even or odd, independently of any other value of n; and, therefore, ----will also leave the same re- r+1 mainder in the isame cases; that is, every odd power of r is of the form m{r + l) — 1, and every even power of r is of the form n{r + 1) + 1. Therefore, in the above expression, we have w + IV, qr z&qm (r+\) — q, pr2 {r+l)+p, crn~ 2 (r+ l)-c, br»- 3 *P.bn' (r + 1) + b, arn a?a«}"(r+ 1) — a, See. &c, And, consequently, Ns*swi"'(r+ l) + w— q+p — e + b — a; and, therefore, when divided by r+ 1, it will leave the same remainder as (m— q +p — c + b — a) divided by r + 1, or as (w+p + b, &c.,)-f-(r+ l)-^(^ + c + a, &c.)-*·(?* + l)·. a. e. d. Cor. 1. Hence, in the common scale, if the sum of the digits in the odd places be equal to the sum 670 623 490 416 760 6 23 139 Therefore its length is 39 feet 3 inches 2' 3". And the same principles are equally applicable, to the extraction of the square root, as is evident by the following example:Oijfereht Scales of Notation, 23? Ex. 3. Having given the area of a square equal to 17 feet ,4 inches 6\ required the length of its side. 82 2 8402 2 15*46(4·202φ 14 ΑψΟίΓ 21 / 1 20000 14804 Proof. 8404ψ 73x800 6x4404 473x8 Therefore the side is 4 feet 2 inches 0/ 2" 10'". And thus may any other numerical operation he performed with nearly as much ease as in common arithmetic. PROP. VII. 127. Every number less than 2n+1, is com- pounded of some number of terms in the series., 1, 2, 2% 2s, 24, 25, &C. 2*. This is made evident by transforming any given number n < an+l into the binary scale, which, from what has been observed at cor. 2, art. 121, will assume the form, N==a.2*+-J.2n“14-c.2n‘9 - —p.2* + q-2 -i- w; where a, δ, c, &c., are each less than 2, and con- sequently either 0 or 1; and as every number less than 2n+l may be thrown into this form, therefore, with the above series, every number whatever,238 Different Scales of Notatiofti within the assigned limits, may be compounded of some number of those terms; Cor. 1. What is said in the above demonstra- tion, not only proves the truth of the theorem, but also points out the method by which it is to be effected; and at the same time it is evident that there is only one way in which the selection can he made. Cor. 2. In the above theorem^ the greatest power of 2 is 2”, and consequently the greatest number that can be formed is 2B+1 — X ; but, if the power of 2 be Unlimited, so also will the number that may be compounded of those terms 5 that is, any number whatever may be compounded of the terms of the indefinite series, 1, 2, 22, 23, 24, &c. Ex. Having a series of weights of lb. lb. lb. lb. lb. 1, 2, 4, 8, 16, &C;, it is required to ascertain which of then! mast be selected to weigh 1719 pounds* First, 1719 i11 the binary scale is expressed by 11010Π0111: the Weights therefore to be em- ployed are, lb. lb. lb. lb. lb. lb. lb. lb. 1 + 2 + 23 -f 24 + 2s + 27 + 29 + 210. PROP. VIII; 128. Every number whatever may be formed by the sums and differences of the terms of the geometrical series, 1, 3, 3s, 3% &c. For, by transforming the given number n into the ternary scale of notation, it will assume the form, N = a3" + i3"'1+ c3e'4 - - - jp3* + #3 + tt>iDifferent Scales of Notation. 230 where each of the coefficients, a, b, c, &c., is less than 3, and consequently they must be either 2, or 1, or 0. Now, in order to prove the truth of the theorem, it will be better to select a partial ex- ample, the reasoning on which will be evidently applicable to every other case. First, then, it is obvious, that, if no one of these coefficients be greater than 1, the question is resolved agreeably to the conditions of the proposition: we need, there- fore, only consider the case, in which some one or more of the coefficients are equal to 2. Let, then, f n = 3" + 2.3n-‘ + 0.3*“® + 2.3"-’ + 3""* + 0.3*“s + t 2.3n'6, &c. ' And, since 3.3"“e = 3"·*, and 3.3"-’= 3“·®, 3.3*·1 = 3% the above expression is the same as (2.3“ + 3”“2 + 3n"4 + 3““5) - (on-' + 3*“s + 3”-e) = (3„+i + 3»-» + 3»-4 + 3-:»j _ (3« + 3»-i + 3-» + 3-«)=N> agreeably to the conditions of the proposition.— a. £. d. Remark. The latter part racfcrs - - J tv « • «J β> Ύ> K *, L· To represent 10, 20, 30, 40, 50, 60, JO, 80, 90, they made 1 . <. use of - - § '■> *? h V> f’ °».·*■» ^ For the hun- Ί itt 300 t^ot> St0 Uo 7*>o 90®· dreds they > p, ; and so on for any other possible combination ; but as regularity tended in a great measure towards simplicity, they generally wrote the characters ac- cording to their value, as in the examples above. In order to express any number of myriads, they made use of the letter m, placing above it the character representing the number of myriads they intended to indicate. Thus, α β y ί · Μ, Μ, Μ, M, &C., represented 10000, 20000, 30000, 40000. Thus,Different Scales of' Notation. 247 Κζ δΐοβ also, M expressed 370000, 'm =43720000; and, generally, the letter m placed beneath any number, had the same effect as our annexing four ciphers. This is the notation employed by Eutocius in his Commentaries on Archimedes, but it is evidently not very applicable to calculations. Diophantus and Pappus represented their my- riads by the two letters Mu placed after the num- ber, and hence, according to them, the above numbers would be written thus : α.Μυ, β.Μυ, y.MU, δ.Mu, &C. 370000 = mu, and 43720000 = δτο0.Μυ. Also 43728097 is expressed by Βτοβ.Μυybfi, And 99999999 - - > hyjfrhe.Mu^O. This notation in some measure resembles that which we employ for complex numbers, such as feet and inches, or pounds and shillings. The same authors, however, ejnployed a still more simple notation, by dropping the Mt>, and supplying its place with a point; thus, instead of δτοβ,Μυ they wrote ; and for Sffyhdj they wrote ; this last number, with the addition of unity, be- comes 100003== 100000000, which was the greatest extent of the Greek arithmetic; and, for common purposes*; it was quite sufficient, because their units of weight and measure, such1* as the talent and stade, were greater than our pound and foot It was, therefore, only astronomers and geometers248 Different Scales of Notation. who sometimes found an inconvenience in this limir tation; thus, for example, Archimedes in hh.jiri- narius, in order to express the number of grains of sand, that might be contained in a sphere that had for its diameter the distance of the fixed stars from the earth, found it necessary to represent a number which with our notation would require sixty-four places of figures; and in order to do this, he as- sumed the square myriad, or lOCjOOOppo, as a new unit, and the numbers formed with these new units he called numbers of the second order; and thus he was enabled to express any number which in our notation requires sixteen figures: assuming again (loopoopoo)2 for a new unit, he could represent any number that requires in our scale twenty-four figures, and so on: so that by means of his num- bers of the 8th order he could express the number in question, which, as .we have said above, required sixty-four figures in our scale. Hence, according to Archimedes, all numbers were separated into periods or orders of eight figures, which idea, as we are informed by Pappus, was considerably improved by Apollonius, who, instead of periods of eight places, and which were named by Archimedes octade§, he reduced to periods of four figures ; the first of which, on the left, were units, the second period myriads, the third double myriads, or numbers of the second order, and so on indefinitely. In this manner Apollonius was able to write any number that can be expressed by our system of numeration; as, for example, if he had wished to represent the circumference of a circle, whoseDifferent Scales of Notation. 249 diameter was a myriad of the ninth order, he would have written it thus: 7. gu)ιε. QcrZs. γφπβ. §&)λβ. γωμ,ς: βχμ*γ. 3. 1415 ()β65 3589 7932 3846 ‘ 2643. γωλβ. βφκο, 3832 7950 5824, Having thus given an idea of the Grecian nota- tion for integer numbers, it remains to say a few words on their method of representing fractions. A small dash set on the right of a number, made of that number the denominator of a fraction, of >vhich unity was the numerator; thus 7-Ύ, V = I= p*a, = vrrj &c.; but the fraction 4- had a particular character, as C, or «<, C, or K, When the numerator is not unity, the denomi- nator is plaoed as we set our exponents. Thus, is*', represented 1564, or^-f; and represented or : also σ-ξγ.γφμδ*^0* = 2633 54433177β = VWVW ·,. This last fraction is found in Diophantus, book 4, question 46. As it was only oiir intention in this place to convey to the reader a connected and general idea of the notation of the Greeks, in order the better to estimate the value of the modern, or, as it is sometimes called, the Indian arithmetic, we have not entered into an explanation of their sexagesi-250 Different Scales of Notation. mals employed by astronomers in the division of the circle, and of which ours is still a representative, as is evident from the following example: O. i/Q' -/]" f /yiv ιβν λατ'== 0° 59' 8" 17"' 13iv 12v 31vi 131. It still remains for us to explain, by a few examples, the method that was employed by the ancients in order to perform the common rules of arithmetic, with this complicated system of nota- tion, and must refer the curious reader, who wishes for more particular information, to an ingenious essay on this subject by Delambre, subjoined to the French translation of the Works of Archi- medes, to which essay we are indebted for many of the foregoing and following remarks. Example in Addition. From Eutocius, Theorem 4, of the Measure of the Circle. yfxa. 847 3921 |. 7)U· 6o 8400 /?)>}. βτχα 9®8 2321 In this example the method of proceeding is so obvious, that it needs no explanation, being per- formed exactly as we do our compound addition of feet and inches, or pounds, shillings, and pence; but it is more simple on account of the constant ratio of ten between any character and the suc- ceeding one.Different Scales of Notation, 251 Example in Subtraction. Eutocius, Theorem 3, on the Measure of the Circle. δ.γχλ? 93636 β.γυ Q 23409 70227 This example also is so simple, that the reader will find no difficulty in following the operation, by proceeding from right to left, as in our sub- traction, which method seems so obviously advan- tageous and simple, that one can hardly conceive why the Greeks should ever proceed in the con- trary way, although there are many instances which make it evident that they did, both in addition and subtraction, work from left to right. In multiplication they most commonly proceeded in their operations from left to right, 4as we do in multiplication of algebra, and their successive pro- ducts were placed without much apparent order, as is evident from the following examples; but as each of their characters retained always its. own proper value, in whatever order they stood, the only inconvenience of this was, that it rendered the addition of them together a little more trou- blesome, ^ , As it is burdensome to the memory to retain in mind the value of all the Greek characters, we have, for the ease of the reader, in the following examples, made the substitutions as below!, by252 Different Scales of Notation. which means their operations will be the more readily comprehended.. Fora, β, f, δ, &c. we.write 1°, 2°, 3°, '4°, See. t, x, λ, μ., kc. - - — l', 2', 3', 4', Stc. f>, + 3ye — zs is impossible, 15. The equation ax6 + 'Jy6 = ze is impossible for every value of a, except those that fall under one of the forms 7n,, or + 1. l6‘. Every odd number prime to 5 is a divisor of any repetend digit, and the number of digits necessary to form the complete dividend will never exceed the number of units expressed by the di- visor. 17. If r be a prime number, then will every prime divisor of the formula an± 1 be of the form 2nx + 1, except only the divisor a + 1, when the ambiguous sign is +, and the divisor a — 1 when that sign is —. 18. No square number can terminate with more than three equal effective digits.Miscellaneous Propositions. 259 ig. The squares of all numbers composed of less than ten units, as 11, 111, 1111, &c., always have the form 1, 2, 3, 4, - - - 4, 3, 2, 1. 20. The equation (x2 + ί/2)2 + (r2 — ?/2)2 = z2 is impossible. 21. The following equations are impossible: 1. Ζ2ϊ/+ϊ/2χ= z2. 2. aibx'1 + alf y* = z2. 3. 2a3bx4 + 2aPy* — z2. 4. 2a3bx* + 4ab3y4 = z2: Required the demonstrations. 22. Find the rational values of x and y in the two equations 2,r4 + 8y* — z2, and Qx4 + 54y* = z2j Or prove that there can be no such values. 23. Every even mimber is the sum of two prime numbers, and every odd number is tlie sum of three prime numbers. Required proof. 24. Every prime number of the form 3ή + 1 is also of the form #2 + 3y2. 25. Let 1, 2, 3, 4, &c. - - - n, represent any continued product of n terms, and let p be any prime number whatever; then will the above product be divisible by such a power of p as has its exponent expressed by the sum of the integral parts of the fractions n > η n — -f “ -f---5" p p p n V &c.· Requited proof. 26. What weights must be selected out of the single series 1, 3, 9> 27, 81, &c,., to weigh 100100 pounds? \ 27· How many terms must be selected out of26ο Miscellaneous Propositions. the single series 1, 2, 4, 8, l6, &c., that their sum may be 17845 ? 28. Let n be any number whatever, and a the difference of n, and the next greater square; also b the difference of n, and the next less square; then will n — ah be a complete square. 29. The expanded binomial (1 - l)" = 1—11 + n(n — 1) n(n — 1) (n — 2) -&c.:^0; and if these terms be respectively multiplied by the series 1, 2, 3, 4, &0. of by any power of these terms, except the nth, as lm, 2m, 3m, 4m, &C. the sum of all the terms thus produced is equal to zero. Required proof. 30. The continued product 1 . 2 . 3 . 4 . &c., (n — l)n = fl()l_ ?il~ n{n— 1)* + -------(w — 2) — 1.2 n. (n— i)(n — 2) 2 . 3 (»· 3)“ + &c. Required proof. 31. If a, b, and c, represent the three sides of a triangle, and c the angle contained by a and b, then, if f a? + ί5 = c2, the < c — 90°; < ai + ab + &2 = c2, the. < c = 120°j t «2 — ah + b* = e2, the m ; and suppose that N, when divided by M, gives a quotient a, and remainder p; then we have N P .Ml — =a + —, and — -----. M M N " a + - M Dividing in the same manner m by p, and supposing the quotient b, and remainder a, we have, in the same manner, M 7 a -b+—, and P 1 P M ' a 5 & + — P P R and a _ 1 & P R * c H aContinued Fractions. s63 a , s . r s=d+ —, and — r r a d+- R & C. &.C, &c. where a, b, c, &c., are the quotients arising from dividing, successively, n by μ, m by p, p by a, &c. And if now we substitute for the fractions &c., their respective values, found as above, we obtain the following expression; M__ 1 _ 1 _ 1 N P l 1 flM d, H—a u—j- 1 m b +—- b +i P c+R. a' JVl and, consequently, the fraction — is reduced to a continued fraction as -required. 135. We are thus furnished with a very simple practical method of performing this reduction, in all such cases; viz. divide the denominator by the numerator, then the divisor by the remainder, and so on, as in finding the greatest common measure of two numbers; and the successive quotients will be the denominators of the fractions, above re- presented by α, έ, ο, &c. Note. If the numerator be greater than the de- nominator, the continued fraction, will be preceded by an integer ,v264 Continued Fractions. 11*1 Ex. l. Reduce ^ to a continued fraction, 1171)9743(8 9368 375)1171(3 1125 46)375(8 368 ~7)46(6 42 4)7(1 4 3)4(1 3 1)3(3 3 0 Consequently the fraction proposed becomes 1171 9743 1 8 + 1 +1+- 1 l+? which is therefore reduced to a continued fraction as was required.Continued Fractions. 265 Ex. 2. Reduce 743 6ll to a continued fraction. 611)743(1 6ll 132)611(4 528 83)132(1 83 49)83(1 49 34)49(1 34 15)34(2 30 4)15(3 13 3)4(1 3 1)3(3 3 And therefore we have for the required fraction 743 1 6n“1 + 4 + ^♦T+l . S + 3 which is preceded by an integer, as stated in the foregoing note,266 Continued Fractions. Remark. We call the integers a, b, c, d, &c., obtained in the foregoing operation, Quotients, being the results of successive divisions; and each of p Q, these, with its depending fraction, as a + —, b + -> R. c + —, &c,, is called a Complete Quotient. PROP. II. 136. To transform a given continued fraction to a series of converging fractions. Let 1 a + τ 1 b + - 1 C + d+ &c. he any continued fraction: it is required to trans- form it to a series of converging fractions. This is in fact performed by the common rules for the reduction of complex fractions to simple ones; thus 11 a a 1 - 1 = *+b _J____b ab+ 1 ~~ ab + 1 b 1 Ί— c 1 i bc + 1 I $ C ~ bc+ 1 1 bc+1 _ be -j- 1 ___ a(bc 4-1) 4- c o[bc4-1)4- c (ab 4-1 )c 4- a' be f 1Continued Fractions. 267 And in the same manner we find i 1 _ (bc + l)d+b ^ + T) + - 1 {(ab + l)c + a}d+(ab + l) C+d But this reduction, when there are many terms in the continued fraction, becomes very em- barrassing, and at the same time unnecessary; for, from what has been alreaidy done, a very obvious law of formation discovers itself, in order to render which the more manifest, let us resume our fore- going results, making also the successive substitu- tions as below; viz. 1 _ l p° a a ~¥ I 1 ■ b p' a+V' ” ~ ab +1 I 1 a+b+i C bc+1 ~ (ab + i)c + a p" 1 a+\ 1 b + -·, c + (be 1 -+■ b — {(ab -f-1} c ti \ d ab -j-1 d xlx II How here it is obvious, that p° =1 - - - - q =-a - p' =bp° - - - q' =bq° +1 p" = cp' + p° - - q" = cq' + q° p”' — dp" + p' - - q'" — dq" + q' j>iv = ep'"+p" - - q" — eq'" + q";268 Continued Fractions. and thus the successive terms of the series of con verging fractions may be obtained as far as Mre please, by means of the given quantities a, b, c, See.·, these terms being 9°’ 9' ’ V &c. 137- Hence we have the following very easy method of reducing any continued fraction to a series of converging fractions. Write all the denominators of the successive terms of the continued fraction in a line, thus a, b, c, d, e, &c.; • then the first fraction will have unity for its nu- merator, and the first term, a, for its denominator; the second will have the second term, b, for its nu- merator, and for its denominator ab + 1and the numerators of all the succeeding fractions will he found, by multiplying the. numerator last found by the corresponding term in the above series, and adding to the product the preceding numerator; and the denominators are obtained in exactly the same manner, as is evident from the foregoing proposition: thus, a, b, c, d, e, Sec. 1 b bc+l (bc+l)d+b a’ ab+ l’ (ab + l)c + a5 { (ab+ l)c + a the last term of which series will be the original fraction first proposed.Continued Fractions, 269 Ex. 1. Transform the continued fraction 1 7+6+\ 1 5 + - 1 2 + 3 to a series of converging fractions; Denominators, 7* 6, 5, 1 6 Conv. fractions, —, —, 7 43 2, 3, 31 68 222’ 487* 235 16*835 the series required. 138. It is also obvious, that we may thus find the series of fractions converging towards any given quantity, without reducing it first to the con- tinued form. For wTe have seen (art. 135), that the denominators α, δ, c, &c., of the terms of any continued fractions, are the quotients obtained from finding the common measure of the two terms •of the given,fraction; and, therefore, having found these quotients, we may immediately ascertain the series of converging fractions, without any inter- mediate step; in fact, the consideration of any quantity under the form of a continued fraction is entirely useless, otherwise than as it leads us to the properties, and the law of formation, of the con- verging fractions; for it is in this form only, that these expressions are at all applicable to any useful purposes.2γό Continued Fractions. Ex. 2i Find the series of fractions converging 39 towards the given fraction y—* 30)187(4 15 6 31)39(1 31 8)31(3 24 7)8(1 7 1)7(7 7 1, 3, 1, / » 1 14 5 39 4’ 5’ 19’ 24’ 187 Quotients, Conv. frac. - which is the series of converging fractions re- quired. 139. If now any series of quotients derived .. M from the fraction — be represented by b c, &c., U, 10, &c.; a f P' P" &C., ? - 1 1 u +f)+p° SPor it is evident* referring to the original form, m __ 1 n a + T b+ &c. 1_ t + - 1 n + — V M+-, z that, by stopping at any particular quotient, and annexing thereto the remainder —, we have the % precise value of the original fraction, as will be still more obvious by turning to the form at art. 134. V Let now u + - = u', then the above becomes % Μ 1 N « + &c. 1 i. u and as the order of formation of the converging fractions does not depend upon any particular values of these quotients, it is obvious that the same law will obtain for the complete quotient u' as for any other; supposing, therefore,272 Continued Fraction& 1 a+ &c. 1 1 1 a + &c. I i t + ~ v we shall have, on the principles of art. 138, 1 a+ &c. 1 1 + - 1 v + —, u } p'u'+p° q'u' + or M N p'(u + j) +p° «'(« + ;) +S' €L# E· D* For example, in the reduction of at any term as below, 711)953(1 711 7u 953 , if we stop 242)711(2- 484 ---x ,15 227)242(1 —· complete quotient 227 ' 15 we shall have the following result: „ · 15 Quotients, - - 1, 2, 1— 15 . 2(1 +—_) + 1 1 2 V 227' Conv. frac. - - p “ +227^ + 1 .2(227 + 15) +227 717 , r .. rf *· ——---------—----= —the original' traction; 3(227+ 15)+227 Q53’ sContinued Fractions. 273 and the same has place for every complete quotient, as is evident from the preceding demonstration. PROP. Hi. p° v' 140. If —n- and be any two consecutive 9 9 terms in a series of fractions converging towards —; then will N , fq'-p'q0-- ±i; the ambiguous sign being -f when and - r, P M when —o < —t q n For let { a, h C, & c. V, U, iSj &c. 1 a\ b ab+ l’ &c. P° r p' p" r p"' q'"’ &c, represent any series of quotients, with their cor- responding fractions; then (art. 139) we have ph' —p"w +p', and q'" = q"iv + q'·, or ■p P W; and, therefore, p'"q" —p'q" -p"q"' -p"q', or p"'q"-p"q'" —p'q"—p"q'. And, in the same manner, since p"=p'u+p°, and q" — q'u + q°, we have p"~p° P' 9' * whence274 Continued FractionΛ p"q' —p0(f =p'q" —p'g°, or p"q' —p'q" —p°q'—p'q°; that is5 —p"q'" —p'q" —p'q'—p'q°—p°q' ί or the successive differences between the products- of each numerator and consecutive denominator, and the product of the denominator and the nu- merator of the same fractions, are equal (abstracting from their signs); hut the difference {ah -f- ϊ) — ah, that is of the first two fractions, is 1, and since the differences are all equal, they are each equal to 1 and, therefore, p°q' —p'q° = ± 1. , p° m , »' M , .But when ^ > —, then —7 < —: and, conse- q n q n i)° >/ va p'q° quently, —z>,> and and, therefore, q q qq qq p°q'> p'q0; that is, p°q'— p'q°= + 1. And, for the same reason, if ^ , then we 5 n have p°q' —p'q'3 — — 1. —a. e. n. 141. It is this property of converging fractions, that renders them so useful in the solution of all in- determinate equations of the first degree; for every equation of this kind has its solution depending upon that of the equation, ax — hi/ = ± 1, as will be shown in the next chapter. Now the solution of ax—by— + 1 is obtained by finding the series of fractions converging towards and assuming for x and y the terms of thatContinued Fractions. 275 fraction, immediately preceding p as is evident from the foregoing proposition. Ex. 1. Find * and y in the indeterminate equation l6* — 41y = l. First, 16)41(2 32 9)16(1 9 7)9(i 7 2)7(3 ΰ Quotients, 2, Conv. frac. 1)2(2 2 h 1, 3, 1 1 2 7 16 2’ 3’ 5’ 18’ 41: Whence we have *=18, and y — 7i which gives l6* — 4\y = 1, or 16.18 —4l . 7 — 1· And it is obvious, that we shall have the same result if we take * = 18 + 41 ?n, and y — 7 ± l6»i; m being indeterminate for this substitution gives also l6(l8±41w)—41(7+ l6m) = l; and by means of the indeterminate quantity m, an infinite number of values of * and y may be oV tained, that will answer the conditions of the equation.2?6 Continued Fractions. If the given indeterminate equation had been a x — inf = — 1, then Vvc must have taken x = 41«i — 18, and y — 16m — 7, which gives l6(41w— 18) — 4l(l6V« — 7) = — I; where the indeterminate quantity in is also the means of furnishing an infinite number of solutions to the equation ax — by = — 1. But as this subject belongs properly to the next chapter, we must dismiss it for the present, and continue our investigation of continued fractions. PROP. xv. 142. If ~~o y ·—, ^7n, &c. be a series of q q q q fractions converging towards any given fraction M . > . , then will these fractions be alternately greater and less than the given fraction; but each ap- proaches nearer to the true value of the original, than the one which precedes it. The first part of the proposition is evident from considering the law of formation of these fractions : for let M .N a + h + - + d &c.; then it is obvious, that because, in order ' ■ a nContinued Fractions. m to have the exact value, we must add a certain quantity to the denominator a (equal to all the other part of the expression): and 1 1 for the same reason; whence it follows, that < M N 5 because, in adding ^ to the denominator a, we make it too great, and, consequently, the fraction too small; and in the same way we find that 1^ a + 1 + c > M N and so on alternately. But, by article 136, 1 1 1 —; — 1 . — 1 (i a —9 a -f- -- 1 Szc b b-α · aye the successive terms of the converging series, being equal to £ £ vl &c., q°* qq'n 9 and, therefore, these terms are alternately greater and less than the original fraction; and hence it follows, that the \nlue of this last is always con- tained between any two consecutive terms of the converging series.2f 8 Continued Fractions. Now, in order to demonstrate the latter part of the proposition, let us consider the difference be- tween any converging fraction, and the original one to which it is an approximation. For which purpose, P° ... P let "q" be the fraction immediately preceding-; and OC let u + ^ be the complete quotient, corresponding V to ~; also> for the sake of simplifying^ put x , u-\— — u 9 z then we shall have, the same as in art. 139, M _pu'+p° n qu' + q° * - from which we derive m p pq—pq +1 n “ ~ q{qu' + q°) ~q{qu'+q°) M p° _ (pq° —p°q)u' _ ± u' ; and N q~ ’qQ(qu' + q°) ?%' + '/)' Whence we draw the following conclusions; 1. Thatandhave always dif- n q N q ^ ferent signs. 2. That the difference ~ — “< which may therefore always be represented by —£■ when d< 1. 3. That — — - is less, abstracting from its N qContinued Fractions. 279 sign, than — — ^; the former being equal to N Q 1 u' -, and the latter to -5—— --5-; that is, q{qu' + q°Y M N M N = —X 1, and q > q°, from the nature of these fractions; much more, of 1 then, is —5->-. Since, therefore, the difference A <1 between any converging fraction and the original is less than the difference between the preceding, one and the original, it follows, that the value of any fraction -approaches nearer to that of — than any one which precedes it. prop. v. 143. To convert the square root of any given number n (not a square) into a continued fraction, and thence to a series of converging fractions, ap- proximating towards the λ/ν. It is evident, in the first place, that this series must be infinite; because the square root of a num- ber not a square cannot be expressed by any ra- tional fraction (art. 18); but we shall find, that the quotient, whence the series of converging frac- tions are derived, will be periodical; and, therefore, the extraction niay be carried on at pleasure. The280 Continued Fractions. method of transformation, in this case, will be better shown by a partial, than by a general ex- ample; and we shall, therefore, first extract the square root of 19, and afterwards show the ap-* plication of the same method to the extraction of any quantity of the form ; n, m, and p, be- ing integers. Extraction of +19 'in Continued Fractions. . λ/19-4 3 1 +19 —4+----------^4+-—77-7-75=4 + +19 + 4 +19 + 4 Vl9±l=2+^L9-2=:2+ 5 ■■2 + · 3 1 Vl9 + 2 y!9 + 2 ~ 5 2 5 =1-1 5 1+ VI9 + 3 = 1+· V19+3’ 2 +19 + 3 ·?. _L λ/19-3_ 4_ 5 — Q _L . i 2 2 Ο 1 +19 + 3 T +/19+3’· 5 +19 + 3 — 1 4 λ/19-2_ •1 4- 3 —— "1 i_ , 1 5 5 ■ I 1 Vl9 + 2 +19+2’ 3 +19 + 2 =24 Vl9-4_ ;2+ 1 2 4- · 1 _ _ Λ λ/19 + 4 λ/19 + 4 V19 + 4=8+^19-4= 1 1 And here, since we have obtained the same ex-Continued Fractions. 28,1 pression as we began with, we may discontinue our extraction; as the quotients 4; 2, 1, 3, 1, 2, 8; 2, 1, 3, 1, 2, 8; 2, 1, &.C.; must necessarily recuy again in the same order, ad injimtum. Now if we substitute for the fractions λ/19 + 4 λ/19 + 2 λ/19 + 3 λ/19 + 2 3 3 &c,. their respective values, as found in the foregoing operation, we shall have V19 = 4 + - 1 λ/19 +4 1 =4 + - 2 + · λ/19- 4 ~~ 1 2 -)— 1 + ■ 1 — 4 + — 1 ________2 H—■ 1 λ/19 + 3 1+3+- 1 —r~ ■ 1+2+l~ , B &c. and, therefore, the square root of 19 has been transformed into a continued fraction, as was re- quired : and hence it is obvious that the same may be converted into a series of converging fractions, as in art. 136; thus, Quotients, 4, Conv. frac. 2? 1, 3, 1, 2» 8, 4 9 13 48 6ι l60 ? 2’ T’ ■ΐΡ 14’ 39 5 each of which fractions expresses the square root of 19 nearer than any preceding one, as is evident from art. 142; and it is manifest, that they may be continued at pleasure to any degree of accuracy required. The operation in this partial example is obvious:2 82 Continued Fractions. we first find the greatest integer contained in νΊ9, which is 4, whence λ/19 = 4 + λ/19-4 and this quantity being transformed to the follow- ing form, hy multiplying both numerator and de- nominator by Λ/1<) + 4, we have 3 1 λ/19-4 vl9 = 4 +--= 4 + V19 + 4 :4 + · a/19 + 4 We then proceed to find the greatest integer con- /I9 + 4 tained in becomes 3 which is 2; hence this fraction V19 + 4 λ/19-2 3 =2- λ/19 + 2 :2 + · 1 V19 + 2 And in the same manner we find the greatest in- teger contained in this last fraction, and so on, till we arrive at the fraction which being the same as the first, all the terms will again recur in the same order, ad infinitum·, and, consequently, the operation from that period may be discon- tinued. And it is obvious that the same prin- ciples may be applied to any quantity of the form λ/Ν + M P 144. The above operation, which is tedious ac- cording to the method that has been explained, and which was necessary in order to show the origin of the rule, becomes extremely simple, by observing theContinued Fractions. 283 following law in the formation of the successive quotient; viz. let yl $ + m n Λ/19 + mi n' = u+ Sec. =?«' + &c. represent any two consecutive fractions in the foregoing example, u and u' being their respective quotients, then will m' = nu — m, and , 19-Wi'2 n =---------■; n so that each value of m', n', and u', is deduced from those m, n, and u, in the preceding fraction: hence the foregoing operation, by means of this law, will stand thus: Vl9 + Q 1 λ/19 + 4 3 V19 + 2 5 = 4 + &c. = 2 -I* See. = 1 + &c. V19 + 3 2 — 3 + See. a/19 + 3 5 = 1 4- &c. &C. &C. 1.4 -.0 = 4; 3.2- 4 = 2; 5 . 1-2=3; 2.3- 3=3; 5.1-3 = 2; 19 ~4S 1 19 -22 3~ 19-3s : 0. 5 19-3* 2 19 -22 ' = 2. Where the calculations on the right hand of the line are set down only to explain the operation, but they are unnecessary when this is once under-284 Continued Fractions. stood; and hence the extraction of the square root by this method becomes very simple. 145. This law has at present only been deduced from observation^ but the universality of it may be demonstrated as follows: Let λ/ν + m λ/ν + ιη — ηιι ■-----— ==«H---------------. and η n λ/ν + ni' -----;-— = ll + &C. n be any two consecutive fractions derived from the λ/ν, n being any integer whatever not a complete square; then., from the nature of the operation, we must have λ/ν 4- m* n -----.—:==-------------—, or n vn — (nu — m) ( λ/ν + m') x { vn — (nu — m)} == nnf, and since this product is an integer, n and n' being each whole numbers, it follows, that m' = nu — my for otherwise the product of the two factors would not be rational; whence again ( λ/ν + mf) ( λ/ν — m') a= n — mn == nn*9 or , n — mn n =--------. n so that the law is universal. And hence the square root of any number N, not a complete square, may be extracted in the follow- ing manner, supposing a to be the greatest in- teger contained in λ/ν, and uf, u'", &c., the greatest integers contained in the respective frac- tions to which they correspond; viz.Continued Fractions, 28S Vn + o = a + &c. VN + nt n VN + m' rT VN+m' : U + &C. =«' + &C. — u" + &c. n" &c. &c. N — m~ 1 . a — 0 —m·, ;--—n. n . u — m —m ; n'. u' — m' = m"; n .w — m = m ; 1 N· 1 S. to n N - m"1 n' N ■ — m"'t =«w. And by continuing thus the extraction, we shall always arrive at a fraction equal to —-■ ; after which, the quotients will recur again in the same order, ad infinitum, as will be demonstrated in the following propositions. 146. Thus the extraction of λ/23 (omitting the calculations on the right hand side of the line, which are supplied very readily as we proceed) becomes, V23 + 0 Ϊ A/23 + 4 7 V23 +3 2 λ/2,3 + 3 7 = 4 4- &c. = 1 + &e. =3 ·+- &c. ^ 1 -f· &c. λ/23 + 4 λ/23 +4 = 8 + &C. — 1 + &C. And having thus arrived at a fraction equal to28 6 Continued Fractions. the second ——γ—-, the operation may be dis- continued, as the quotients after this recur in the same order as at first; and hence we may calculate the series of fractions converging towards V23 to any degree of accuracy required; thus: Quotients, 4; 1, 3, 1, 8; 1, 3, 1, 8; 1, &C* „ ' 4 5 19 24 211 Conv.frac. -, -, —, y, —, &c. Scholium. Numbers falling under any of the following forms, viz. P'±h f±P> °r/±-^p, have their square roots very readily extracted by continued fractions, the period of circulation never exceeding three terms: thus, for examples, -y/17 + O 1 λ/17 + 4 ϊ = 4 + &C. — 8 4- &G. which last quotient will be repeated, ad infinitum. V15+0 1 J\ 5 + 3 6 3 + &c. l + &c. Λ/15 +3 1 =6+ &c. the two last of which quotients will be repeated as before. And it is the same with all numbers falling under any of the above forms.Continued Fractions. 287 PROP. VI. 147· The series of quotients arising from the extraction of the square root of any number N, not a square, will be periodical. We have already seen, that this has been the case in the partial examples which we have given in the foregoing proposition; and it is here pro- posed to demonstrate, that this law must ne- cessarily have place for every possible value x, when it is not a square. p° p p' First, let us suppose to be any con- secutive fractions, converging towards the λ/ν; and let u°, u, be the corresponding quotients, u' being supposed the greatest integer contained in . . . VN + m the complete quotient n so that, in the following expressions, u°, u, u', p° V ν' r r ν' up + p( q' uq+q°* And if instead of u we take the complete quo- tient —N + m, whence u was derived, we shall n have, in the place of the foregoing equation, λ/ν + m 0 p--------+ f =---------------- (art. 139), n + m + q°«5 |*β 28S (Continued Fraction#; which becomes, by reduction, p λ/ν + pm τ p°n —- -----· ■ · q vn + qm + q°n ’ whence we draw the equation qm + λ/ν (qm + q°n) —p vn +pm + p°n; and since here we must have the rational part equal to the rational, and the irrational to the irrational/ we obtain the two following equations: qm —pm +p°n, p — qm + q°n. Multiply the first by q°, and the second by p°, gives qq°m=pq°m-tpYn, pp° =qp0m+p0q°n; then, by subtraction, qq°m — pp° = (pq° — qp°)in, and pp- Nqq = (pq° - qp°)n; this last being derived in a similar manner, by multiplying the first equations by q and p. Now, by the property of continued fractions (art. 140), we have pq°-qp°= + 1, if^> v-n; pq°-qp°=-l, if|< VN- Whence it appears^ that pq° — qp° has always the .cP pp same sign as pp — nqq; because, if - > vn, — > n ; q qq and, consequently, pp > mqq ; and the contrary, if < VN; and hence again it follows, that n is alwaysContinued Fractions. ■positive; because pp — ^qq = (pq° — qp°)n^ and pp — sqg, and (pq° — qp°), have always the same sign. And this furnishes tis with the means of ascertaining the limits of m and n; for, since , n — mn . «' =----— (art; 145), n ' arid n and n' are positive, it is evident, that m'2 < n, or m! < y'N; also, m' being an iriteger, it can never ex- ceed the greatest integer contained in yN. And since m = nu — m' (by the same article), or m-vrri — nu, arid m and rri are each < yN, it follows, that neither n, nor u, nor nu·, can be greater than m + m'·, and we have seen, that neither m nor m' can exceed a (supposing a to be the greatest in- teger contained in yN), therefore, neither n nor it can exceed 2a; or, which is the sarnie, 2a is the limit both of fn and the quotient u. And hence it appears, that in the transformation of yN into continued or converging fractions, which (from art; 145) has always the form VN + O 1 — a + &c. VN+m . . " =u + &c. n VN + m _ , ——u + &c. n yN + m n Sec. ==u" Λ- See. See. since m, n, arid u, caft never exceed certain iimits; that is, m not >a, n not >2a, and u not >2a: u'ύ$0' Continued Fractions. also, the expression itself being infinite, the same values of m and n must necessarily cotne together aft infinite number of times; and thus form a series of periodical quotients, which will Continue to be repeated ad infinitum, as we have seen in the partial examples in art. 143. — &. e. d. prop. vii. 148. In any series of quotients derived from vn', the second is that which first recurs, and com- mences the Second, and all the other periods of circulation; that is, the quotients always recur in the same order as at first, excepting only the first a, which expresses the greatest integer con- tained in λ/Ν. In order to demonstrate this (since we know that the quotients recur in periods), we shall suppose the first period to be a; tt, β, y, δ, &c. - — %,u, u', u", See.; and - - - - - - -· - tv, tt, u', u", &e., part of the second period ; apd then prove, that λ = iv, the quotient preceding λ = that preceding w, and so on to a; which must, therefore, ne- cessarily be that quotient which commences each of the periods. Let, then, fa; α, β, y,- - - λ, %, u', u”, See. w, u, u’, u"; I £ &C - - - £p- - - - - -· v. l’ ' q°’ g’ a0’ a’ represent any series of quotients, and their corre- sponding converging fractions; also, .letContinued Fractions. 29 i aA + a aA + m n° VN + m n a/N + MI a/n + m be the corresponding complete quotients. Then, from what has been demonstrated (art. 145), we have n -nf = nn°, and N-?n2 = nn; whence n° — n; and we shall also have (by the same article) m ± Xrf — m°, and m==wn — ni; Whence we dratv rtf - m n — λ — w. But (art. 1.47) 0 P <]°n , . p . qm + q n —p, or m — —-----------; and since — is an ap- q q q ** proximate value of «/x, we must have - == a + & <1 7* fraction — (d being, as above the greatest integer in a/n), and hence result a — m- q 11 - And since if < q, froni the nutate of continued fractions, we shall hate a — rri the second of which gives by division (u-a). + q q whence again it follows, that u — a is the greatest p integer contained in -; but as this fraction is an q approximation towards vn, the greatest integer contained in it is a: we have, therefore, u — a = a, or a = 2a·, that is, the last quotient in the period = 2a. — a. e. d. PROP. IX. 150. The equation p1 — n<£2 = 1 is always possible in integers, if n be any integer number whatever not a square. For, by the foregoing proposition, the complete quotient answering to the last quotient in any . _ VN + W D · . Vi, period, as——- = u.+ &c., is such, that u=2a (a being the greatest integer contained in λ/ν) ; and, consequently, as we have seen, n= 1, becauseContinue If, now, we represent by and -, two converg- ing fractions, the latter corresponding to the quo- tient 2a, we have also (by art. 147) ]i> - Tsf = (p(f -jp(j)n; but, in the present case, n= 1, andpj®— p°q— ± l.i by the property of continued fractions; therefore, p4 — N (f = ± 1; the upper sign having place when — > λ/ν, and V the lower one when ~ < λ/ν. But all the converging fractions in the evei\ places are > vn, and all those in the odd places < v'N^ as is evident, because they are alternately greater and less than vn, and the first is always less than vn ; hut since these periods of quotients recur ad infinitum, if the first fraction answering to the quotient 2a, be not in an even place* it must necessarily be so when that quotient recurs again; and, consequently, the equation jf — n q2 = 1 is always possible, n being any integer number not a square; and there are an infinite number of values, that maybe given to p and q, which answer the conditions of the equation; viz. every fraction P - standing in an even place, and corresponding to the quotient 2a. — a. e. d.Continued Fractions. Gor. 1. It appears, from the foregoing proposi- sition, that the equation /-N,/= -1 is also possible in all cases where the quotient 2d occurs first in an odtl place, and that there are likewise an infinite number of values that may he given to p and q, which will answer the required conditions; but if 2a occur first in an even place, then the equation ’ is impossible. Cor. 2. Hence also the indeterminate equation o o e x~ — ay~==,& is always possible in integers; for the equation x* — aip — 1, gives this equation, therefore, is always solvible in in- tegers ; which has in fact been otherwise demon- strated in Part l. J51. It will not be amiss to illustrate what has been demonstrated in the foregoing propositions by a few examples: Ex. 1. Find the values of x and y in the equa- tion a?*-15 y*=i. Here we have, by the conversion of ^15, V15 + 0 1 -v/15 + 3 = 3 + &c. 6 V15 + 3 = 1 + &c. =6+ &c. 1 .3-0 = 3; -15 32 =6*. 1 „ 15 - 32 6.1 -3=3; ---7t— = 1. 6 1.6-3=3; &c.Continued Fractions. m Quotients, 3, 1, 6, 1, 6, &c. Fractions, —, &c. Now the first fraction y, which answers to the quotient 2a, is in an even place; we have, therefore, x = 4, and y = l, which gives 4a— 15. 1- Ex. 2. Find the values of x and y in the equa- tion First, V17 + Q. 1 V17 + 4 if — l Ty- = j. λ/17-43 = 4+&c. 1.4-0 = 4; -------= 1. λ/17 + 4 o V17-43 = 8+&c. 8-4 = 4; * 1 -=!, = 8 + &c. 1 .8 — 4 = 4; &c. Quotients, 4, 8, 8, 8, 8, &c. . 4 33 .fc ructions^ ■ 0 · 1 o And here, the first fraction corresponding to 8 f>eing in an odd place, we employ the second, which gives „r=33 and y—8, whence 332-17·83= 1.m Continued Fractions. Ex. 3. Find the values of x and y in the equa- tion x1~’ 13y* = 1. First, V13 + 0 1 a/1 3 + 3 ■■3 + &c. : 1 + &c. λ/13 +1 , — = 1 + &c. 3 λ/13 + 2 ~3 ' λ/13 + 1 4 λ/13 + 3 ί ' = 1 + &C. • 1 "1“ &c. • 6+ &c. ■13-3* 1.3-0=3; -------= 4. 1 13-l2 4,1-3 = 1; --— =3. 3.1 - 1 = 2; 3.1-2 = 1; .1-1=3; 4 13-22 3 13-1* 3 13 — 32 4 = 3. = 4- = 1„ 1.6-3 = 3; &.C. which last gives the quotient 2a, or 2.3; we have, therefore, for Quotients,3,1,1, 1, 1, 6; 1, 1, 1, 1,6; 1, &c. . 3 4 7 11 18 11.9 137 256 393 649 Jj Factions, , ,1 5 , , , , ■, , · * 1 1 2 3 5 33 38 71 109 180 Now here again the first fraction, answering to the quotient 6, being in an odd place, we proceed till we meet with 6 a second time, which will ne- cessarily be in an even place; and the fraction 649 corresponding to it is —so that x = 649 and y = 180 are the least values of x and y, that answer the conditions of the equation ar — 1 8 ?/ = 1. If the proposed equation had been x°~ 13/= - 1,Continued Fractions. 299 we should have had, a· = 18 and y = 5, for the least values of a? and y that satisfy this equation. But if the first fraction answering to the quotient 2a be not found in an odd place when it first occurs, then it follows, from what has been demonstrated, that the equation ar — ny2 — l is impossible, as we have before observed. 152. Scholium. The solution of the equation = 1 is one of the most important problems in the in- determinate analysis, it being necessary to the solution of many other interesting questions of this kind; and, notwithstanding the method we lrave given is direct and simple, yet the least values of X and y in many cases being very great, the task of finding them is very laborious: thus the least values of x and y that solve the equation x1 — 2\ \y‘ — 1, are x= 278354373650, and 19l62f05353, And the equation xq — 56581/ =s= 1 has the least values pf x and y as follows; viz. _ f 16610072525797731839820799846220132 x~ i 4702014613503. _ f 6982536164167704S7157775940222021OO y~ { 2391003072. These circumstances have induced a few cele- brated mathematicians to form tables of the values »f x and y, necessary for the solution of the equa- "lonf — n/=1.300 Continued > Fractions. Euler first undertook this task, for all values of N from 1 to 100, which was afterwards doubled by Lagrange, both of which tables are given in the second volume of Euler’s Algebra. But Legendre has extended the same to upwards of 1000, at least for the solution of the equation a;2 — N?/2 s= + 1; and he has shown the method of deducing from them the solution of every possible equation a;2 —n«/2= + a, whether the numbers in his table give x1 — n^*=+1, or a;2 —· xy2 = — l: a part of this table is subjoined to the present work, which will be found useful in many cases.—■ See Table II. PROP. X. e P 153. Given the difference between- and λ/ν; viz. % p δ '---λ/Ν = —, ? '/ δ being less than unity, to find the necessary P conditions for the value of δ, that ^ may be a fraction arising from the extraction of ^n. Let the given fraction ^ be converted into a series pf converging fractions, giving the Quotients, a, b, c, - — u. a ab+1 P° P Conv. frac. bContinued Fractions. 301 p Now if ^ be a fraction converging towards vn, it follows, that all the quotients a, b, c, &c. are likewise obtained from the same development; and, consequently, that the quotient u is followed by others, u', u", u'", &c. Let now the com* plete quotient, answering to the fraction be Λ/Ν + m then we have, on the same principles as in art. 1 A’J, p- y"N = — 9 VN + m n λ/ν + m n + P° + 9 O Or, by substituting for the complete quotient a/n + m the above expression will become λ/Ν = PfX+p° 9^ + 9°’ Whence, by substituting for vn, we have ... P_P°9-P9° _ l 9 9{9^ + 9°) 9(9Ρ + 9Ύ s which last expression must be equal to —; that is, ±-____I___ - or g - tf q^w+q0)’ qp+q0' Now since μ. is the complete quotient corre302 Continued Fractions. P m Bponding with the fraction it must be positive, and greater than Unity, and, therefore, S < and hence, conversely, if > h the value of jx must necessarily be positive, and greater than unity j and, consequently, ^ will, in this case, be a frac- tion converging towards vN· That is, if - be any fraction, and the difference P 9 λ/Ν: and δ < 9 + 9 , v then is — a fraction, which arises in the develop· ment of into converging fractions. Which is the condition required to be found. PROP. XI. 154. If the indeterminate equation ar — n?/2 = ± a, be possible (a being < ^/n), a must be found in the denominator of one of the complete quotients, arising from the development of vn. It appears from art. 147, that when a is found in the denominator of any complete quotient, as! tftat is, when n — A, and be two convergeContinued Fractions. 3 03 frig fractions, the latter corresponding with this complete quotient, we shall have pi — aq!>:!=A(pq0—p0q); or, since pq°—p°q= ±1, we obtain , p— ytq = + A. And it is here proposed to demonstrate, that this equation can only have place when a is thus found in the denominator of one of the complete quo- tients, derived from v/n, a being always sup- posed less than ,/N. Now, first, from the equation /r—= ± a, we obtain + A p-q = -or P ~Vq λ/ν P ± A q . + n) And, if we represent, as in the foregoing article, p , % . ---VN by —, we have q J q·’ δ +Α . ±A q — = ------------ or o=-------- T 9\P + ? Vn) p + q v*. p Let now ^-be the converging fraction preceding P : ·, in the series of fractions arising from the develop- V ment of then, by the preceding article, we have to prove that304 Continued Fractions< P + q νκ q + q0 > p for in that case it necessarily follows, that - is a 9. fraction arising from the development of vn (art. 153). Now, since-— vn = -v, we have q q*> * · ·; p — q -ί/N + -; again, ir Aq q . . ----i— < —^ so is also P + q vn q + q A(q + q6)<(p+q w), or, substituting for p, it becomes δ A(q + q°) < (2q VN + -). Now this inequality is readily demonstrated; for it may be put under the form δ 2q vn- a(^ + q°)+~> 0, {q + q°){ VN-a) + (q-q°) ^* + ->0; H and since λ/ν > a, and q > q°, the whole of this expression is positive; and, therefore, >0, at least when δ is positive; and if δ were negative, we should have evidently 8 (q-q°) VN>-; and, therefore, in either case the inequality is established; that is,Continued Fractions. SOS xq ,0 * p + q V N q + q' and, consequently, ^ is found among the fractions converging towards the VN. Therefore, when it is required to find the values of x and y in the equation x2 — ny* == + A, a being < λ/ν, we must convert n into a continued fraction, by the forms given in art. 145; and if A be found in the denominator of any one of the com- plete quotients obtained by this development, we shall have the solution sought, by finding the con- verging fraction answering to this quotient, which solution will give x‘ ■ ay*= a, or Λ* — nyt = — a, according as the fraction is found in an even or odd place; and if a be not found in an odd place, the latter equation is impossible; attd if a be not found in the denominator of any of these complete quo- tients, we may be assured, that the proposed equa- tion is impossible under either sign. Ex. 1. Find the values of x and y in the equa- tion ’ Xk-23y*=2. First, by the development of ^23, We have ^23 + 0 1 λ/23 + 4 7 S23 + 3· ■ss4 + &C. = 1 + &c. = 3 + & c. X306 Continued Fractions. and in this last fraction, 2 being found in the de- nominator, we have Quotients, 4, 1, 3, &c. Fractions, —, —, &c. 11 the last of which, answering to the quotient 3, gives a? = 5, and?/ = l; so that ar-23/ = 2, as was required. Ex. 2. Required the possibility or impossibility of the equations .r — 1“/— 3, a;'—17/= 2, a?' — 17/— —3, a;2 —17/ = - 2. , by the development of Vl7> we have .-^ = 4+ fa. _ν07 + 4 1 &c. —- S "f· &c. &c. whence it follows, that since neither 2 nor 3 enter* into the denominator of the complete quotients, the equations are all impossible. Cor. 1. The indeterminate equation ar- (or+ !)/= + a . is always impossible, if a > 1 and < */(a?+ l); be- cause the complete quotients arising from Λ/(α2 + 1) have only unity enter for a denominator: we must of course except those cases also in which a is aContinued Fractions. 307 Complete square, as these will always be possible' from the equation a?2 — ny~ = 1. For, by the forms art. 145, we have - V(a2 + l) + 0 . sj, o2 +1) +a ϊ = 20 + &C. which last complete quotient will be repeated to infinity. Cor. 2. The indeterminate equation Χ2-(θ2-l)/= ± A is also impossible; under the same limitations, be- cause, by art. 145, we have ^/(e4— 1) + 0 - J = (a—l) + &c. yjd1- 1) + (a- I) 2 a —2 y(oa- 1) + (a- l) 1 = I + &c. -2{a— 1) + &c. arid these two last complete quotients will be re- peated ad infinitum; and, consequently, only 1 and 2 (a — 1) will ever be found in the denominators of them. Cor. 3. The indeterminate equation af — {a1 + a)yi = + a is always impossible, if a > 1 and < o, excepting, as before, those cases in which a is a complete square. For30& Continued Fractions* */{a%+ n) + a a 2 + &c. <\/(a^ + a) 4* a ' 1 . = 2a+ &c. The two last of which fractions will continually recur; and, consequently, the equation is always impossible under the above limitations. Cor. 4. The indeterminate equation x1— («* — a)y* = + a. is always impossible, if a > 1 and < (a — l), except the cases in which A is a complete square. For V(e? —«) + 0 ΐ (—l=y. x b xb Let, then, p and q be the terms of the fraction preceding ^; then, if aq — bp= — 1, we may convert it into +1, by making x — hm — q, and y — am—pj which evidently gives a(bm — q) — b(pm —p) = + 1. And, on the contrary, if aq — bp— + 1, it may be converted to —1, by a similar substitution; and it is evident, that, by means of the indeterminate letter m, an indefinite nixmber of solutions may be obtained in both cases; and when we require no change in the sign, then we have x — bm + q, and y = am +p. Ex. 1. Find the values of x arid y in the equa- tion lbx—l*Jy=l. First, by the rule for continued fractions, 15)17(1 2)15(7 1)2(2' Ouotienfs, Conv. firac. 1 8 17 1’ r 15 ’ whence p — 8, and q — 7, which give 15y-l7g.= +li319 of the First Degree. therefore, the general values of x and y are x=\>jm + 8, andi/ = 15m + 7; and by assuming m — 0, 1, 2, 3, &c., we shall have, for the corresponding values of x and y, as follows: « = 8, 25, 42, 59, 7'6, 93, 110, &c. y=7, 22, 37, 52, 67, 82, 97, &c. Ex. 2. Find the general values of x and y in the indeterminate equation 13A?.-9y = l. Here, by the rule for continued fractions, 9)13(1 4)9(2 1)4(4 Quotients, 1, 2, 4. Conv. frac. —, —, 13 ΊΓ whence p=3, and q = 2, which gives I3q-9p= - l; and, therefore, the general values of x and y are x — 9m — 2, and y — \3m — 3·, and assuming m= 1, 2, 3, 4, &c., we have the corresponding values of x and y, as follows: x= 7? l6> 25, 34, 43, 52, 6l, &c. ' y= 10, 23, 36, 49, 62, 75, 88, &c.' These two examples, with what has been before done in the preceding chapter, will be sufficient to render the student ready in the solution of any equation of the above form, which is the more ne-Indeterminate Problem^ cess ary, as we shall see that every indeterminate equation of the first degree, which has any possible solution, depends upon the solution of the equation ax—by=z±\. prop. ir. lfio. To find the general values of x and y in the equation ax — by = + c. First, with regard to the limits of possibility of this equation, it may be observed, that a and h must be prime to each other, or, if they have a common divisor, c must have the same, for others wise the equation is impossible; and if each of these quantities have a common divisor, the whole equation may be divided by it, and thus reduced to another, in which a and δ are prime among them- selves ; for, if this cannot be effected, the equation cannot obtain in integers. Supposing, then, a and b to be prime to each other, and q and j> the least numbers that fulfil the conditions of the equation aq — bp— + 1, determined by the foregoing proposition, .then it is evident-that we shall have a. cq — b. cp=±c', making; therefore, x = cq, and y = cp, we shall have the solution required: but it is obvious that the same result will be obtained by writing x = mb + cq. and y => ma + cp,which gives also ‘ a{mb + cq) — b(ma + cp)= ±C; where, by means of the indeterminate m, an in-321 bf the First Degree. definite number of values of X arid y may be obtained. And we may always convert the value of the equa- tion from + c to —c, or from — to +, by taking cp and cq negative, and, in this case, tn positive, iri order that x and y may be so; for; if a. cq — b ,cp = +c, then a(mb — cq) — b(ma — cp) = — c; arid if a. cq — b . cp = — c, then will a(mb — cq) — b(ma — cp) = + c. So that the general values of x and y are; x — mb± cq, and y = ma + cp, the upper sigri having place for cq and cp, when the expression aq — bp has the Same sign with c in the given equation; and the lower one when it has a different sign. Ex. 1. Find the values of x and y in the in- determinate equation 9#- 13//= 10. First, iri the equation 9q— 13p= + 1, We have q — 3 and p — 2, which gives +1, the same sign as 10 in the proposed equation ; arid, therefore, the general values of X and y are, i=i3m + 30, arid ?/= cp» + 20· Therefore, assuming successively m— —2; —1, Ο; 1; 2, 3, 4, &£., we have the following corresponding values of a? and y, which are all deduced from the first two, b^ adding successively to the values of x the co- efficient of y, and to y the coefficient of x. γIndeterminate ProbteniS 3,22 *=4, 17, 30} 43, 56, 69, 82, &C„ .y=2, II, 20, 29, 38, 47, 56, &C. Whence we obtain the following solutions i 9 . 4-13. 2 = 10, 9.17-13.11 = 10, 9.30-13.20 = 10, 9.43-13.29=10, &c. &c. Ex. 2. Find the values of x and y in the in~ determinate equation 7*-12*/ =19. First, the equation 7q-l2p=.±l, gives q — 5 and p = 3, from which is derived' 7 ϊ 1 “P ~ 1, which is a different sign from 19 in the given equa- tion ; therefore, the general values of x and y are, x = \2m — 5.19, andy — fm — 3.19; or x=12m — 95, and y==fm — 57. Whence we obtain the corresponding values of ,t and */, by assuming m = 9, 10, 11, &c.; and it is obvious, that we cannot take m < 9, be*- cause we should then have x and y negative; and these values of x and y are deduced from each other, as in the foregoing example, by simple addition. *=13, 25, 37, 49, 6l, 73, .85, &c. y- 6, 13, 20, 27, 34, 41, 48, &C.bf the First Degree. 323 PROP. III. 46l. To find the general values of x and y in ihe equation ax 4- by = c, and to ascertain the number of possible solutions, that the equation admits of in integers. In the foregoing proposition, where; the dif- ference of two quantities was the subject of investi- gation, we found, that the number of solutions was infinite, providing a and b were prime to each other; but when we consider the Sum of the two quantities, as in the present case, the number of solutions is always limited, and in many cases the equation is impossible; we have, however, de- monstrated (art. 4l), that this equation will al- ways admit of at least one solution, if a and b be prime to each other, and c > ab — (a + l·); and it is proposed, in the present proposition, to ascertain the exact number of solutions when the equation is possible, and to point out more accurately the limits of possibility. The solution of the indeterminate equation ax + by~c depends, like that in the foregoing proposition, upon the equation aq —bp — ± I, though its connexion with it is not so readily per- ceived. For let p and q be the terms of the con- verging fraction, immediately preceding thei* we shall always have either aq — bp~ 1, or bp·*- aq = 15324 Indeterminate Problems and, in this case, it is indifferent which of the two terms is the leading one* because we are only con- sidering the sum ax -f by — c. Let, then, aq — bp~ 1, then we have also a.cq — b.cp~c; and it is evident, that we shall have the same result if we make x = cq — mb, and y^cp — ma^ for this still gives a(cq — mb) — b(cp — ma) — c: assuming, therefore, for m such a value, that cp — ma may become negative, while cq — mb re- mains positive, we shall have a(cq — mb) + h(ma — cp) — c; and, consequently, os—cq — mb, and y~ma — cp? but if m cannot be so taken that cp — ma shall be negative, while cq — mb remains positive, it is a proof that the proposed equation is impossible in integers. And, on the contrary, the equations will always admit of as many solutions in whole numbers, as there may be different values given to m, such that the above conditions may obtain. And hence we are enabled to determine, a priori, the number of solutions, that any proposed equa- tion of the above form will admit of ; for, since we must have cq > mb, and cp < ma, the number of solutions will always be expressed by the greatest integer contained in , cq cpο/’ the First Degree. 325 as is evident, because m must be less than the first of those fractions, and greater than the second; and, therefore, the difference between the in- tegral part of them will express the number of CO different values of m, except when ~ is a com- plete integer, in which case, as m< we must take the next less integer; or, which is the same, we must consider γ as a fraction in this case, and b reject it; but this must not be done with the other quantity, because m > Ex. 1. Required the values of x and y in the equation 9a? + 13?/ = 2000, and the number of possible solutions in integers. First, in the equation 9?-13>=l, we have at once 7?= 2, and q — 3; therefore, the number of solutions will be 2000 x 3 2000 X 2 ^ -^-—^- = 461-444 = 17. Which are-i’eadily obtained from the formulae x —cq- mb, and y — ma — cp; or X = 6000 - 13 m, and y = pm — 4000; in which, assuming- m = 445, in order that Qm > 4000, we shall have the following solutions, each of which is deduced from the preceding one.826 Indeterminate Problems by adding successively 9 for the values of y, ao4 subtracting 13 for those of #. # = 215, 202, 189, 176, 163, 150, 137, &c. y=T 5, 14, 23, 32, 41, 5Q, 59, &C, That ig, 9.215 + 13. 5=2000, 9.202 + 13,14 = 2000, 9.189+13.23 = 2000, &c. &c. &c. Ex. 2. Let there be proposed the equation 11#+13#= 190 to find the number of solutions, and the values of x and y. First, in the equation 11 q-13p = l, we have q — 6, and p—b·, therefore, 190.6 13 190.5 Π 87-86= 1: whence it follows, that there is only one possible solution, which we readily obtain frorr, the formulie x=cq — nib, and y — ma — cp; or #=190.6— 13m, and# = llw—190.5 : where, by taking m — 87, in order that 11 m > 190.5, we have # = 9, and y = 7, which gives 11.9+13.7 = 190, as was required. Ex. 3. How many different ways may IOOOZ. be paid in crowns and guineas ? Putting # for the guineas and y for the crowns, and reducing IOOO/. to shillings, we have32f of the First Degree. 2lx +5y-20000; and it is required to determine the number of $o«* lutions that this equation admits of in integers, For this-purpose we have the equation 21q-5p~l, which gives q = 1 , and p — 4; then we have cq cp __ 20000 80000 h a 5 21 190: that is, 1000 (d—cz)p;329 of the First Degree. by attending to which, all the possible values of x, y, and 2, may be obtained: but as these questions generally admit of a great number of solutions, the object of inquiry is not so much to find the solu- tions themselves, as to determine, a ■priori, the number that the equation admits of in integers. JNow we have seen (art. l6l), that in the equation ax + by =c, the number of solutions is generally expressed by the formula cq cp b a’ g and p being first determined by the equation aq — bp=±l. V, therefore, in the equation ax + by = d — c%. we make successively s=l, 2, 3, 4, &c., the number of solutions for each value of 2 will be as below; viz. ax Λ by— d— c, num, of solu. ax Λ-by = d—2c, - - - αχΛ- by — d—3c, - - - - (d— c)q (d— c)p b a ■■ id- - 2c)q id- -2c)p b a ' id- -3c)q id- - 3 c)p b β ' &c. &c. The sum of which will be the total number that the given equation admits of; and, therefore, in order to find the exact number of solutions in any equation of this kind, we must first ascertain the330 Indeterminate Problems sum of all the integral parts of the arithmetical series, . , (d-c)q id— 2c) q (d-3c)q , ld-4c)q , b + b + fe-+&c-an« {d-c)p (d-2c)p (d-3c)p (d-4c)p ——:----- -i----r— i------^ -r *——---r- occ.: a a a a and the difference of the two will be the exact number of integral solutions. Now, in both these series* we know the first and last term* and number bf terms; for, the ge- neral terms being (d—cz)q. b~~^ and (d— c%)p. a A we shall have the extreme terms by taking the ex- treme limits of z; that is, % = 1 and % < -; which last value of % also expresses the number of terms, in the series. Hence, then, having the elements of the pro- gression given, we readily find the sums of the two whole series; and if, therefore, we also find the sums of the fractional part of the terms in each, we shall have, by deducting it from the whole sum, that of the integral part of the series, as required. The latter part of this problem is readily effected; for, the denominator in each term being constant, the fractions will necessarily recur in periods, and the number in each can never exceed the denomi- nator: it will, therefore, only he necessary to find the sum of the fractions in one period, which, being multiplied by the number of periods, will give the sum of the fractional part of the terms, and these*331 of the First Degree. taken from tlie total sum, will give the sum of the Integral part of the series; then, from what has been before observed, the difference of the two sums will be the number of solutions required. It should also be observed, that, when the number of terms does not consist of an exact number of periods of circulation, the remaining terms, or fractions, must be summed by themselves, which is also readily effected, as they will be the same as the leading terms of the first period: and it must also h . be remembered, that ^ is to be considered as a fraction in the first series, but not - in the second, ■ a ' g,s is explained at page 325. Ex. 1. Let there be proposed the equation + llz = 224 to find the number of solutions which it admits of in integers. 224 Here the greatest limit of *< — is 20; also in the equation bq-Tp^l we have q = 3, p^27 a = 5, and b~r*J;· ami, there- fore, the two series of which the sums are required, beginning with the least terms, will be 3.4 3.15 3.26 3.37 0 3.113 ■1st, —-^ + —+ —+ &c. ——. 7 7 1 7 i Λ 2.4 2.15 2.26 2.37 , 2 A13 ' 2d, —- + -T— + —— + —— + &c. —7 * The common difference in the first being 3.11 ~T~’ and332 Indeterminate Problems in the second —, and the number of terms in 5 each 20; whence we have 930, for the sum of the first, and 868, for the sum of the second. Also the first period of fractions, in the first series, is 5 3 1 6 4 2 7 ^nd, in the second series, the first period of frac- tions is, 3 2 4 1 —b 0 -f — -f — —=2; 5 5 5 5 7 - being considered as a fraction in the first 5 (art. 162), but not - in the second. ί) Now the number of terms in each series being 2% we have 2 periods and 6 terms of the first series, = 2.4 -f the first 6 fractions =11, for the *sum of all the fractions; and, therefore, 930—11=919, which is the exact sum of the integral terms, first series. And, in the second, we have 4 periods, = 4.2 = 8: and, therefore, 868 — 8 = 860, the sum of the integral terms of the second series: and hence, according to the rule, 919-869=59 is the number of integral solutions. Remark. This example is the same as prob. 11, page 191, Simpson’s Algebra, where the number of solutions is said to be sixty ; but, upon examination, it appears, that one of the333 of the First Degree. sixty wliicb he has given cannot obtain; that is, £ = 14, ,r = 10, and y —14; which error being corrected, gives 59 for the number of solutions, as above. Ex. 2. Having proposed the equation 7.r +9^ + 232 = 9999, it is required to determine the number of its solu- tions in positive integers. 9999 Here the greatest limit of z <-----= 434; also b 23 in the equation 7q-9p=i, we have q — 4 andp = 3, a = 7 and £ = 9; also 9999-23.434 = 17: therefore, the series whose sums are required are as follow; viz. , . 4.17 , 4.40 , 4.63 , ; 4.9976 1st, — 1 -----1-----(- &C. 9 9 . . 3.17 3.40 3.63 0 2d, ———H------1 —^-+ &c. 9 3.9976 The common difference in the first being 4.23 2 -----=10-; 9 9 and, in the second, 3.23 6 7 ~97; also the number of terms in each-, 434, that being the greatest limit of z. Hence we have the sum of, 3 First series, —963769-» Second series, =929349·334 Indeterminate Problems Also the first period of fractions in the fi series, 5 7 9 2 4 6 8 1 3 -+-+-+-+-+-+-+-+-=5: 9999999 99 ■. 434 2 . , anu —— = 48-, or 48 periods and two terms 9 9 5^3 = 5.48+- + -=241-.· 9 9 9 And, in the second series, the first period off fractions will be 6 4 -3 - + - +O+- + - + - +- = 3;· 7 7 7 7 7 7 and 434 -■62, or 62 periods: therefore, 62.3-1S6. Hence, „ ... 3 3 ■ f integral terms, first se- 9 9 l ries; and f integral terms, second 929349 -186 =9291631 g^ieg_ Whence the 1 >=34365,- dinerence, J is the number of integral solutions required. 163. Cor. We have at present only considered the ease in which two, at least, of the given coefficients are prime to each other; and, when this is not the case, the following transformation will he requisite* and which will be better explained by a partial than by a general example. Let it therefore be proposed to find the number of solutions that the equation33 3 of the First Degree* 12.H+15^+ 202 = 100001 admits of in positive integers. By transposing 20s, and dividing by 3, we have 2—1 Ax + by = 33334 - 7z + —; ^ Λ ““ i and as this last must be an integer* put —-— == or £ — 3^4- 1, and this being substituted for 27 the above equation becomes 1 Qx 4 1 by 4 20(3u 4 1) = 100001 ; 017 dividing by 3, and transposing Ax + by + 20 u = 33327, the number of solutions in which will be the same as in.the original one, but in this u may become 0, as we shall in that case have % = 1. Now here, the greatest limit of u < 33327 20 l666, and the equation 5q — 4p = 1 gives q— 1 and p = 1; whence the series will each consist of 1667 terms, because u may =0, and their sums will be 7 27 47 67 n 1st, H—"“4·.-—+·'—+ &c. 4 4 4 4 7 27 47 67 5+T + T+T+ &c· 33327 _ 1 - = 6945972-; 4 4 33307 „ 4 —--= 55S6777-; o a and the fractions of the first series will be 3 3. 3 0 - + - + -+ &c.; 4 4 4336 Indeterminate Problems where, each term having the same fraction, th# sum will be 3 .........l -x 1667= 1250-. 4 4 In the second series the fractions are 2 2 2 . 4+i+i+&c· the sum being fx 1667 = 666^ 5 5" and hence we have 6945972-- 1250^ = 6944722, 4 4 555G777~-666 j=5556lllo 5 5 Whence the total number of solutions, } 1388611. When there are four or more unknown quan- tities, the number of possible solutions is found in a similar manner. prop’. V. 164. Having given any number of equation^ less than the number of unknown quantities which enter therein, to determine those quantities. Let there be proposed the two equations ax + by + c z = d, a'x + h'y + c’% — d'\ to find the values of x, y, and z. Multiply the first by a' and the second by a, whence, by subtraction, we obtainof the First Degree. 33 f (a'b — ab')y + (a'c — ac')z — a'd— d'a; Or, dividing each of these known coefficients by its greatest common divisor, if they havei any, and representing the results by b", c", d", this equation becomes br>y + c'}z = df'. Find now the Values of y and % in this equation, and these being substituted foil· them, in the equa- tion d—cz — by x- a will give the corresponding values of X; of which those that are fractional must of course be rejected,· and also those that render (cz + by) >d. Ex. 1. Giving { 3x + by + 7* = 56o, 9# + 2by + 492 = 2920, to find all the integral values of x, y, and a. Multiplying the first by 3, we have f 9a? + l5#+ 21a =1680, l 9# + 2by + 49a = 2920, whence, by subtraction, we have lOy + 28z= 1240; or, %+14a = 620: and here the values of y and z are found to be, ?/= 110, 96, 82, 68, 54, 40, 26, 12; *= 5, 10, 15, 20, 25, 30, 35, 40. And of these the" only two that give 560 — 7z—.by x=- an integer, are the following; viz.338 Indeterminate Problems ( z— 15, andi/ = 82; whence x = 1 \ z = 30, and y = 40; whence x = 50. Remark. Though this method of solution never fails of giving all the possible values of x,- y, and z, in equations of this kind, yet we may frequently shorten the operation in particular cases, in the following manner: Having obtained, as above, the equation 5 y+ 14a — 620, we have, by division, y=124 + 3*-|, so that z must be a multiple of 5; make, then, z — 5m, and we obtain y= 124 — 14m; which values of y and z, substituted in the first equation, give 3x —35#= —60, or 3x — 35u — 60, whence u must be divisible by 3; take, therefore, u — 3t, and we obtain x = 35t — 20, y= 124 — 42#, ands=15#: the value of y limits t not to exceed 2; assuming, therefore, t— 1 and 2, we have exactly the same solution as above. proK VI. 165.' To decompose a given numeral fraction having a composite denominator into a number of simple fractions having prime denominators. This is in tact only an application of the fore- going propositions to this particular case; for letof the First Degree: 339 in * — be the given fraction, and suppose, in the first lih instance, that its denominator consists of two prime factors^ or n — ab, it will then be to find m p a —?==- + y> or ab a b aq + bp = m; in which equatiori, having determined the values of p q p and q, we shall have — and 4 for tile fractions re- r 1 a o quired; and as many different ways may any such fraction be decomposed into others, as the above equation admits of integral solutions. 7ΪΙ If the given fraction be then we may first resolve it into two fractions, and one of these into two others; thus, let m p q abc ~ ab c3 then we have abq-y cp—m·, and having, from this equation, found the values of p and q, we shall have Again, let whence we obtain m __p q abc ab c° p r s ab ά + V as + rb=p·, Z 2340 Indeterminate Problems find r and s in this equation, so shall we have m _r s q abc a b c’ as required. And we may proceed in the same manner to decompose any given fraction having a composite denominator. Ex. 1. Find two fractions, the denominators of 19 which are 5 and 7, whose sum is equal to ♦j 0 Make 1 ?_!9 7 5 35’ which furnishes the equation 5^ + 7? = 19; and here we have p = 1 and q = 2: therefore, — and — are the fractions required; for these give 1 2 = 19 7 + 5 35‘ Ex. 2. Find three fractions, the sum of which shall be equal to . 1 315 Having first found the denominator to be equal to the product of the three factors 5.7 · 9* it follows immediately, that these three numbers must be the denominators of the fractions sought; and if the given fraction cannot be decomposed into three fractions, having these denominators, it is in vain to seek the decomposition in any others. Supposing, then, in the first place,of the First Degree. P y_401 35 + 9 ”"315* we have 9/> + 35§ = 401; which give -l3ql=l, which, by art. 151, -are found to be p — 649 and q— 180, whence x = =? 7.180a = 2268ΟΟ, which is the least integer having the required con- ditions. 169. Case 2. To find the values of x in thq indeterminate equation ' mixt + bx-\- c=3s.348 indeterminate Problems Assume z = mx +y, then we have mV + bx + c s= mV + 2 mxy + y~. Whence, by cancelling nvx1, τ= %f~c #.v i » b — 2my P Or if, in order to generalize, we make y this becomes P ~C(l b(f — 2 mpq ' In which expression p and q may be assumed at pleasure. Ex. 1. Required the value of x in the equation 9λ·2 + 7« + 5=22. Here, since m = 3, b = f, c = &, the values of X are contained in the expression _ *'■ 7 % 5 in which, by assuming p == 4 and q = 2, we have 4 1. a?=—, or which fraction answers the condition of the equation. Ex. 2. Required the value of x in the equation <)x* + 5=z\ Here m — 3, b — 0, c = 5; therefore, the general value of x is x —6pg5 or x 5q*-f 6pq By taking q~ 1 and p—\ we have x- , whichof the Second Degree. 349 fraction will be found to answer the required con- ditions. 170. Case 3. To find the value of x in the indeterminate equation ax2 4- bx + W?— z2. Assume z = m 4- xy3 then we have ax2 4- bx 4- m? = m* 4- 2mxy 4 Whence b — 2my Which is exactly the reciprocal of the expression deduced for the value of x, in the preceding case, ex- cept that we have a instead of c; and if, in order to V render it more general, we make y = - it becomes, b(f — 2-mpq X - Ο o ? — acf which latter equation may always be resolved in integers, because o o ^ p- — aql =1 is always possible (art. 150); and this is true whether b be positive or negative, as we have only, in the latter case, to make either p or q negative; or, which is still the same, assume z — m — xy in- stead of z — m+ xy. Ex. 1. Find the integral value of# in the ecpia- tion 5xq + 7# + 1 — z1. Here we have a— 5, b = 7, m—l, whence _1q*-2pq p*-5q\' Now, in order that we may have350 Indeterminate Problem4 p*-5f=l, we find, by art. 150, p = $ and g = 4, whence x — 40, which number will be found to answer the required conditions. Ex. 2. Find an integer value for x in the equa- tion 7-** — 5.r +1 = sr. Since a — 7, b= — 5, andm=l, we have r_-5f + 2pq f-Vt ' And, in order that this denominator may be equal to unity, we must find, by art. 150, the values of p arid q, such that this condition may have place, which are p = 8 and q = 3; whence is obtained x — 3, which makes 7 · 33 — 5.3 + 1 = 72, as required. Hence it -follows, that when integral values of x are required, we must have particular values of p and q, obtained by means of art. 150, or of the table subjoined to this volume; but when only fractional values are required, then we may assume p and q equal to any numbers at pleasure. 171. Cor. The foregoing solution will hold, whe- ther m be known or unknown; in fact, when m' is indeterminate, it may, in the expression ___________________bql — 2mpq X — £ - y p — aq be assumed at pleasure; and the solution is still more general, if m also enters into the middle term with a?; thus, for example, if we Write y for m, the equation under the latter supposition will beof the Second Degree. 351 ax* + bxy + f = sr; which may always be found in integers by assuming f x = 2pq + bq%, I y = f -aq\ in which expressions p and q may be assumed at pleasure; and, consequently, integral values obtained for x and y ad libitum (art. 101). Ex. Find the values of x and y in the equation 3a'“ + bxy = %2. Here a = 3 and b = 5; therefore, f x = 2pq+bqt, l y=p%-3r· And, assuming p = 3 and q — 1, we have x= 11 and y = 6, which values, substituted for x and y in the proposed equation, give 3.11*+ 5.11.6+6' = 2f. And other values may be found by changing those of p and q. If b = 0, and the proposed equation be ax1 + y~ = sr, then the values of x and y are (x=2pq, \y= p*-aq\ Which is exactly the result obtained at cor. 2, art. 54. Remark. A few other partial rules might have been added here for particular cases or conditions of the coefficients; blit, as the principles explained in the following propositions embrace every pos- sible form of equation, a multiplicity of rules for conditional equations seems to be both unnecessary and improper.352 indeterminate Problems PROP. II. 172. Every indeterminate equation of the second degree, containing two unknown quantities, may be reduced to the form if — Af = Bi Let ax~ + bxy + cif + dx + ey -\-f= d represent any indeterminate equation of the second degree, in which x and y are the two indeterminates, and «, b, c, d, e, and f, any known integers posi- tive or negative, or zero} then I say, that this equation may, in all cases, be reduced to the more simple form Ut-Af = B. For, first, multiply the proposed equation by 4a, which makes 4 ax'* + 4abxy + 4acy" + 4adx + 4aey + 4af— 0; add ffy* + 2bdy + d? to both sides, and transpose: so shall we have (2ax + by + dy~ {by + dy — 4a{cy‘l + ey +f); or 2 ax + by + d = A/{(by-\-dy — 4a(cy2 + ey +/")}. And, in order to abridge the latter expression, let V{(by + dy-4a(cy + ey+f)}= t, A2 — 4ac - ----- - = a, 2bd-4ae- - - - - — = 2g·, d2—4a/*- - - - - - — h; that is, A will represent the multiples of y*} 2g the multiples of y, and A the absolute quantity which contains no indeterminate letter. Now these sub-of the Second Degree. 353 stitutions will furnish the two following equations % Viz. f 2m + by +d=t, l a/+2gy + h = f. Multiplying this last by a, we have A-f + 2 Agy + Ah - A f ·, adding g~, this becomes (\y + gf = A(f-- h) + g~, or (Ay + gf-At^g^-Ah. Substituting again ay + g — n, and g- — aIi == B, we have tr — At'“ = b ; and, therefore, the proposed equation has been transformed, as required. And it is obvious, that the values of x and y in the proposed equation will be immediately deduced from the solution of the transformed equation tf-Af-H. For we have *y+g- "> °r y - u-g ; and 2ax + by + d=t} or x- t—d—by 2 a pr, substituting for y, in this last, we have __(t — d) a — (u·— g)b x —1 ~ » 2«A And since, in the transformed equation, u and t enter only in the second power, we may take t and u, in these last expressions, of the values of cc and y> either positive or negative, at pleasure;354 Indeterminate Problems and when the solution of the proposed equation is required only in rational numbers, it is obvious, that we shall have it immediately from any rational values of t and u in the equation u5 — a t" = B; but if the proposed equation is to be found in in- tegers, then we can only employ such values of t and u as, when substituted in the expression for x and y, render these quantities integral; in conse- qnence of which restriction, the solution, in these cases, is generally very tedious, and frequently im- possible. Ex. l. Let it be proposed to transform the equation + Sxy — 3y~ + 2x — by —110, or 3x2 + 8xy - 3/ + 2x - by - 110 -0, to another of the form Ul — At' — B. Here a —3, b = 8, c— — 3, d=2, e— —5, f— — 110. Whence bd—2ae=g=46 \ d~—Aaf= h= 1324 j and f bl — 4ac = \ = 100 \ g1 —ah = b = — 130284; therefore, the transformed equation is Ui-^l00f= -130284: and, if we had the values of t and u in this equa- tion, those of x and y in the original one would be readily obtained by means of the formulae 2 a * y-355 bf the Second Degree. but the solution of the equation U*-Af = B / belongs to the following proposition, the trans- formation being all that is required in the present case. Ex. 2. It is required to transform the equation 5a’2+ 3*+ 7 = i/', or SoV2 — «/2 + 3x + 7 = 0, * to another of the form id — At~ = Bi Herea = 5, b — 0, c= — 1, d=3, e — 0,f=^. Whence we have bd—2ae=g = 0, 1 , i J b2 — Aac = a — 20, dd — Aaf= h= — 131, J ant f g2 — ah = b = 655: therefore, the transformed equation is id — 20ί2 = 655: and the values of u and t being found in this equa^ tiou, by the method explained in the following pro- position, we shall have those of x and y in the one proposed from the expressions V u-g and x t —by —d 2 a Remark. The general equation above given in- cludes every possible form of an indeterminate equa- tion of the second degree, and we have seen that this is always reducible to the form id — At2 = B; and, consequently, on the solution of this last depends that of every possible case which can arise in indeterminate equations of this dimension: but 2 A 2356 Indeterminate Problems it should be observed, that, in the solution of the equation iv — \ f = n, , / cl? we may have t and u fractional, as m = — and t whence the above becomes x* ' y* = B, or l z’ * x%— Ay* = bst; an equation in which the indeterminates x, y, and %9 are all integral,' as well as the known quantities a and b j and as all possible eases are reducible to this form, the solution of it becomes the more particularly interesting. prop. m. 173. In all cases in which the equation of — Ay* = BZ* is possible, it may be transformed to another of the form x/Q,—yn=-cr/~. We have already (at art. 53) given a rule for judging of the possibility or impossibility of every equation of the form Χ* — €ψ*=*ΒΖ*, or, more generally, of the form } AX*—By and z, as integers prime to each other; also y prime to b and z prime to a: which being premised, we shall proceed to theof the Second Degree. 359 transformation of the given equation, as announced in the head of the proposition. 174. Transformation of the equation — At/" = B.&2 to the form x^—y'i — czn. In the first place, let us suppose b > a, for, if it were otherwise, we should have only to put the equation under the form a?2 — nz~ = Af, in which we have a>b; so that we may always suppose the coefficient of the second side of the equation to be the greatest of the two, unless in case of equality, which case forms a separate pro- position. Let then, in the given equation, 0 0 0 ar — Ay* = Bz, b > a, and since we have y prime to b, by the foregoing article; we may make x = ny — By' (cor. 1, art. 40), n and y' being two indeterminates; which expression being substituted for xy in the above equation, it becomes ηγ — ‘Itmyi/ + bH/~ — ay1 = b z2 ; or, dividing by b and transposing,' pi~_ —-—y2 — 2nyy' + byn = ; and, since b is prime to y, it is evident, tluit w® — a must be divisible by b, for otherwise the equality could not obtain, all the other part of the equation being integral; in fact, our rule for ascertaining the possibility of the proposed equation (art. 53) depends upon the condition of the equationβ6θ Indeterminate Problem§ n2 —a B e being an integer: let* then. n B k2 being the greatest square contained in the quo- tient of n — a and this value being; substituted for ΎΙ — A in the above equation, gives b 'Wy2 — 2nyyf + By'1 = Now we have before shown (art. 44), that all squares are of the same form to modulus b, as thp squares 1% 2s, 32, &c., (4-bY and, therefore, the least value of n will never ex- ceed ±b: we shall, therefore, be sure, by trying all the intermediate numbers from 1 to 4-b, to fall upon the possible values of n between the limits 1 and ^B, that render the above expression an in- teger. Suppose, then, we have found one or many values of n within the above limits, which have the required conditions of rendering —-— an integer, avc must, with each of these values, continue the transforma™ tion in the following manner; viz. Repeating again the equation b 'Ify2 — 2nyy/- + By'1 = $r, and multiplying by b'k1, we'haveor of the Second Degree. 36l v^lty* — 2im'k%yy' + b'bA*yn (B'lry — ny')2 + B'ntfy'2 — ivy and, since it — a = b'bA2, this becomes (b 'k2y — ny')2 — aif* = b'Jcz2 ; or, making B'Jry — ny' — x'y and Jrz2 = z'2, we have, for the transformed equation, x'2 — ayn — b V2, which is exactly similar to the equation first pro- posed, except that in this f/ < ^b, for n < -|b, therefore n2 — a < ; and, consequently, since we must have b' < We have, therefore, transformed the given equation to a similar and dependent one, of which one of the coefficients is less than in the equation proposed; and if, in this new equation, b' be equal to unity, or to any square, the equation is trans- posed as required. And the values of x\ ?/, and being determined in this, will give us the values and z, in the original. For we have 1st, x — ny — by\ 2d, of — b'/tv/ — ny\ ' 3d, z' — kz. From the second of these we have x' + ny' which therefore thus becomes known. And the first gives362 Indeterminate Problems And the third reduces immediately to But if neither of the above conditions have place, that is, if b' be neither equal to a square nor to unity, we must first ascertain whether b' be still > a, and if it be, we may proceed in the same manner to transform: this last equation to another similar one, x"2—Ay"1 = b"z"2 ; in which last b" < \b' : proceeding thus, by suc- cessive transformation, we must be finally brought to an equation in which or c < a. Having arrived at this equation, by transposing, we shall have x"- — cz”'- — Ay"2, in which now, we have a > c. If, then, we represent this new equation by a;8 — cj/2 = as8, we may, by proceeding exactly as above, reduce this to another similar and dependent equation, x'2—Cy'2 = A d. Representing this anew by X2 — Dj/2 = CZ2, we may proceed to reduce it in a similar manner to the preceding one. Since then, at every step, we reduce the coefficients, so that a < b, c < a, d < c, &c., it363 of the Second Degree. is obvious, that we must finally arrive at one that is equal to unity; for it has been seen that these coefficients are always positive, and they can never become ==0. Now it is impossible for a series of f integers to go on continually decreasing, under these conditions, without one of them becoming at last equal to unity; and hence it follows, that when the equation X2 — Ajf = Β2Γ is possible, it may always be transformed into another equation of the form Xn -y2 ;= pz'% which last equation is always resolvible (by art. 54); and, from the values of the indeter^ minates in this last equation, we may proceed by successive steps to those of x, ?/, and z3 in the original one proposed, by means of equations analogous to those in the preceding part of fhi§ proposition; viz. x! -f mf s' Z~T' 175. From what has been explained in the* above article, we derive the following simple method of arriving at the final equation, without absolutely performing the operation, which it was necessary to explain in order to show the principles pn which the transformation was effected?364 Indeterminate Problems It appears from what has been siiid, that, in order to effect the successive reductions, we must make the following calculations, n~ —-A , .·. XT — AJ/'=B ST, --------= B ΛΓ. ■a., o / a 3r — Ajr=B % r * 9, // o .r —Ay =b: —Ay2 = B(m)£2, or B n'~ ~ a B -bT2 B &C. &c. till we have c < a ; then, transposing, our calcula- tion must be cai gs above : thus, tion must be carried on again in the same manner G O O ΛΓ — Cy = A z~9 $r — ey = a zy X* — cy2 ~ a'V, cT2 — cy2 = a(w)*2j or D£2, n —c = a' A" &C. -α"Γ% &c. in which expressions /r, /2, &c., are the greatest integral squares in the quotient — : which re- ductions and transformations must be continued, till Are arrive at the equation required; that is, in which one of the coefficients above represented'by A, b, c, &c., becomes unity; and having found the value of the indeterminates in this last, we shall arrive at those of a*, y, and %9 in the original equa- tion, by means of equations analogous to the following;pf the Second Degree. 365 x -\-nif X = 711) — bj/. in the above forms, the accents of x, y, and z, are omitted, to avoid confusion; but the reader will be aware that these letters are not of the same value in any two of the preceding equations. It may also be proper to add, farther, that though, in order to show the successive transformations, we have employed several forms, yet there are few practical cases in which these are numerous. Ex. 1. It is required to transform the equation and hence to determine the values of a1, y, and z. in the equation proposed. Here we have by assuming ft = 4, whence b' = 1; and, therefore, the transformed equation is O . w O m O x~ — by' = l\z to another of the form Now the general values of xf and y' in this equa- tion are (by art. 54) xf = pq + 5m9=14, *' = f -bm?=z 4, y' = 2pm = 6, V366 Indeterminate Problems that is, by assuming p = 3 and m = 1; whence those of x, y, and z, in the equation proposed, are oc' + ny' 14 + 24 r- b'A» = 38,- x.—ny — v,y' = 4.38 — 11.6 = 86^ zt 4 z~~k =ϊ = 4· which numbers answer the required conditions, for 86s-5.382=11.42. And, by giving different values to p and m, various other integral solutions may be obtained. Ex. 2. Required the values of x, y, and z, in the equation a.·2 - 12/ = 13%°. First, it — 12 ~ΕΓ = ·ΐ=1> by assuming n — 5; whence we have, for the trans- formed equation, x'~ — 12yn — z'~, or xn — zn = 12j/'*, in which we readily find x' = 4, %' =2, and y' = 1; whence x' + ny' 4 + 5 =— = 9> x = ny — Br/' = 5.9— 13.1 =32, 2 %~T =1 . = 25 which numbers answer the required conditions, for 32s-12.92=13.22.of the Second Degree* 3&J In these two examples, we have arrived at the equation required by the first transformation, in which cases we readily find integral values for a?, y, and but if two or more transpositions be re- quisite, then we must be satisfied with fractional results, at least we cannot always obtain integral ones, under those conditions. Ex. 3. Required the values of x9 y, and £, in the equation a?2 — 1 Oyq = 31 z~. First, 10 31 ■=B T = by assuming n= 14, and thus we have b' = 6. If ~ I, so that the transformed equation is xn — 10yn = 6z'~, or xn— 6z'*—l0y'\ Then again, w7'2_ = cT= 1, { 10 by assuming n' = 4, whence c : transformed equation is 1, and the new { ■6z"*= y"\ or y !=6*Λ in which last we find readily x" = 5, y"—\, and z" = 2. And having thus obtained the values of the indeterminates in this equation, we readily deduce those of x', y', and and hence again those of x, y, and z, in the equation proposed. Thus,368 Indeterminate Problems x" + n'z" 5 + 4.2 = 13, cT- l x' = n'z' - cz" =4.13- 10.2 = 32, , y" 1 9=ST =7 = 1: which numbers answer the conditions of the equa- tion oyn^6z'*-y and hence again we have af + ny' _ 32+14 Λ _23 r- 6 23 , — 229 w = w# — 3\y —14. -- — 31.1= —y z' k 13 T 13, or 89 and these fractions, or their numerators only, will answer the required conditions ; for 22 1, we must proceed in the same manner to reduce it again to a similar equation, in which A//<4rA/; and it is manifest, that, by thus con- tinually decreasing the values of a, a', a", &e., we must, at last, arrive at a term equal to unity; and then the equation will be transformed, as required. And from the values of the indeterminates in this last equation, we arrive, by successive steps, to those of x, y, and z, in the equation proposed, for in each of these we shall have analogous equa- tions to those first obtained; viz. z' + m/ y =»£ —Ay, whence# =----------; A λ %' — A'k^z ~ ny', - - - y —nz — Ay'; x” '~ -r and x = ax'. x'k = x" x‘of the Second Degree. 371 Ex. 1. Find the values of x, y, and 2, in the equation af- 13/ = 132®. First, making Λ’ = 13x', the above equation becomes, after division, y1 \%l — 13χ,ςί; also by assuming w= 5; whence a'=2, and #*=1, so that the transformed equation will be y+2/2=2«//a. Now here we have a known case, namely, ■when y'= Ϊ, 2'= 1, and x" = l ·, whence again 2*+ny' _ 1 + 5.1 _ . Z~ A ir ■ 2 * y == nz — Ay' = 5 .3 — 13.1 = 2, . . 1 X~T ~7 “1; and, consequently, x — 13x' = 13, y = 2, and 2 = 3, are the values of x, y, and 2, required for 13*-13.2"= 13.3"". 177· Scholium. This problem may also be resolved upon principles entirely different from the foregoing: for it is demonstrated (art. 105), that the Sum of two squares can only be divided by num- bers that are also the sums of two squares; and, consequently, when the equation is reduced to the form y* + 2a = AXn, it is evident, that both a and x'* are the sums of two squares, because yl + 2® is divisible by each of 2 B 2Indeterminate Problems 3?2 those quantities; and, farther, it has been shown (art. 91), that the product of two numbers, each the sum of two squares, is of the same form. Hence, then, we have the following solution: assume a —jr + cf, and x'°—pn -1- q'2, then will (// + l to X* — 2f = A32. „r2-h •V = A / blf = AS", For as these formulae can only be divided by numbers of the same form as themselves (art. 105, et seq.); therefore, when any of these are possible, a is of the same form as the first side of the cor- responding equation, .2* being likewise so; and then again, the multiplication of two formulae of this kind, as (p2 ± aq*) (p'2 ± aqn) ^ x* ± ay*: that is, they give a product of the same form. Hence, then, representing the above equations by the general form x* + ay* = A%2,374 Indeterminate Problems the solution may be pbtained as follows: Find f p2 + cuf =A, \pn + aqn = zi3 the latter of which equations is readily found by art. 54, and the former is always possible if the equation be so; and having thus found the yalue§ of p, q, //, and q', we shall have (pq + cuf) (pn + aqn) = A z* == (PP' + aqq'f + a(pq'+p'qY- Whence again we derive, by comparison, f x=pp'±_aqq\ 1 y =pq' + p'q· Ex. 1. Required the values of x3 y, and z, ip the equation x^Jt2yi—Qzi. Here, a is of the form x1 + 2y1, for a = 6 = 22 + 2.12; therefore, p — 2 and q = 1. Also, assuming z1 — 3°, we have s2 = 3°'=1+2.22; therefore, p’—l and q'=2. Whence, ( x=pp' ±2qq' — 6, or 2-, t y=pq'+ qp' = 3, or b. Which numbers answer the required conditions; for, / 62 + 2.32=6.32, l 22 + 2.52 = 6.32. And various other values might be found by assuming any other square for s2, which has theof the Second Degree. 3 Jo form + 2qn; and this may always be done by squaring any number of the same form. Ex. 2. Required the values of x, y, and z, in the equation af — by1 — 1 lx2. Here, a being of the form x1 — bif. or a=11=42-5.12; therefore, p — 4 and q — 1. Also, assuming z1 = 22, we have Z5 = 22 = 32- 5.12; therefore, p'~3 and q' — l. Whence, f x=pp' ± bqq' - 17, or 7; \y=pq'± qp' — 7, or 1. Which numbers give the following results : f 17*-5.7e=11.3% { 7*-5.1®= 11.2\ And various other values might be obtained, by assuming other squares for x2=p,y/2— 5 qn. It will be observed . here, that the ambiguous signs in the compound expressions for x and y are ± and + ii| the first example, but ± and ± in the second; that is, the ambiguous signs are ± and +, when the connecting sign is + in the proposed equation, but ± and + when that sign is — the reason for which change will become ob- vious by considering the nature oi the two products. Cor. 2. In the above equations, we know im- mediately, from the/form of a, whether the equa- tions proposed he possible or impossible; as in the former case, a must have the same form as the first side of the equation with which it is connected,376 Indeterminate Problems at least with an exception in the two last, when a is pven (see art. 108 and 109). And thus far these equa- tions may he considered as forming a separate class^ but in other respects the same principles may be employed for any equation whatever; that is, when a is of the same form as the first side of the equa- tion in which it enters; but when it is not of that form, it does not imply the impossibility of the proposed expression, as is the case in those we have been considering. prop. v. 178. Every equation of the form OC* — A?/2 = B£2, . Λ . Λ m — b η2 — a . m which —-—-, and — , are both integers, is resolvible in rational numbers. We have before investigated this theorem (art. 53), but it will be observed, the rule thence deduced, although perfectly correct, is deficient in this, that it is not demonstrated, when an equation falls under a possible form, that it admits of a rational solution; the only certain conclusions being with regard to impossible forms:. it is therefore proposed, in the present proposition, to supply this defect, by demonstrating the absolute possibility in the former case, the truth of which, or of the above theorem, results as an immediate conse- quence of the transformations effected-in the pre- ceding propositions, and the demonstration of art. 52. But in order to render the investigation as simple and conclusive as possible, it will beof the Second Degree. 377 proper to resume here the forms of reduction, as in art. 175; viz. X1 — Af = B 2T, · X2 — AJf — bV, x1 — A?/2 = B(,n) z*5 or c*2, B n'2 — a B'# &c. = b' k~, = B 'Ύ1, &c. . In which last equation, b(w°, or c < a, Then again, x1 — cf — a *2, — c A = A//% a?2 — cf = a V, a?2 — ey* = a(w)£2, or D£2, = a"Z/9, &c. till we have d < c; and so on, as has been before explained, in art. 175, Zr, A*/2, &c?, Z2, Z'2, &c., beings as in that article, the greatest squares con- tained in the respective quotients, arising from the division of n2 — a, nl — c, &c., by b, a, &e. And it is to be demonstrated, in the present proposition, that, if it be possible to find—, and ——, both integers, that all the other above analogous forms are also possible in integers; and, therefore, that the equation x2 — Ay2 = bz2 is reducible to a dependent equation of the form xn — yn=zczi9 in which last form the solution may always be ob-378 Indeterminate Problems tained, and whence the values of x, y, and z, in the original equation, also become known. Now, first, if n — a B = B 'If, we have evidently, by transposition, W2 —A B ■ b¥·, 71 — A but if —jp— be an integer., so likewise is (n — mb')' — a b' ’ where the indeterminate u may be so assumed, that . (w — mb') < -1-b' ; therefore, calling (n — ub') ~n'} we have nn — a and, consequently, nH — a ■■B'k'2, an integer; and here again we have also an integer; or, making n' — un" = n", :B'T, rin- A an integer, and so on, for as many transformations • ^ ^ ^ as are requisite; that is, if ——■ gives an integral JB quotient, so likewise will all the other analogous formsof the Second Degree. ηη-κ ft"2 — A 0 -, &c. 379 r-, t ) B till we arrive at p— a __ i(«) c. And it therefore now only remains to be shown* __jj 7JZ ^ C that, if be an integer, —------is so likewise 5 A A for, this being demonstrated, it will follow, from what is shown above, that the analogous forms m'2 —c m'n— c A/ > —, $tc, A ' ' A" will also give integral quotients; and, therefore, that the possibility of the original equation de- pends upon the two conditions of ft2 —a , in1 — B ’-----, and --------, B A being integral. In order to demonstrate this, let us repeat again our first equations, ft2 —A ... . nr - b b' ¥ --------- = b re , whence---------------= 1; nn — a 1 b'b"¥ i B* ft"2-A — 15 ft * n — a B"B"'¥n B" “ Ji· K ^ A — A j &c. &c. t 1 t ?? p &c. 1 >■ sT II - b(’°cF B(») a — 1; and, consequently, bb'¥, b'b"/c'2, b"b'"A"*, &c., are found amongst the remainders of the squares380 Indeterminate Problems n2, nn, n"% &c. to modulus a; that is, + p-A -l· bV'A'®, &c. are all possible forms, when compared with A as a modulus. But we have demonstrated, that the product of a possible and impossible form always produces an impossible form (art. 52), therefore, B and b7 must be of the same kind, with regard to possible or impossible, to modulus a; for if b was a possible remainder to modulus a, and b7 an im- possible, then would bb'A:2 be also impossible, as is evident from the proposition above quoted: but we have seen that bb-'A® is a possible remainder, and, consequently, b and b7 are both of the same kind, as to possible or impossible, to modulus a. And, in the same way, we shall have b7/ of the same kind to b7, and, therefore, also to B; and the same of the other quantities b77, b777, &e. to c: therefore, VV^pA + B, we shall have also ' /λ2^α + €; and the same reasoning will apply to every trans- formation. Now, since the solution of the pro- posed equation, OOP λγ — = Bar, depends upon its transformation to the form and this transformation depending upon the pos- sibility of the integral quotients above stated; also if be an integer, or a381 of the Second Degree. these having their possibility involved in those of the two conditions ri' — λ . nv — b ------, an d-------,‘ B A being integers: if follows, that when these two obtain, the solution of the proposed equation may always be obtained in rational numbers, v-α. E. d. Cor. And ip the same manner it may be shown, that the equation AX" — ¥>y~ = CZ" is always possible, if it falls under a possible form, according to the method employed at art. 53. Lemma. 179. We have, in the foregoing propositions, given a general method for solving all possible in- determinate equations of the second degree, and of ascertaining their impossibility when they admit of no rational solution; and, in the following article, it will he shown, how, from one known case, an infinite number of others may be obtained; but, before we proceed to this, it will be advan- tageous to the reader to collect, under one point of view, all that has been demonstrated in this and the foregoing chapter relating to the equation —Ny®=±A· First, then, it has been demonstrated, in art. 105, that the equation x°" — wf - + 1 is always resolvible in integers, providing n be not an exact square. As to the equation382 Indeterminate Problems Λ* —Ν^®=—1, it is only resolvible in certain cases; that is, when the fractions arising from vn recur in periods, consisting of an odd number of terms. Also the equation Λ’2 —Kf/= + A is always possible, if a be found in the denominator of any of the complete quotients arising from the VN; but the equation x® — Ny®— — A is only possible when a is found in the denominator of one of the complete quotients, that occupies an even place; and, consequently, the corresponding fraction an odd place. In all these cases, however, if there be one solution possible, there are an inde- finite number of others; and, in the following pro- positions, it is proposed to find general expressions in which the values of x and y are contained for each of the above cases. PROP. VI. 180. To find the general values of x and y in the equation X~ — N7/® = + 1, from the values of p and q in the equation joe_N?a= + 1. The present problem divides itself into three cases, on account of the ambiguous sign ±, which are as follow; viz. To find the values of x and y in. the expression a® — ny1, under the following con- ditions :383 of the Second Degree. 1 st, x1 — ni/ = 1, from the known case // — N(/ = 1; 2d, x* — sf = l, - - - - - - p°- — 'sq1——li 3d, x* — n«/2= — 1, - - - - p2 — ■$(/ = —l. Case 1. Resolve the two equations p-— n^s=1, and — isf — 1, into the factors f + 1, l («*+# v*){x-y vn) = i; then we have also (/> + q v'N)'"(/) — q λ/ν)·* = 1“= 1: equating these with the factors in x and y, we obtain X+y ^/N = (j9 + f/ λ/ν)"*, a·-?/ λ/ν= (p-i/ vn)m. Whence again, by addition and subtraction, (p + q -v/N)” + (j» - j. vn)” ®” 2 (p + q λ/ν)*- (p-q λ/ν)* 2 λ/ν ’ which values of x and y will always be integral, and will, therefore, be the general values sought; and these are evidently infinite in number, because m is indefinite. Case 2. The same .method may be followed here as in the preceding case, except that the powers represented by the exponent m must be even, in order to convert — 1 into +1, as is obvious from inspection; and, therefore, the general values of x and 3/, in this case, are _(p + q V^Ym + (p-q vNym 2 ■i384 Indeterminate Problems _(p + g v^Ym -(p-q λ/ν)m y — 2. VN * Case 3* Here again we have evidently the same result as in the former cases, except that the powers of m must how be odd, for every odd power.of — 1 = — 1; therefore, the values of x and y are now _ (P + q Vn)"w+1 + (p — q λ/ν)2μ+1 x~ a _ (p + (] y'N)am+1 —{p — q λ/ν)μ+1 2# The number of values of x and y being indefinite, as in the former eases. Let us now illustrate these rules by a few ex- amples. Ex. 1. In the conation ]f— 14q‘2 = l, having given p =15 and q = 4, to find the; general values of x and y in the equation χ- — 14y1 = 1. Here, making m — 2, we have (15 + 4 +14)* + (15-4 λ/14)2 X =----------—-----—-------;-----= 449, (15 + 4 +14)2- (15-4 +14) _, _ TJ — " — 12uj 2 +14 which give 449^ —14.120* — 1; and other values may be found by assuming any other power instead of the second. Ex. 2. Given p — 4 and q — \, in the equationof the Second Degree» 385; to find the values of x and y in the equation α?-ΐγιρ=·+ι. Here we have again ^_(4+ vi7)°- + (4- ^7)\_ZZ) _(4+ „/17)*-(4- ^17)a__ y 2 vi7 whence " 33a — 17 ..8B'= 1; and other values may be found by assuming any other even power instead of the second. Cor. This method of deducing the values of x and y in the equation x? — yy° = 1, from those of p and q in the equation p1 — yq1 = — 1, is very useful in finding those values of x and y'j being much more ready than continuing the ex- traction by continued fractions; because, whenever1 the minus sign arises, it always happens before the plus sign; this will be evident from the following example. Ex. 3. To find the values of x and y in the equation 3?— 13ϊ/2 — I. We find in five terms, proceeding by Continued fractions, that, in the equation /--13?2=-1, p= 18 and q — 5; therefore, without pursuing the extraction any farther, we have, by means of the above formulse, 2 QS&6 indeterminate Problems (18 + 5 Λ/13)4 + (18-5 νΊ3)* :64$, χ = - 2 (18 + 5 Λ/13)2-(ΐδ-5 λ/13)* υ = -------—-------------— = 180; ' 9 2 ^13 which, as we observed above, is a much readier1 method than carrying on the extraction, as in ex. 3, art. 151. Ex. 4. Given p~4 and q= 1, in the equation /_!7q°=-l, to find the values of x and y in the equation #*-17/=-1' Assume, (4 + Vl7)’ + (4- V17)S x = - ■ 268. (4+ λ/17)’-(4- λ/1 7)’ ^ y=-----------2—7---------= whence 26 s3- 17.67*= -1: and other values may be found by assuming any other odd powers instead of the third. PROP. VII. * 181. To find the general values of x and y in the equation X1 — Ni/3 = ± A, a being < λ/ν. We have already shown how the first values of x and y are to be obtained (art. 154), and . shall therefore now suppose that these values are known, or that we have found the values of m and n in the equationof the Second Degree. fetid also those of p and q in the equation J94—,N§®== ± 1. Then it is obvious, that {p1 — aq*) x (?na — mf) = ± a ·, but, by art. 95,· (// — vqq) x (m°“ — NH2) — f (pni + Tsqn)2— s(pn +qm)q, or 1 (p»i — xqn)t — 's(pn — qm)i} whence we have, for the values of x and 2/,· 38/ f x=pm±yqn, l y y = pn + qm. But the general.values ofp and 5 in the equation f~K(f=i± i· are., by the foregoing proposition, (P+9 V*ja+(p-q p=~' 2 * _ (/>+ (i ^)M - IP - 9 vn)" . q ~ 5 in being even or odd, as the case requires; which general values of p and q, being transferred to the formulae f χ£ϊρηι± tiqrtj ί # = -pn ± qm, will furnish the general values of x and y in the equation proposed. Cor. If the known case be · - a, and vi e wish to deduce from this x1 — 1vy® = + a, we must find p and q in the equation and then 2 C 2388 Indeterminate Problems ( x =pm ± tiqn, 1 V —P*1 ± ψη) will be the values of x and y required. And generally, if the known case have a dif- ferent sign from the equation proposed, then we must employ the equation p‘ — Ts(f = — 1. But if it have the same sign, the equation p* — n (/ = + 1 is that from which the general values of. x and y are to be obtained. Ex. 1. Having given the values of m and n in the equation nr — 7»s = 2; viz. m = 3 and η — 1, to find generally the values of x and y in the equation **-7/=2. First, in the equation f-7q~ = l, we have p = S and q =3; whence the above formulae give x=pm±Tsqn> 1 , . ( x=3, or « = 45; y=pn ±qm; j ^ \y=l, or# =17; and it is obvious, that, by finding the other values of p and q in the equation /-7f = l, we should readily deduce those of x and y in the equation proposed; but it is perhaps as well to consider the values of x and y just found, as new values of m and n, and then we have immediatelyof the Second Degree. 38$ x=pm±Kqn,\ Q1. f # = 717> y=pn±qm·, j \y = 271·, and so on for other values, ad infinitum. Ex. 2. Find the general values of x and y in the equation a?-13/=-3; having given those of m and n in the equation m*-13n9 = 3; viz. m = A and η = 1. In this equation, since the absolute quantity has a different sign from the known case, we must em« ploy the equation f-l3q2=-l, which gives p = 18 and q = 5 ; whence x=rm±r,qn, 1 thati . (*=7, or 137s y=pn±qm ; J f t/ = 2, or 38; which are two of the values sought, for f 7*- 13. 22= -3, jl372-13.382= -3; and other values are readily found, as in the fore- going example, PROP. VIII. 182. To find general and rational values of x and y in the equation of — n/ — ± A, A being any number whatever. In this case we have no direct method of finding integral values for x and y, as we had in the fore- going propositions, unless a fall within the limits prescribed in the last problem, in which case the3Q0 Indeterminate Problems Solution belongs properly to that article, and we have, therefore, in this place, only to attend to the case in which a > yTsr. Now if a, though greater than -v/n, be made up pf any number of factors, as a', a", a'", &c., each of which is less than the „/n, the solution of the , equation • X® — N?/® = Hr A, when it is possible, may be deduced from those of the equations m2 N·»2 == ± a', m'~ — n n'2 = + a", m'n-m"2= ±a"\ &C. &.C. because the continued product of factors, each of the above form, is itself also of the same form (art. 95) i and we shall therefore have (m® — n«®) (mn — nm'®) (mm—jm"~) = x% — Ny2 = ± a, where the values of x and y will be always deter- minate, and integral functions of m, n; m% n'·, pi'% n"} &c, For, by the same article, (to® — n»®) x (m'- — nη?) — (mm' ± n nn'Y — κ (mn' + m'n)*. And making now f 7nm' ±κηη'=ρ, : l mn' ± m'n = a, f?e have (m* — n n2)(m'2 — n»'4) = p* — n a2. ’ Again, (p2-Na5)(m"2-Nw//2) = (pm" ± ήομ")2 - k(p n" + am")5;of the Second Degree. 39X makings therefore, f vm" ± = xy | pra" ± a m”—yy we have (ml — n »s) (#i/2 — nw/2) (m"* — nw"2) = x9, — nt/· = ± a. Also, substituting for p and a, in the foregoing values of x and y, we obtain f cT = w!\mni + nw·^) ± nn"(mn' ± m'ft), (_ y == (sram' ± nww') ± m"(mn/ ± wz'w).. Whence a? and y are determinate and integral functions of ?ny ft; m', ft'; fti", ft"; &c, 5 which are known integral quantities, Having thus found one integral value of x and y in the equation X2 — NT/2 = + A, %ve shall have the general values of those quan- tities from the equation ± 1, as in the foregoing propositions; that is, calling the values of x and y, found as above; m and ft, the general values of x and y will be x=pm±i*q?iy y—pn ±qm; the general values of p and q being expressed by _ (P + ? VN)W + (p-q λ/ν)“ p-------------2 , (p + q v$)m-{p-q a/n)”1 q~ 2 a/N . ’ the indeterminate power m being even or (>dd, as the case may require.p9% Indeterminate Problems Ex. 1. Required the values of x and y in the equation x°-\31/2 = — 9r First, having resolved — 9 into the factory + 3 x —3, we must find the values of m and ηΛ and w! and n', in the two equations f nf — I3ri2 = + 3, \ m'*-13n?= -3; and also the values of p and q, in the equation f- 13f=l, and then the general values pf x and y may be de- termined as above; thus, in the present case, we have, from example 2 of the foregoing proposition, m = 4 n = 1, ^== 7 whence the first values pf x and y are f x =mm'± N?m'= 2, or 54; \y~m'n± or 15. And by means of these values, and those of p and q, in the equation p2 — 13 vn, and cannot be resolved into factors, each of which is less than */**, we have no general ^method of solution for integral values; in fact, the equation will not always admit of such values, although there may be fractional ones that will ob- tain, which is not the case if a< ^n, or resolvible into factors that are < λ/ν. We must, therefore, in this case, employ a different method of solution; that is, we must find the values of t and u in the equation . f — Ntt8 = ± AZ1, by art. 176, and then, dividing the whole by we have f uq ρ·~ν=±Α; or. making- = w and- = w, this equation becomes 0 2 ' z 1 m1 — n rf ?= + a ; and, calling this the known case, we shall have the general values of x and y, by means qf the equation jp4_Ngr«5= + 1, as in the foregoing article; that is, f x=pm±Kqn, \y = pn±pm·, only in this, the general values may he fractional instead of being integral, as in the former case. Hence it appears, that in all cases when one solution is given, as many others may be deduced from it as we please; and when the given case is integral, all the other solutions will also be integral;S94 Practical Examples. and when the first is fractional, all the other de- pendent solutions will be fractional likewise. Cor. The methods that have been explained, in the preceding proposition, for finding the general values of x and y in equations of the form χ- — ny* — A, are equally applicable to equations of the form ar — n?/9 = az", as is evident; because this equation being mul- tiplied by τγίΐΐ leave the second side of it the same as at first. Practical Examples. 1. Find the least integral values of x and y in the indeterminate equation x* — by2 = 1. Am. .r=9; y — A. 2. Find the integral values of x, y, and zx_ in the indeterminate equation x”~ — by1 — 13x8, or prove that there are no such values. Ans. Impossible. 3. Required to ascertain the possibility or im- possibility of the equation δ#2—7yq— ii»V ' Ans. Impossible. 4. Find the least integral values of x and y in the indeterminate equation xy7lf=l> and also in the equationPractical Examples. β95 ^-7/=-ι, if the latter be ppssible. Ans Γ * = 8, y = 3, 1st equation. \ Impossible, 2d equation. 5. Find the two least integral values of x and y in the equation a?-\3yi=l. Ans X * = 6*9, 84?431; yms· =180, 233640. 6. Find the least values of x and y in the equa-. tion x°~— 13i/2 = 4. ' Ans fx=119> ' \y=z 33. 7· Required the least integral square that, when pmltiplied by 113, shall exceed another integral square by unity. Am. {* = 1204353,, \y= 992.96. 8. Required the least values of x and y in the equation 79^-101/ = !.^9β CHAP. IV. On the Solution of Indeterminate Equations of the Third Degree, and those of Higher Dimensio?is, PROP. i. 184. To find rational values of x in the equation axf + bx® + cx + d= z2. It is only under one partial condition of the ab- solute term d, that this equation admits of a direct solution, that is, when d is a complete square, as d—f1, in which case the equation becomes ax? + bx°~ + cx +f* — z~; and, when this condition has not place, we have no other method of proceeding but by trial; and even when it has, we can find but one solution at a time, which is obtained in the following manner: Method. Assume 'ζ=/+ψι'·> then, by squaring, we have & ax3 + bx* + cx +f* 2 + cx + -^,-x2, or & ax* + bxq ~ whence, x = : Indeterminate Problems, &:c. 3$f Rx. 1. Required the value of x in the equation x3 + x* + 3x + 1 = z%. Here α = 1, b—l. c = 3, andy. (4 l/JV-c-f And having found the value of y in this last, we shall have x—y + to; and, therefore, x, in the proposed equation, will thus become known. Ex. 1. Required the value of x in the equation x? + 3 = z\ Here the known case is m—1, which gives l3+3 = 22, thereforey= 2; also a =1, 6 = 0, c = 0, d= 3. And, applying these values as above, we have fa - - - - = α' = 1, < (3am +6) - —V —3, b (3am* + 2bm + c) = c' == 3. Whence the new equation is ys + 3y* + 3y + 4 =z*.400 Indeterminate Problems And the value of y, by the first formula of the •eceding proposition > is c2 — Ahf1 9—48 -39 3 4ap 16 16 Therefore, -39 . - 23 x=y+m= : 16 +1= 16 5 the value sought. Ex. 2. Required the value of x in the equation 3a3 4“ 1 ^ a2. Here we have a known case, m = 1, which gives 3 m1 + 1 =j‘~ = 4, therefore f—2·, also a — 3, b = 0, c = 0, and d = 1 j and hence {a - - - =«'=3y Sam:+ b - =b'=g, Sand + 2bm + C= c' =Q; Whence the new equation is 3y3 + 9y* + 9y + 4 = z\ ■ And here the value of y by the first formula (art. 184) is __ci — 4hf!t _ — 21 ^ 4 of* And, consequent] γ, 16 x—m+y^i+ 21__— 5 ιϊΓ_Τ6^ Which value of a’ will be found to answer the required conditions of the equation proposed. If we had employed the second formula instead of the first, we should have found -1Q52 , -62{)of the Third Degree* 401 And it is obvious, that we might now consider either of these results as the value of m in the known case., and thus proceed to find other values of x, providing the equation admitted of more answers. But this is not always the case, as it often happens that from one known value of x we cannot derive another; and this is still not owing to any defect in the method we employ, for it is demonstrable, that some of these equations admit of only one answer; such is, for example, the equa- tion Xs 4~1 == 2Γ, which obtains when x = 2; but there is no other value of either integral or fractional, that will fulfil the conditions of the equation, as may be demonstrated on similar principles to those enn* ployed in Part I* chap. v. prop, in* 186* To find rational values of x in the inde- terminate equation ax4 + bx3 + + ώ + 6 = aV This proposition, like the preceding one, only admits of a direct solution in particular cases; viz* 1st, When e is a complete square, as e =/\ 2dy When a is a complete square, as a = wz\ 3d, When both the foregoing conditions obtain* And when no one of these circumstances has place, a direct solution cannot be obtained, there being, in fact, no other means of proceeding but by trial ; if, however, in this way, one solution is402 indeterminate Problems found, a variety of others may commonly be deduced from the one known ease, as is shown in the fol- lowing proposition. 187- Case 1. To Jmid a rational value of x in the indeterminate equation ax4 + bx3 + ex*+ dx +f2 = z*. Assume z =ρχ~ + qx+fi then, by squaring, we have ρ-χ* + 2pqx? + (/ + 2pf)x* + 2 qfx +f2 — ax4 + bx3 + ex2 + dx+f*. And now, by making f 2 qf =d, · \q*+2pf=Cf we have p\x4 + 2pqx3 = ax? + bx\ or (p1 — a)x* — {b — 2pq)x3. Whence, from the latter equation, x=b-2pq^ p*-a But the preceding equation gives d c — q2 ? = ψ and p = 4cf~~d2 or P- 8/3 Which values being substituted in the foregoing expression for x, we have (8//4 — 4cdf2 + ds)8f2 X- 16c/4 - 64a/6- 8cdf* + d4 3 and this formula will always render the proposed equation a square.403 of the Third Degree. Ex. 1. Required the value of x in the equation x* + Xs + x* + x + 1 = z2. Here, since a — 1, b = 1, c=l, d=l, andjT=l> Yre have -40 -8 X~ 55_~ΪΤ’ which fraction, being substituted for x, gives vi2r’ as required. Ex. 2. Required the value of x in the equation 2xi — 3x + 1 = a®. Here we have a = 2, b = 0, c = 0, d= — 3, and ■*= 1: whence x = 216 which fraction, being substituted for x, will be found to answer the required conditions. 188. Case 2. To find a rational value of x in the indeterminate equation mV + bx* + cx1 + dx + e = Assume 2 = ma? + px + q 5 then, by squaring, we have mV + 2m:px' + (// + 2mq)x* + 2pqx + q2~ mV+ bx? + ditions, the former making S76'’ and the latter, on the same principles, giving s2 = (—-)2 576' · 2(7 Method of solving the indeterminate equation mV + bx3 + ex'2 + dx+/2 = z~. Assume ζ = ηιχί+ qx+f, as before; by which means we have again407 of the Third Degree. mV + 2mqx3 + (q~ + 2mf)x3 + 2qfx +f3 = mV + bx3 + or b + 8m J — 4 m~c 4ndd + 4mbf b2 — 8 mf— 4m~c ’ the second formula being obtained as before, by supposing f negative. Ex. 2. Required the values of x in the equation a:4 — 3# + 4 = V Here m=l, b = 0, c = 0, d= — 3, and f=s2·, .therefore, _ Anvil — 4 mbf _ — 3 ^ b°- + 8mf— Anvc 4’ 4m°d + Amhf _ + 3 έ2 — 8mf— 4mlc 4s either of which fractions, substituted for x, will be found to answer the required conditions. Remark. It will be observed again here, that the formulae which we have thus found, all fail under the same circumstances as before; viz. when b = 0 and d=0: we must, therefore, in all such cases,408 Indeterminate Problems endeavour to find one value of x by trials; and, if this cannot be done, then it is in vain to attempt the solution of the equation, which, as we before observed, may not admit of one; but, if one value can be found under any circumstance, then we may deduce others from this one, as in the following proposition, PROP. IV. 190, Having given the value of m in the equa- tion am4 + bm3 + cm2 + dm + e = f~, to find values for x in the indeterminate equa- tion ax34 bx3 + cx* + dx + e—z*. Assume y + m~x, then we have ay4 + 4amy3 + Gamy* + 4am3y + am4 = ax4s - - by3 + Shiny* + 3bm*y + bm3 = bx3, - - - - - cy* + 2 any + end — cx*, ------ - dy +dm = dx - - - - - - - - - e = e. And now, writing - =a', 4am +b —V, 6am* + 3bm + c - - - - _ =c', 4am3 + 3bm* + 2cm +d - - - — d', am4 + bm’ + cm* + dm + e - = f*. we have a χ4 -f■ b x3 -f- c 3d d x e — z*, oi* a'y4 + b'y3 + c'y* + d'y +f* = z*. And now, the last term of this formula in yof the Third Degree. 409 being a square, we have, by case 1 of the pre- ceding proposition, (8b f* - Ac'd'f + dn) 8/2 V ~ 1 6c'/4 - 64a'/6 — 8c'df" + d'1 ’ and, consequently, since x=y + m, we shall have the value of x as required. Ex. 1, Required the value of x in the equation 5x4 —l=z% the known case being m = 1, which, in the equation 5 m4 — 1 =/s, gives /® = 2*. Here we have a— 5, b = 0, c = 0, d=0, and e=—1; therefore, a'= 5, b' — 20, c' = 30, d' = 20, J—2‘, whence (8&/4-4c'd/* + d,3)8/2 -24 ^ “ l6c'/4 - 64a/6 - Sc'd'/2 + d'4_ 11’ nnd - 24 -13 *■=y + m=—7Y +1 =—;· And, since a? enters only in the fourth power, we may likewise take x positive as well as negative; and, therefore, the value of x sought is 13 x= -1--, ~ ll’ which fraction will be found to answer the required conditions, making % ~ 121 ." Ex. 2. Required the value of x in the equation 2a:4 —1=2®. Here we have a known case, viz, m — 1, and/= 1;410 Indeterminate PrbMems therefore, since a— 2, h — o, c — 0, d— 0, f= 1, and m— 1, we have a' — 2, b' = 8, c'~ 12, d' = 8, and jf= 1; whence {8hT-^c'dT-Vdn)8p V ~ TQc'T -64 a/6 - 8e'd'-J~ + d'4~12’ and, consequently, y + m, or ar = 12 + 1 = 13; which paay be taken either + or — 1, because only the fourth power of x enters into the proposed equation, and this number answers the required conditions for 2.134-1 =(239)2. w e might now, in both the foregoing examples, consider these known values of x as new values of m, and thus proceed to find others; but it is ob- vious that we should soon be led to very compli- cated fractions, which would render the practical operation very laborious, prop. v. 191. To find rational values of x in the indeter- minate equation ax* -f bx* 4 cx-\-d = z5. This equation, in its present general form, will not admit of a direct solution, this being only ob- tainable under the three following conditions; viz, Case 1. When d is a complete cube, or d=f3. Case 2. When a is a complete cube, or a = m?. Case 3. When both these conditions have place. 192. Case 1. To find the value of x in the inr determinate equation ax3 + bxq + ex = *3.of the Third Degree. 411 Assume px+f=z; then, by cubing, we have //V + 3pfx‘ + 3pfx +f — a x3 + hx* + ex +y5. And πολύ, making 3pf~ — c, or p = -^p, we ob- tain ax’ + bx~=ρ3χ* + 3p~fxi, or x = - 3//-δ a—p° And, by substituting here the above value of p, this expression becomes (c2 —3δ/’)9/3 x- 2 7«/δ Ex. 1. Required the value of x in the equation 3 Xs + 2x + 1 = £3. Here « = 3, b = 0, c = 2, and/=l; whence (c*-3bf)9f3 36 x- 27 a/6 73 which fraction answers the required conditions for 3(|)'+2l)+i=f)'· Ex. 2. . Required the value of x in the equation x3 — 5x — 1 = s3. \ ^ Here a—l,b = 0,c=—o, and/= — 1; therefore, (c*-3bfs)9f -225 x- 27 af 152’ which fraction answers the required conditions, making 223412 Indeterminate Problems 193. Case 2. To find the value of x in the in- determinate equation nix' + hoc9 + cx + d=z\ Assume z = mx +p·, then, by cubing, we have mV + Snipx9 + 3mplx + p* =s mV + bxr + cx + d. And now, making Snip = b, or there remains Whence cx + d^= 3mp*x + p3. · p —d x= --------- c — 3mp J Or, substituting for p its equivalent ^5, we have _ V — sydni X (3cms — b9)Qm3 ' Ex. 3. Required the value of x in the equation ry$ Q ryyt I /yj -- r^3 Uit }" it ““** Αύ · Here m= 1, b= —3, c = l, and d=0; therefore V — Q.’jdnt6 1 X (3cms — b9)Qm3 ~~ 3’ which fraction gives z3 = (-)3, as required. Ex. 4. Required the value of x in the equation x' + x — 7 = 23. Here m = l, b = 0, c=l, and d— — 7; therefore, _ b3 — 2pdm6 (3 cm3 — b-)Qml which is the value of x required.413 of the Third Degree. 194. Case 3. To find the value of x in the in- determinate equation mix3 + bxq + cx+ f3 — z3. Under this form the equation belongs to each of the foregoing cases, and may therefore be solved by either of them; it also admits of another so- lution on the following principles: Assume z = mx +f, then, by cubing, we have m3x3 + 3 rtdfx1 + 3 mfx~ + m3x3 + bx*+ cx + f3. Whence, bx* + cx~3mfx2 + 3mf*x, or _3mf2 — c X b—3mf‘ We have, therefore, for indeterminate equations of this form, the following distinct solutions; viz. Bv easel x-^~3bf3^f3 ^ ’ 2*Jnif3 — c3 _ b3- 27f3me J (3 cm — b )(} m _ 3mf2 — c J ’ b—3mj That is, by writing m3 for a, in case 1, and f3 for d, in case 2. Ex. 5. Find three values of x in the equation x3 — 3x? + 1 = 2s. Here /« = 1, b = three values of x, formulae, are x- - 3, c=O, f= 1; whence the US determined by the above 2 , -rl Ϊ' ■3, x=~, and x= .. «i.414 Indeterminate Problems Ex. 6. Required the values of # in the equatiorl x3 — 3x — 1=«*. Since m= 1, b = 0, c— —3, f= — 1; therefore^ the required values of x are. x — —and X— 2. Remark. The above are the only cases in which the proposed equation admits of a direct solution, and even these all fail when b and c are both zero at the same time; that is, in equations of the form ax3 + d—z3) which are, in fact, frequently impossible, as we have seen in Part I. chap. v. But if in any proposed equation of this kind one solution is known, others may be deduced from the known case, according to the following proposition. prop. Vi. 195. Having given the values of m in the equation am3 + bnf + cm+ d=f3, to find the values of x in the indeterminate equa- tion ax3 + bx* + cx + d=z3. Assume y + m=x; then we have ay3+ 3amyi + 3amty + am3 = ax*f - - byi + 2bmy+bmi = bxif - - - - - cy + cm = cx, ■ - - - -. » -d =d.415 of the Third Degree. And now, writing a ----- - =af, 3am + b - - - - - =&', 3ami + 2bm + c - - - =c% am3 + bm~ -f cm + d - =/% a x3 + b x* + c x + d = or a'lf + b'tf -1- c'y + fs = ■ ^ 9 which last equation being of the form of that in the first case of the preceding proposition, we have (c*-3bf)9f3 ^ 27afs-ci 5 and, consequently, since x—y + m, the value of this quantity will also become known. Ex. 1. Having given m—l in the equation 2 m3 - 1 - f\ it is required to find the values of x in the equation 2a;3 —1=.'ϊ3. Here, since 2 m3 — 1 = Is, we have m— 1, f— 1, a — 2, b—O, c = 0, d= — lr, therefore, a' — 2, b' — Q, c' — 6, and f—l. Whence jc'°--3bT)9f5_ ^ 27 a'p -c's ’’ so that x=y+m=— 1 + 1=0, or x — 0; that is, we cannot find a second value of the indeterminate equation: and this is no imperfection in the rule, for the proposed equation is impossible, except in the particular case of m— 1, as may be readily de- monstrated by the principles contained in art. 69·4lf> Indeterminate Problems Ex. 2. Required the value of x in the equation X2 ~l~ «X14~ 1 ^ 2»3, the known case being m— — 1. Here « = 0, b — 1, c = 1, d= 1; and, therefore, by the foregoing formula, we have a' = 0, V — 1, e'= — I, and/=l; whence (c^-3bT)9fs -18_ y 27a'/6 — cn ~ + * whence x=—18 — 1 = — 1$, which number will be found to answer the required conditions for 19^-19 + 1 =73· We shall here conclude our investigations with regard to those indeterminate equations in which there enters only one unknown quantity, and pro- ceed to those in which two or more indeterminates are concerned, which, notwithstanding their ap- parent difficulty, frequently admit of general so- lutions, aa will be seen in the following propo- sitions. PROP, VII. 19&* To find the general values of x and y in the equation xq -f axy -f hyq = zs. We have demonstrated (art. 100), that if m and n are the two roots of the quadratic equation φ2 — αφ + b = 0, the product of the two formulae (x -f my) and (x + ny), will be equal to x2 + axy -I- by1; or, writing t for x, and u for y, we have (14- mu) x (t + nu) = f + atu -f bu\417 of the Third Degree. it also follows from art. 101 (retaining f and u instead of x and y), that the product of any number of factors of the form t + mu is also of the same form; thus (t + mu) (t' + mu') = τ + mu, by making τ — tt' — btm', and u = tv! + t'u + aim'; and, in the same manner, we have (t + mu){t" + mu") = τ' + mu', where τ' — rt" — huu", and ν' = tu" + t"v + avu" % whence again, (t + mu) (f + mu') (t" + mu") = τ' + mu', and (t + nu) (t‘ + nu') {t" + nu") = τ' + nu': and, therefore, the continued product of these sht factors gives (τ' + mu') (τ' + nu') = τ'2 + ατ'υ' + ύυη. Now if in the above six factors we make t = f — t", and u = u' — u", our product will become {t + ?nu)3(t + nu)s — τ'2 + ατ'υ' + hu'*, or {f + atu + buf = τ'2 + ατ u' + bv"; that is, we shall have τ'2 + ατ'υ' + δυ'2 a complete cube; and it only remains to find the values of τ' and u in terms of t and u. Now, for this purpose, we have τ = tt' — huu', and u = tu' + t'u + auu'; or, since t= t' = t", and u—u' = u", these become T = f — bu2, arid u = 2 tu + au°. But we have again, τ' = τt" — huu", and u' = τα" +1"u + auu"; and making t" = t, u" = u, and substituting the above value of τ and u in this expression, we have418 Indeterminate Problems τ' = f — 3b tu1 — abu3, u' = 3 fu + 3atuQ + (a2 — b)u3. Hence, then, we have the general solution of the equation τ'2 + ατ'υ' + bvn = z3, or x* + axy + by1 = x2. For we have only to assume x=f — 3 btu* — abu3, y=3 fu + 3 atu2 + (a®—b)u3; and in these expressions we may give any values at pleasure to the indeterminates t and u, and we shall thus have x3 = (tf + atu + bu^f. Ex. 1. Required the values of x and y in the equation ,r2 + 3xy + by% = Xs. By the above formulae, we have a =3 and b— b, whence x = f — Ibtir — 1 bu3, y = 3t°~u + 9 for + 4u3. And here, if we assume t= 1 and u=l, we have x= 1 and y— 16; but if, in order to obtain a dif- ferent value for a?, we take t—b and u—\, then the formulae give x = 3b and 3/ = 124, whence Xs + 3xy + by* = 4b3; and it is obvious, that we may thus obtain an inde- finite number of values of x and y, by only chang- ing those of t and u. Ex. 2. Required the values of x and y in the equation f x* — xy + 2y-—%3.of the Third Degree·. 419 Here we have a = — 1 and h — 2, so that the general values of x and y are X—t — 6tu° + 2 U3) y — 3t*u — 3tu? — u3. And here, assuming t — 3 and «=1, we have ;i' = 11 and?/=17, whence < x1 — xy + 2 y* = 83. Ex. 3. Find the values of x and y in the equa- tion x*—7 y^—z3. Here we have a = 0 and b = — 7> whence the general expressions become x~f + 2ltu*i y — 3fu + ^iC. Assuming now i = 3 and «=1, we have x = QO and 3/= 34, which gives X2 — 7y* ss 2s. And an indefinite number of other values of X and y may be found, by changing the values of t and u* PROP. VIII. 197. To find the general values of x and y id the equation Λ?® + axy + by* = z*. By the foregoing proposition* we have (t + mu)3 = ϊ' + mv', and (t + nu)s = τ' + ηυ'. In which expressions we have τ' =f — 3 btu* — abu3, u·'=3 fu + 3atuq + (a* — 6)w!. 2 e %420 Indeterminate Problems Now, from what has been before demonstrated, (τ' + ηιυ') (t+ mu) = τ" + mu", (τ' + nu') (t + nu) — t" + nu", in which we have τ" = τ't — bv'u, and ϋ" = τ'u + iu' + αυ'ιι 5 and since (t + mu)3 = τ' + ηιυ', and (t + na)3 =τ' + ηυ', we have (τ' + ηιυ') (t + mu) = (t + mu)* = τ" + ηιυ", (τ' + nu') (t + nu) = (f + n«)4 = t" + ηυ".- And hence (by art. 101) we obtain (# + muY(t + nu)* = τ"2 + ατ"υ" + δυ"2, or (jf2 + atu + bu*)4 — τ"3 + ατ"υ" + δυ"'2. It therefore only remains to find the values of t" and u" in terms of t and u. Now we have t" = T't — bv'u, and u" = t'u + ίυ' + αυ'ιι; but τ' — f — 3 htir — abu3, and u' = 3fu + 3atu3 + (a3 — b)u3; and substituting these values of τ' and u' in the above expressions, we have t" = f — 6b fit1 — Aabtid — («2δ — δ3)η4, u" = 4fn + 6a f if + 4(a2 — b)tus + (a3 — 2ba)u*. Hence, then, we have the general solution of the equation, τ"3 + flT'u" + bv"\\or x° + axy + by- = s4; having only to assume for x and y, as above; viz. x = f — Qbfu1 — Aabtu3 — {orb — b')u*, y 4fu + 6a fid -|- 4(«3 — b)tv3 + (a3 — 2 ba)id,421 of the Third Degree. in which expressions we may give any values at pleasure to the indeterminates t and u, and we shall thus have %4 = (f + atu + bir)4. Ex. Required the values of x and y in the equa- tion a? + 7y» = *4. Here we have a = 0 and i = 7^ whence the gene- ral values of x and y will be X—t4 — 42/V -f 49^ y = 4fu—2&tu*. And as it is indifferent whether x and y he negative or .positive, we may assume t = 1 and u = 1; whence x = S and y = 24, which gives x2 + 7y1=s4. And' it is obvious how other values may be ob tained by changing the values of t and u, PROP. IX. 198. To find the general rational values of x and y in the equation x* + axy + by1 — zk. As the quantity x1 + axy + by1 is formed from the product of the two factors, (x + 7ny)(x + ny)9 in order that it may become a power of the dimen- sion ft, each of its factors must be also complete powers of the same dimensions. Let us therefore make (a? + my) = (t + mu)k, and (x + ny) = (£ + nu)'\422 Indeterminate Problems From the development of the first of which ex- pressions, we have, by writing 1, α, β, γ, δ, for the coefficients of the expanded binomial, (t+'muf=x + my = tk + a. tk~l (mil) + βΐ* ~%muy + ytk~s (mu)} + &c. Now, since m is one of the roots of the equation φ2 — αφ + b — 0, we shall also have nr — am + b — 0; therefore, rrt — ma — b, ni — in'a — mb — (at — b)m — abf "because a — (m + n), and b — mn, whence (it — b)m — ab = nt; and, in the same way, we find irt—(ct — b)nt — mab — (ct—%ab)m — ctb-sirVt·, ' * and so on for the other powers of m. We shall therefore only have to substitute these values in the preceding formula, and then we shall find that the expression will be compounded of two parts, one wholly rational, and the other a multiple of m; equating therefore the first with x, and the other with y, we shall obtain the general values of these quantities. And if, in order to simplify the result, we make a' = 13 = Q, a" =«, b" = b, a'" = ακ" — bit, b"' II — 6b', Aiv = «a'" v ^ V yet attempted, and is deserving of particular attention, not only on account of its difficulty, but because a general solution would be obtained with very great difficulty, if indeed it be at all possible to arrive at it by any other method. Here we must consider the product of these three factors (t + mu + nrw) (t + nu + n~w) (t +pu+p%w), m, n, and p, being the three roots of the cubic equation φ3 — αφ2 + ίφ — c = 0; and, consequently, m + n+p = a, mn + mp + np — h, and mnp — c. Now by the real development of the above fac- tors we obtain (t + mu 4- m%w) (t + nu + rtw) (t+pu + phv) = f + [m + n + p)tu + ' {m~ + it +p3)fw + {mn + mp + np)fot +426 Indeterminate Problems (win + nip + rim + rip +p°m +p%n)tuw + (rriii + nip* + n*p*)tuf -f (mnp)u3 + (ninp + rimp +p*mn)u*w + (nimp + nipqn + njfm)uivq + (m*n'pq)w3. And, since m + n+p = a, mn + mp + np — b, and mnp = c, : shall find that 1. --------- = 1, (m + n+p)- - - - - - - -a, (mf + ii’+p2) - - - - - - ^a*-2b, (mn -f mp -f- np) - - - - - = b, (mQn + mp + nm + rfp + p*m + pq n) = ab —3 c, (wfri1 -f m~p* + ηγ*) - - - - —i®—2«c, {mnp) - -- -- -- - = Cy (rtinp + rimp + p*mn) - - = aCj (rririp + nip~n + rijhn) - - - = bc, (m*n*p*) ------- = if. Therefore, making these substitutions, the product becomes f + afu + (a® — 2b) fw 4- btvf + (ab — 3 c)tuw + (If — 2ac)tw* + cii + aciiw + bcwuf + chv3. But any two factors of the form (t + mu + rriw) (if + mu' + mfiv'), always produce a product having the same form as each of those factors; and, therefore, the above formula will have this pxoperty, that, if we multiply together as many similar formulae as we please, the product will always have a similar form.of the Third Degree. 427 Suppose, for example, that it were required to multiply the above by a similar formula, t'3 + at'-u' + (a® — 2b) t'lw' + bt'un + {ah — 3c) t'u'w' + (δ2 — 2 ac)t'wn + cu'3 + acunw' + hcu'iv'2 + cw's. Since this last may be supposed to be generated from the multiplication of {t' + mu' + wV) {f + mi' + «V) (f +pu'+jrw), we have only to seek the product of the six fol- lowing factors, f (i + mu -I- m~w)[t + nu +rn%iv){t +pu +fw) \ (t'+ mu'+ mho')(f + nu' + rfw')(t' + pu' +p~w'): and, first, let us take two of them; viz. (i + mu + mlw) [t' + mu' -j- mrw), the product of which is ( tt' + m[tu' + ut') + mq(tw' + wt' + mi') + m3{mo' + \ wu') + m4ww'. Now, m being one of the roots of the equation φ3 — αφ® + δφ — c = Ο, we have m3 — anf + bm — c — 0, or m3 = amit — bm + c·, whence m4 = am3 — hm1 + me — (a* — b) nr — [ah — c)m + ac; so that substituting these values, and, in order to simplify the result, making τ = tt' + c[uw' + wu') + acww'} u = tu' + uf — h{uw' + wu') — [ab — c)ww', w = tw' + ivtf + uu' + a{uw' + wu') + (a® — b)ww', we shall have {t + mu + nfw) ψ + mu' + mV) = τ + mu + mV;428 Indeterminate Problems and, in the same manner, we obtain (i + nu + n?iv\ f + nu' + nrw') — τ + «u + wHv, (t +pu + phv)(t' +pu' +]f'w') =t +pv +p!w. And, therefore, the product of the six foregoing formulae is the same as that of the three, (t + mu -t νι2ηήί'τ + nv + »V)(t +ρυ +j?vr) — τ3 Jr ατ'υ + i d3 — 2&)xV + δτυ24 (ab — 3c)tuw + (¥ — 2ac fry?* -f cu3 + acuhv + beuw2 + c3w3. Now, making t — f, u = u', w = w', this last formula becomes equal to the product (i + mu + mhof(t + nu + n"wf(t +pu + prw)2, and is therefore a square, and the values of τ, u, and w, before determined, now become τ =t~ + 2cuw + aciv% . u = 2 tu — 2 burn — (ab — c)w~, w = 2tio + 'it + 2 auw + (a* — b)io". We have, therefore, the general solution of the equation above given; viz. τ* + ατυ + [a1 — 2b tw + έτυ2 + (ab — 3c)tuw + (b2 — 2«c)tw2 + cu3 + acifw + bcvw2 + cV = z2. But in order to make this apply to our equation Xs + ax'y + bxy2 + cif — z%, we must take x=t, y — u, and 0 = w, which re- duces it precisely to our case. There! ’bre5 when it is required to find rational values of x and y in the equation x3 -f ax~y -f bxy*"+ cy3 == s2r we must have, firsts 2tw + ir + 2 auw + (a2 — b)w2 = 0; or u~ + 2 auw + (a2 — b)w2 fr — — ~ ■ · 2 wof the Third Degree. 429 Then we obtain, by writing x and y for τ and u, x = f + 2 cim 4- acid1, y = 2tu — 2 hum — (ah — c)id\ in which expressions u and to may be assumed at pleasure, but the value of t will depend upon the equation id -f 2auw + {d2— b)uf ί t = 2 w Cor. When any of the coefficients a, b, or c, become zero, the result is modi simplified; thus, if a~-0 and b = 0, or the equation takes the form xs -f cy3 = .d, then the values of x and y will he expressed by the formulae - and 2w X = f -]- 2CWW, y = 2 tu -f cur; or, by substituting for iv, we have cu3 { a?- cu* y — 2tu + -jjj , or At x = 4t* — 4cu*t, y — Stu + cit; the last formulae being found by reducing the two first to the common denominator 4f, and then re- jecting it in both the values of and y. Ex. 1. Required the values of x and y in the equation xs + y=sh430 Indeterminate Probtems Here we have c— 1, and therefore the values of x and y are expressed by the formulae f x=4t* — 4ust, \y—Sfu + u4t where t and u may be assumed at. pleasure; if we take t — 1 and u — — 3, we have x = 112 and y = 57» which gives x?+y5 = z®, or 1123 + 57®= 126ls; and other values of x and y may be found by changing those of t and u. Ex. 2. Required the values of x and y in the equation .τ’ — 3y3 = zs. Here c = — 3; and, therefore, the general values of x and y may be represented by the formulas ( x = 4f + 12lit, \ y—'8fit — 3m4; where, by taking t = 2 and u= 1, we have x — 88 and y = 6l, which give 88’ —3.6l3 = 23s; and if we had taken £=1 and u= 1, we‘should have had t= 16 and y — 5, whence 16’ —3.5’ = 6ls; and an infinite number of other values may be found for x and y, by changing those of t and a. Ex. 3. Find the values of x and y in the equation x’ + δί/3 = 2®. Here c = 5, and hence the general values of x and y are f x~4t* — ‘lOiv’t, \y— &fu + bu*i,431 of the Third Degree. by taking t — 2 and u= 1, we have x — 24 and y = 69, whence 24* + 5.69s = 1287*; and, by changing the values of t and u, an infinite number of integral values may be found for x and y. Ex. 4. Find the values of x and y in the equation 4- 2x*y + 2xyl + y3 = z*. Here we must have recourse to our first values of .rand?/; viz. if + 2 mm + (a® — b)nf 2 w In which expressions we have a = 2, b — 2, c= 1, and u and w indeterminates that may be assumed at pleasure; by taking u = 1 and w = l we have l+4+2_ 7 ' 2 ~ 2' x—- 49 4 ' +.2- 65 4~’ y = 7-4-3= -14; which values of x and y answer the required con- ditions of the equation, as will also the integers x — 65 and y — — 56.432 Practical Exampled; Practical Examples. 1. Required the value of x in the equation •t3 + 3 = »2. -23 1873 Am. or—. 2. Required the value of x in the equation 3x3 +1 = z3. , -5 8 -629 Am. *=-g, or-, or— 3. Required the value of x in the equation 2x4 — 3x3 + 2 = z~, and x4 — x3+l = z3, or prove that such values cannot be found, except in the obvious case of x= +1. , Ans. Impossible. 4. To ascertain the values of x in the equation A'4 + 8x’2+1=^. Ans. ar= 2, or 15 58 —···, or---. 28’ 2911 5. Required the values of x in the equation x3 + 2 = z3. 383 Ans. x = 5, or 1000 6. Required the possibility or impossibility of the equation ,r3 + l=s2, except in the known case of x—2. Ans. Impossible.Practictil Examples. 433 To find the values of x in the equation 3x3 + 3 = s3. Am. x — 2, or -20 8. Required the values of x in the eqiiation -1090 .A + 4 = z3. Am. x=ll, or 9. To find the value of x in the equation ^ + 7 = z4. Ans. x= 27 367 144* 10. Reqiiired the integral value of x and y in the equation λ?3 + 7xy + ^y~—%^· 11. To find the integral values of x and y in the equation x't-\-7ytt=z>· 12. Required rational and integral values of x and y in the equation of + 7y%—%*· 13. Find integral values of x and y in the equa- tion 2xi — 7if — 16z\ 2 S434 CHAP.Y. On the Solution of Indeterminate Equations of the Form xn— b = u(a). Or the Method of de- termining x> such that x" — b may be divisible by «. prop. r. 2O0V . To ascertain the possibility or impossibility of every equation of the form a?-*· b = u{a),. and the number of solutions in the former case, a being a prime number. First it is obvious* that, if b — a, or any multiple of a, the equation admits of an infinite number of solutions, by assuming x — a, or any multiple of a; we shall therefore only consider those cases in which b is prime to· a. Let, then, b be prime to a; and suppose, first, that n and a — 1 have a common measure w; that is, suppose n = riw, and a—l — a'zc, then I say, that, if the equation admits of one solution, it will also admit of w solutions,, and no more. For, if the equation be possible, we shall have xx” — b = M(ay.· But, since o is a prime number, \ — Μ,(α). (art. 87)·..indeterminate Equations, &;c. 435 And now, if by art. 159 we find two other num- bers, p and q, such that n'p — a'q=\, ov n'p = a?q+li We shall have, by means of this and the foregoing equations, which, by rejecting the multiples of U, may be written thus, xn'wtt; *b, and x“'w tfc l, the following results; viz. b^x^^xf^^^x^, or xwzp,b!J; whence x’e — bp = m (a). Therefore, when the equation is solvible, x must have such a value that xa, when divided by a, shall leave the same remainder as br divided by a; but (by cor., art. 87) the equation χύ — c = M(a), r will have w different solutions, and no more; arid, consequently, when the proposed equation is pos- sible, it will have also w solutions, and no more. Now, with regard to the possibility of the equa- tion, it will depend upon that of o-l b w — 1 =m(o) ; rt *-1 that is, if b w —l be divisible a, the proposed equa- tion will be possible, but otherwise it will not. For, since xn'wztzb and xaa wl, we have ¥ mxn>a'wefc lB' ep. 1, or ¥ 1; * This character indicate^ that x\ is of the same form as b to modulus a, or that their remainders are equal when divided by a. 2 F 2436 Indeterminate Equations but a — l = a'w, therefore a' = ——-; and, const'-- w a- 1 quently, b w tfc 1, or a- 1 & ω — 1 ==Μ(α), which equation must necessarily have place when the proposed equation is possible; and, therefore, by means of this, the possibility or impossibility of the proposed equation may be readily ascertained; and, in the former case, the number of its solutions Λγϋΐ be to, as we have seen above, the whole of which are contained in the equation xw — bp = And it is obvious that, when one of these solu- tions is obtained, the others will be found by mul- tiplying the known root by each of the roots of the equation x*— 1=μ(ο); for if rn divided by a leave a remainder by and r'n divided by a leave a remainder 1, then will rnr'n divided by a, also leave a remainder b; therefore, if r be one root of the equation * % xn — b~u(a)y and r', v/f, rA//,&c., be roots of the equation af—1=m (a)y the other roots of the first equation will be Ty rr'y rr"y rr'", &c. We shall, therefore, after illustrating what ha» been taught by two examples, proceed to the solu- tion of this last equation. Cor. 1. If n>a — 1, We need only consider theof the Form xn — b = u{a). 437 remainder arising from tlie division of n by a — 1. For since χ?~ι — 1 — Μ(·α) (art. 87), or, according to our contracted notation, we shall have ^m(a~ . that is, will leave the same remainder- as . af, when both are divided by a. Cor. 2. It follows also from the above propo- sition, that when n is prime to a — 1 the equation is always possible. For in this case £0 = 1, and, there- fore, x^p.bv, the exponent p being deduced from the equation pn — q{a— l)==l. Ex. 1. It is required to ascertain whether the equation i»_.l.l=M(29) be possible in integers. By the above proposition, if this equation be possible, so also must α- 1 hw — 1 = m (a). Now, in this case, a = 29, 5 = 11, and tv — 7; and, therefore, this last becomes 114-1 = m(29); which equation being impossible, the proposed equation is impossible also. Ex. 2. Required the number of possible solu- tions that may be given to the equation £δ^2 = Μ(3ΐ),438 Indeterminate Equations Here a=31, b = 2, and zo = 6; and since we hav$ 31-1 2~"F~-1=m(31), the equation is possible, and admits of six solu- tions. prop. ir. 201. To find all the values of x in the inde- terminate equation xn— l=M(a), a being itself a prime number. Case 1. When n is prime to a — 1. Here we shall have (by art. 87, and by writing r instead of x) ra~'— 1=m(o); and, consequently, r(«-i)»_ i =Μ(α)? or 1 = m(«) ; and, therefore, (ra~1)n = xn, or x = ra~' = l; that is, x = l, which is the only possible solution in this case. Case 2. Let n he a divisor' of a —\. Since we may here make a—l—a’n, we shall have (by art. 87) ra’n— 1 =m(«) ; and, consequently, ra'n = xn, or * = /, where r may be assumed any number whatever prime to a. If now we make ra'^r', r' being the remainder arising from the division of r"' by a, then, since r,n — 1 = μ(«), we have also r,M-l=M(a);439 of the Form wn~b~M(a). therefore, if r' be one root, r/m will be another, whatever value we give to m9 and since the equa- tion r'n - 1 = μ(λ), can have but n solutions, or roots, these will be found, either wholly or in part, in the series, r', r'% r'3, r'\ &c. r^1; that is, this series, or the remainder of each term when divided by a9 will furnish all the roots of the proposed equation, if these remainders be all dif- ferent from each other, but they will give only a part of the n roots, if any two .or more of them leave the same remainder. Remark. When the root r' is such that the terms of the above series leave different remainders, then r/ is said to be a primitive root of the equa- tion x11 — i = m(o) ; and as we shall have frequent occasion to em- ploy these quantities in chapter vii., it will not he amiss to demonstrate here some of the principal properties of these roots, after which we will give a few examples by way of illustration. 202. tion prop. in. If r be a root of the indeterminate equa- xn — i = M(a), and such that rm — 1 be not divisible by a (m being any divisor of ?i)9 then I say, r is a primitive root; or, which is the same, all the roots of the above equation will be contained in the series440 Indeterminate Equations r, r2, r3, r4, &c. rn~'; or the remainders of these, when divided by a, will he all different from each other. For, if possible, let any two terms of this series give equal remainders, and let them be denoted by rp and rq, then it is obvious that we shall have rp —rq = u(a), or rp~9 — l =m (a); or, making p — q = s, it becomes ' r‘—1=m(«): and let the common divisor of n and s be k, which will be unity, when n and s are prime to each other; and if now, as in art. 200, we resolve the equation np- — sq' — k, or np' = sq' + k, we shall, as in that article, have this result, rnpf z&r$q'+k; and since, by hypothesis, rn— l=M(a), and r‘— l=M(a), we shall have, by rejecting the multiples of a, rn^t 1; therefore, rnp'^rfq'+h^rk^l; that is, rk— 1=m(«). Now, since s—p — q must necessarily be less than n, and since k is the common divisor of s Ύί and n? we may make n = mk^ or — = and s — \ and^ consequently, rm- 1 = M (a),441 of the Form xn — h = M (a). which is contrary to what we have supposed; there-* fore, no two of the terms in the series r, r3, r3, r4, &c. r”"1, can leave the same remainder, and, consequently, r is in this case a primitive root of the proposed equation; and, therefore, all its n roots will be found in the above series. Cor. It follows from this demonstration, that, if n be a prime number, every root r, of the equa- tion xn—\ = u(a), is a primitive root, and will give, by its successive powers, all the roots of the proposed equation. Thus, for example, since 3s —1 = m(1 l), we shall have also 9*’ — 1, 5s —1, 45— 1, each dir visible by 11; or, which is the same, 3, 3a, 3s, 34, or their remainders when divided by 11; viz. 3, 9, 5, 4, for the roots of the equation Xs— 1— m(ii). PROP. IV. 203. If m, p, q, &c. be different prime divisors of n, then will the number of primitive roots of the equation xn — 1 = m(o) be expressed by the following formula, m— 1 p — 1 <7—1 n x----x -----x ---, m p q &c.442 Indeterminate Equations For (by cor., art. 88) there are only n values of x, that satisfy the equation xn— 1 =m(o) ; 71 and there must also, bv the same article, be —- values only that fulfil the condition of the equation n * xm —i = m(u); and, consequently, out of the n first roots, there 71 are n — —, that will not answer the last condition; 771 and, in the same manner, we find there are n — that will not fulfil the conditions of the equation n xp — 1 =M(a); and proceeding thus with all the factors of 71, we ascertain, finally, from the same principles as those employed at art, 24, that the whole number of primitive roots is expressed by the formula Til — 1 p — 1 Q — 1 n x------x in 1 q — x — &c. P a. e. d. Cor. 1. If n be a prime number, every number that is prime to a is a primitive root. Cor. 2. If n be any power of a prime number, as n~ mp, we must assume such a root r for x> that the equation — 1 ~ m (a) has not place, then will the successive powders of r be the roots sought.443 of the Form x" -~b — m{a). ob /3 *Y jnake m —p = f, q = μ.", by rejecting tire multiples of a, if these quantities are > a, and then resolve the separate equations u, jj,* v x — 1=m(a), x — 1=m(u), x — 1==m(«), Now supposing the roots of these equations to he r, r', r”, the root of the proposed equation will be rrV'; and the other roots will be the successive powers of this last quantity. Ex. 1. Required the seveq values of x in the equation «7-r 1 = m(379). Since 379 — 1=7-54 we have x = ru, where r in ay be any number prime to 379 (art. 201). Assume, therefore, r— 2, and we have, by reject- ing successively the multiples of 379, riaefc306, r**s*:23, r48=fci50, r5,^l25. Therefore, x — l 25; and since the power 7 is a prime number, this root is a primitive root, and gives, by its successive powers, or by their remain- ders, all the seven roots of the proposed equation ; that is, x= 125, 1254, 125s, 1254, 125s, 125°, 1257; or «=125, 86, 138, 195, 119, 94, 1; which last are the seven roots required. Ex- 2. It is required to find the values of x iu the equation x63-i = m(379). Since 63 = 7 - 9 we may (cor. 3, above) resolvethe. two equations444 Indeterminate Equations χΊ — 1 = m(379), and x9 — 1 = m(379) ; the roots of the first being r = 125, and of the second rf—180; and the product of these, rejecting the multiples of 379> is 139, which is one of the roots of the proposed equation, the others being contained in the series 139, 139\ 139s, 1394, &c. 1399. PROP. V, 204. To find the value of x in the indeter- minate equation x9'1 + 1 = m(«), β being a prime number, and An a divisor of a — 1. Find the general value of x in the equation X4n — 1 =m(«) by the foregoing propositions, and let this general root be represented by r™, then will r2p+' be the general root of the proposed equation χ'π + 1 = m(«), where p may be taken any number whatever, For, rm being any root of the equation X4b-1=m(«)j it follows, that r2* is a root of the equation X°"— 1 ; because rm, being substituted for x in . x4B — 1 =m(«), is the same as rim, substituted for x in the equation χ·π~ i =jvr(a). Now Λ'4" - 1 - {x2B - 1) (x2b + 1) ; and since the first of these factors has for its roots445 of the Form xn — b~te[a). all the even powers of r, there remain all the odd powers of r for the roots of the other factor, which is the equation proposed. Ex. Required the values of x in the equation x56 + 1 =m(433). First, the solution of the equation #72 — 1 = m(433), by proposition 2, gives # = r6, because 433-1=432 = 72x6. And by assuming r=5, we have 56 = 3/, reject- ing as before the multiples of 433; and, therefore, 372/,+1 = .r, is the general root in the proposed equation* which, by assuming /; = 0, 1, 2, 3, &c., and re- jecting the multiples of 433, we have the following solution: r== f ±37, 8, 127, 203, 79, 99, 2, 140, 159, \ 128, 133, 216, 35, 148, 32, 75, 54, 117, the sign ± being common to each of the roots. PROP. VI. 205. To find the values of x in the equation af — b — M(a), h being such that bm ± 1 is divisible by a, and m a divisor of a— 1 n This proposition divides itself into two cases, viz* first when n and m are prime to each other, and second, when these quantities have a common measure.446 Indeterminate Equations Case 1. When n and m are prime to each other* Find two other quantities* p and q, such that pn — qm = l, or pn = qm+ 1; then will x = bpy be one root of the equation sought* y being itself a root of the equation y—(± i),=m(a). For, by making x — bpy, we have xn ;*» bpnyn bqm+lyv byn and, consequently, af — 5=M(a)- Case 2. When n and m have any common divisor w* Let n = n'tv, and find the values of p and q, such that pn'—qm = 1, or pn' — qm + 1 ; then we shall have xw = bpy, or xw — bpy = M(a), y being one of the roots of the equation y'-(±l)’ = M(a). For, by making here xw = bpy, we have S*s bpnY efe bqm+Y ete by* t* 6 : and, consequently, xn — b = u{a)9 Remarl·. By means of the aboTe proposition, we are enabled to convert a number of equations, ?uch as xn — b = M(a) into others of the form af±l=M(a). It furnishes ns also with the means of resolving, in an infinite number of the cases, the equation 3;n — b = yi(a)}of the Form xn — b~M{a). 447 into n' equations of an inferior degree* as will ap- pear from the following examples. Ex. 1. Required the values of x in the equation a?3 4-49 = M (223). First* since 223 — 1=3.74* and (-49)74- 1 =m(223), the proposed equation is possible (art. 200); which fact being ascertained* we have m = 74* and it now remains to find 3p — 74y = l* or 3ρ=74<7 + 1* which equation gives p— 25. Whence x={-A^Ty, y being a root of the equation 3/3-1=m(223), the general form of which (by art. 201) isy = r74* where r may be assumed at pleasure; and* there - fore* the required root x* of the proposed equation* is * = (-49)°-Vr74* the remainders of which* when divided by 223* will be the simplest form of the root sought; thus we find the required roots are x=— 36* —66* +102. Remark. We should have obtained this solution more readily by first solving the equation x3 4.7 =m(223)* the three roots of which* squared* would have fur- nished the roots of the equation proposed. And this method may be employed in all cases448 Indeterminate Equations in which b is a complete power; for, generally, if r be any root of the equation χπ — JiM(e), we have rk for a root of the equation xn — bk — u{a)i -Ex. 2. Required the value of x in the equation x6 + 20 = m(6i). First, since 6l — 1 =6.10, and b = — 20, we have (— 20)10—1 ==m(6i), or (— 20)5 + 1 ==m(6i); therefore the proposed equation is possible (art. 200); and since this last exponent 5, or (m), is prime to that proposed 6, or (n), it follows, from the first case of the preceding proposition, that x = bpy, p being first found from the equation 6p — 5q = 1, ov&p — bq + l, and y from the equation yn+ 1? = Μ(βΐ); therefore p — 1 and q = 1; also (by ai't. 204) the general root of y is 2,92,+I; and, consequently, the* general value of x, in the proposed equation, is x = -20.29ai:+I, which, by involving and dividing, gives •£=±7, ±24, ±30. Ex. 3. To find the values of x in the equation x10 — 5=m(601). Here we find be + 1 =m(6oi), and since 10 and 6 have a common measure 2, xvti shall have, by the second part of the above pro- position, x® = bby, or x2 — bsy = μ (6o 1),of the Fotm xn — b = m(«). 449 rj beiiig a root of the equation /-1=m(6‘oi), the general root of tvhich is y = (— i6»9)*; and thus the proposed equation may be transformed into the five following ones of the second degree; viz.. —120= m(6o1 ), i.,? — 154— m(601), X1 — 276 = M(601), X® — 234 == M(6‘01), a·5 + 183 =»i(6‘0l). s PROP. VII; 206. To find the value of x in the equation xn — b = M(a), in which if — 1 = u{a) j .a—1 w bemsra divisor of n Let; x — rm be the general root of the equation . a'"c-l = m(«), now, since b is found in the series rn, r**, f9B, rin, &c., #**-·>■> . let the term ip which it is contained be r^n, then will the general root of the proposed equation be x — r V+VifJ' For, since rmw+'J'=x, we have £>i = γ,ητηιν+η{ί ^ and, consequently, It therefore' only remains to be demonstrated, that b must necessarily be found amongst the re- mainders of the series r”, r~x, r9”, rin, &c., r'u'~l'n. 2 G450 Indeterminate Equations Now, because rm is the general root of the equa- tion — l = m [a), we shall have (,.«»)»_ X=M(«). that is, ·/·”’* is a general root of the equation — 1 = M (a) and since, also, if —l = m(«), it follows, that b must necessarily fall amongst one of the remainders corresponding with r,an; that is, in one of the terms of the series r’% τΐη, r3", r4n, &c., Remark. There is no exception to this method of solution, but it will sometimes be very laborious to find b in the above series of roots. Ex. Required the value of x in the equation a·10—5 = m(6oi). We have already considered this example, and have decomposed it into five equations of the second degree ; we shall now attempt the solution on the principles of the last proposition. Since b= 5, we have, by rejecting the multiples of βθ1, — 1 and Z>'st£l; thus «> = 12. Now the complete solution of the equation yao-i=to(6oi), found by article 201, is «=( — 140)m, and, con- sequently, « in the equation «'*— 1 =m(6oi), ■ · is « = ( — 140)wm =120”; therefore, b ought to be contained in the formula 120”1, and we find thisof the Fohn 0?~bi*M(a)i 45i Succeed in taking m—5. Therefore, the complete! Solution of the proposed equation is # = ( — 140)5+ia”, or *=214.(169)*, from which expression result the values *— ±214, ±106, ±ll6, ±229, ± 237·. Remark. We might have-pursued this subject much fartlier, by finding the value of * in similar equations, in which the divisor is any power of a prime number; and, filially, for any composite number whatever i but what has been said will enable the ingenious reader to arrive at the solution of these cases, and Others, that may arise, by the application of the rules and principles laid down in the foregoing pages. 207. Scholium. In all the propositions which have been hitherto the subject of our inquiry, we have been able to pursue the investigations, and derive the results of ortr operations, by means of certain rules and principles, as direct and satisfactory as in any other branch of algebra; but in what follows; few Or no rules Can be given, and consequently much ifmst necessarily be left to the skill and ingenuity of the analyst himself: still, however, the results that have been obtained in .the preceding chapters Will be found of essential service in Our future inquiries, nothing more being requisite than a judicious application of them to the various cases that may occur; and it will therefore be Convenient, for the sake of reference, to have exhibited here, in the form of a table, such of the foregoing resulting formulae as are most commonly employed in DioK phantine researches. 2 g 2452 Table of Indeterminate Formula?, Table of Indeterminate Formulae. FORM I. Equation ttX—by—± e. General value of x = mb ±cq, - ~ ~ ~ ~ y — ma±cp./ In which expressions tn is indeterminate, arid the values of p arid q result from the solution of the equation ap — bq—±\ (art. l6o)» ii. Equa. ax + by = c. General .value of x — cq — mbi - - - - - y = ma — cp. Nuin. of solutions —. b a The quantities p and q being ascertained as above > also m indeterminate (art. l6'l). III. Equa. ax 4- by + cz = d. General value of x=(d— cz)q — mb, - ~ - — - y==Ma—{d—cz)p, The quantities p and q being found as above j . (1 also m indeterminate, and % any integer <- (art. 162).453 Table of Indeterminate Formulae. IV. Equa. x~ — ay1— z^. General value of x =p- + aq‘, " - - - " U = ‘P(b - - !■ r - z =p2 — aqa% In which expressions a is given, and p and q are indeterminates (art. 171). v. Equa. + ay1 — z9. General value of x=p* — aq~, - - - - - y=2pq> - - - - - 2=jp2 + aq2; β being given, and p and q being indeterminate* as pbqve (art. 171), VI. Equa. ax' + bxy + yi = z:. General value of x — 2pq + bq*, - - - - - - - - - 2 =p* + bpq + a 8).454 Table of Indeterminate Formulas. VIII. Equa. + + General value of x p — cq% bq% — 2mpq} _ mp* + mcF—bu3, - - - - - z = F + biF. Note. This is deduced from the foregoing one* by making «=0, XXI. Equa. sF + axy + by*=z*. Gen. value of x=F — 6bFiF — 4abfu3—(a^b—$*)«% . ( 4fu + 6«#V + 4 (a® -rp) tu3 + (a® V- ^ — 2ab)u*i - - - z = f + atu, + δη®. Where a and b are known quantities, and t and u indeterminates which may be assumed at pleasure (art. 197). XXII. Equa. x* + by*= z*. Gen. value oTx = F — GbfiP + b3u*3 - - - y = 4fu — 4btu3i - - - z—F+btf. * Note. This form is deduced from the foregoing one, by making a = 0.Table of Indeterminate Formulae. 45$ XXIII. Equa. xq + by1 — zm. Gen. value of x — tm — fit"l~‘1u'>b + &c., τ'- s- y = utm~'u — ytm'3u3b + stm~iuhb~ — - - - z = f + bit2. In which expressions b and rrj, are known quan- tifies, as are also l, a, β, y, δ, s, &c., these letters representing the respective coefficients arising iron* the binomial (t + u)m; but t and u are indetermi* »ates that jnay be assumed at pleasure (art. 198)» ' XXIV. Equa. x1 + cy3 — z*t General value of x = 4f ^ 4ctu5, τ - - - * ■ y = 8fu + cu*} - - r - - 2 = £3 + CM3. Where c is given, but t and u are indeterminate» (cor., art, 199). xxv. Equa, 3? + qx"y + bxy* + cy3 = z*. u9 + 2«mv -f (a9 — 7;) 10® Particular value pf i= 2 iv General value of x = f + 2cuw + acw~, - - - - - ys=2tu — 2buw—(ab — c)tvl. In which expressions a, h, and c, are any given quantities, u and tv indeterminates; but t is limited by the above equation, and depends upon the values of u and w (art. 199)·4βθ CHAP. VI, The Solution of Diophantine Problems. PROB. X. 208. To divide a given square number into two other square numbers. Let a" represent the given square, and x® and y% the required squai'es; then we have only to satisfy tjxe equation { d—x®+y~, or d — if — a;2. Jn order to which, let us assume px •**m? ax a-y=—. P From which we readily deduce ,

JiQ.— "r ■. j* n p p(i _px qx ^ (p1 — “ + q2 may be a divisor of 6‘o, as in prob. i. prob. v. 212. To find a number such, that two given squares being each subtracted from it, the two re- mainders may also be squares. Theorem. Let dr and ¥ he tl>e two given squares, and resolve ~— into any two unequal factors, m and m', and ~~~ into any two unequal factors n and r/, then will (»»* + n%mn + n'1) be the number, required. Demonstration. For, by art. 9L , ϊ, βΛ / m f {mm' + ηηΎ + {mu' — m'nY, v ’; ’ ( {mm — nn ) + [pin + mn). And since = mm', and ——— = nn', therefore a = mm' + nn', and b = mm' — nn'; and, consequently, the square of each being taken from the above product, will leave a square re- mainder, 2 h ‘2468 Diophantine Problems. Cor. Hence, when the two given squares are? both even, or both odd, the question will admit of one or more solutions in whole numbers, according to the number of different ways in which ———, and may be resolved into unequal factors. Suppose, for example, 18s and 22 were the tw@ given squares; here’ 18 + 2 ' = 2x5, or 1 x 10; and 2 18 -2 = 2x4, or 1 x 8. Then the number of solutions will be four, which are as follows; viz. (22 + 2s) X ( 52 + 4s) =328, (2a + 12) x ( 52 + 8') =443, (1® + 2a) x (10* + 4a) = 580, (l2 + l2) x (102 + 8) =328. Two of which values of the required quantities are equal, because the first factors of m and n are equal. So that, in fact, we have only three solutions ; namely, . Γ 328 —2* =18*, 328— 182 = 22, < 445 — 22 = 212, 445 — 182= 1 la, t. 580 — 2a = 242, 580— 182= l62. But fractional solutions may be found ad libitum. PROB. VI. 213. To find two integral numbers such that, unity being added to each, as also to their sumDiophantine Problems. 4f>9 and difference, the four results shall be complete .squares. Let x and y represent the required numbers, it is required to find {'x +1 = wi®, y + 1 = n~, a; ++ 1 x—y + 1 Now here it is obvious, that the three squares r~, m2, and p2, are in arithmetical progression, their common difference being y. Let us, therefore, represent these three squares, according to prob. iii., by C s1 = (4pqr-p2-2q2f, < ml = (p2 + 2ip — 2pq)2, L r'2 = (p2 + 2(f)2. Then we have, for their common difference, y = 4psq — 12p2(f + 8pq3; and all· that is required is to find this quantity, plus 1, a square, or 4psq — 12p2q2 + 8pq9 + 1 = ri\ Assume, therefore, η — 1 + 4pq3; then we have, by squaring and cancelling the like parts, 4p'q — 12p~cp — 16p*qe. . Whence, p-4qs + 3q; in which expression q may be assumed at pleasure. And thus the general values of x and y will be determined; viz. by first making p = 4q5 + 3q, and then470 Diophantine Problems. ( x= (jo2 + 2j® — 2pqY — 1, \y = (l+4pq5y--l. By taking q — 1 we have p~7> whence x —1368 and y~ 840; which numbers answer the required conditions: for { 1368+1 =37% 840+1 =29% 1368 + 840+ 1=47% 1368-840 + 1=23% PROB. VII. · 214. To find three or more numbers, such that the sum of their cubes may be a square, and if from this sum the square of each of the quantities be subtracted, the remainders shall be squares. Let x, y, and z, represent the three numbers, then the conditions required are as follows; viz. x? +y* + z3 = or < φ2 = 5® + y", Φ4—x2= f-,j &c. &c. &c. &c. Which, in their present form, appear to involve considerable difficulty; they are, however, rendered very simple, as follows: Assume any square number, a% and, by pro- blem i., resolve into two square numbers, as many ways as the problem requires; thus, a 2 = a" +b% A* = a'*+b'\ a* = a!n + δ"2. &c. &c. In which equations all the quantities that enterlyiophantine Problems. 471 are known; but these expressions will obtain also, if we introduce any indeterminate square rtf·. thus, A®_ cf ¥ rtf rtf nr* A® an V* rtf rtf "** rtf ’ A® a"~ b"2 O 0 H- 2 * m m m See. & c. And these will evidently answer the required conditions, if we make ¥ a ψ l·"3 —=y1, nr J nf = r providing m be so assumed that b"3 A® ¥ V3 O ~~3 + m m m + Sec.» which gives ni- ff + bn + bm + &c. And thus we have immediately the solution of the problem proposed. Ex. Assume a = 65, then, by problem i., we have 65®=63®+l6®, 65® = 56s + 33®, 65® = 6θ®+25®, 65®=52®+ 39®. Whence, b= l6, &' = 33, b"=25, b"'= 39, and ____l68 + 33® + 25s + 39s _ 114977 m~ 65® 4225 ‘ And, therefore, from the foregoing formula:,472 Diophantine Problems. l6x4225 67600 . τ *= 114977 = 114977’ 33 x4225 139425 114977 “ 114977’ ___25 X 4225 ^1D5625 *= 114977 = 114977’ 39x4225_164775 w~ 114977 — 114977* Which are four numbers, such that the sun* of their cubes is a square; and if from that sum the square of each be subtracted, the four remainders are squares. Remark. This solution deserves particular at* tendon, as it would be perhaps difficult to solve the problem in any other way; it is also applicable to various other questions of this kind. PROB. VIII. 215. To find three integral square numbers, such that the sum of each two, with double the other square, may form three perfect squares. Let af, if, and represent the required squares, and we have to find f of +1/ + 2z' = rs, < x% rh + 2?y" = S', (. y~ + 2Γ + 2x2 = t~. Since these quantities are all integers, it is evi- dent that we may suppose them prime to each other; they must also be all odd numbers, as will appear by considering the possible form of squares to modulus 4.Diophantine Problems. 47S Let then, y==x +2p, and z = x + 2q, and we shall have, from the first two formulas, of + y* + 2 s2 = 4x4 + 4(p + 2 q)x + 4 (// + 2 q*), χίι + z° + 2y~ = Ax' + 4( 2p + + 4 (2p~ + which expression must be equal; make, therefore, f / + 2q- ~/3 = 2f + 92- g2, \2f-p^2q =2g-2p-q, from wliicli equations we readily deduce the fol- lowing general values of f and g; viz. + 3p); g = ±(5p + 3q). And by substituting these values for f or g9 in the above expressions, forx, Λνβ obtain _ 7p~ ~ 30pq x~ Ήρ+ν) This value of x will satisfy the first two com ditions, and we shall have, by means of this, the corresponding values of y and z, because y=x + 2p, and z=x + 2q; so that, by multiplying each of these quantities bv the common divisor 8 (p + q), we have x= 7pl — 30pq+ 7<72, y = 23p2 — 14pq + 7q*, Z == 7P* — 14pq + 23 ^2,4/4 Diophantine Problems. And with these expressions, in which p and q are indeterminates, it l’emains for ns to fulfil the third Condition, i/2 + + 2x° — f. Now, in order to simplify, make p = q + $q, and we have vr = (7Φ3 — ΐβφ — #=(23φβ+·32φ+ l6))q, which, being substituted for p in the above equation, we have y* + z* — x‘i = q4(<,p4 + 8φ3 + 20φ3 + ΐ6φ + l). Whence, again, from form xv., di±8mf3 — 4cf*_—23 ^ 4bf2 + 4md,J‘ 4 ’ T>ecausem=l, b = 8, c = 20, d=l6, andjf= 1. — 15 Butp=(2 + <$>)q, or p~ ——q; therefore,p—15 and q = — 4: whence x=p* + qi =241, y =p2 + pq — = 149, z =pq —pq — f = 269.4f6 Diophantine Prohleftts, Which numbers answer the required conditions, and others might be found, by the other formulae given at form xv., as also by changing the value of q*. Practical Diophantine Problems. 1. To find a + x and β — χ both squares, and to point out the limits of possibility with regard to the form of a. Jins. atfcP + M2· 2. To find x1 +y + s’ = <ρ3 a cube. Am. + Τ + 3. To find two numbers whose sum is a square, and also such, that each being added to the square of the other shall be a square. Am. Any two numbers whose sum is 4. To find three numbers in arithmetical pro- gression, the sum of every two of which shall be a square. Am. 120X, 8404-, 1560-1. 5. To find three numbers such, that the pro- duct of every two, plus the sum of the same two, may lie a square. Am. 4, q, 28. " it was intended to have added here the solution of several other Diophantine problems, but this work having already ex- ceeded the limits which the author had prescribed to himself, he is under the· necessity of cancelling the solutions of several questions, and placing them among the following practical ex- amples.Diopharitine Problem^. 6. To find three such numbers, that each being added to their product shall be a square. Ans. -, —< 9 9 9 7· To find two numbers, whose difference is equal to the difference of their squares, and* the *um of their squares a square. Ans -i 7’ or anf two fractions the L sum of which is unity, 8. To find two such numbers, that their product, added to the sum of their squares, may be a square. Ans. 5 and 3. 9. To find three rational right angled triangles having equal areas. Ans. 10. To find three squares, whose sum is also a Hyp. Base. Perp„ 58 40 42 74 24 70 113 15 112* square. 144 Ans. 9, 16, —. 11. To find a quadrangle inscribed in a circle, of which the sides and area are rational. Ans. Sides 80, 45, 100, 63. 12. To find an oblique angled triangle such, that its three sides, perpendicular, and aline bisecting the greatest angle, may be all rational numbers. · Ans. The sides are 875, 870, 145. 13. To find a triangle such, that its three sides, perpendicular, and the line drawn from one of the angles bisecting the base, maybe all expressed'in rational numbers. Ans, 48®, 299, 209.478 Diophanfme Problems. 14. To find two triangular numbers such, that their sum and difference shall he both triangular numbers. 15* To find two such squares, that their pro- duct added to the square of each shall be a square. 16. To find three square numbers in harmonica! proportion. 17. To find three numbers in arithmetical pro- gression such, that the sum of their cubes may be a cube. 18. To find three numbers such, that their sum may be a square, and the sum of their squares a fourth power. 19. To find a cube number, which, added to the sum of its divisors, shall be a square. 20. To find a square such, that the sum of its divisors being subtracted from it the remainder shall be a cube. 21. To find a square such, that the sum of its divisors being subtracted from it the remainder shall be a square. 22. To find a square such, that being added to the sum of its divisors the sum shall be a square. 23. To find two squares such, that each added to the sum of its divisors shall give the same num- ber. 24. To find two square numbers such, that one of them, and its divisors, shall be equal to the di- visors of the other.*7 9 C H A P. VII. On the Solution of the Equation xn —1=0, n being a Prime Number; with its Application to the Analytical and Geometrical Division of the Circle. PROP. I. 217. All the imaginary roots of the equation xn - 1 = 0 are contained in the general formula 2for x — 2 cos.------(-1=0, n h being any integer not divisible by n, and ?r re- presenting the semicircumference. It is a known trigonometrical property, that if 1 1 2 cos. y = x + —, 2 cos. ny = xn + —, from which two equations, viz. , 1 2 cos. y = x + -, ■ x 1 2 cos. ny—xn + —, are readily deduced the two following, xi —2 cos. y.x +1=0, a?2” — 2 cos. ny. xn + 1 = O, which must necessarily have one common root, being both derived from the same value of x; and480 Analytical diid Geometrical since these are both reciprocal equations, if x he one root, ·- will be another: they have therefore two oc common roots ; that is, the two- roots of the. first equation are also roots of the second; and, con- sequently, from the known theory of equations, the former is a divisor of the latter. If, now, we make y—of try— 2far, these equations become » 2 Tiir — 2 COS.----X +1=0, n xin — 2 cos. 2forxn +1=0. But the cos. 2for = 1, 2π representing the whole circumference; therefore, the latter equation now reduces to xin — 2xn + 1 =0, or (xn— l)2 —O, having still for its divisor the other formula 2 kir x" — 2 cos.---#+ 1 =0; n that is, the roots of the equation (xn— l)s=0, oraf—1=0, are all contained in the formula . 2far x —2 cos.-----a?+ 1=0; n and, therefore, by giving to h the successive values k=l, 2, 3, +(n— 1), the following formulae will be obtained; viz.Division of the Circle. x —2 cos. —x +1=0, n 431 X —2 COS. (ri— i)rt· n ■X+ 1=0; Which contain among them all the η — 1 imaginary toots of the equation x“- i =o. Cor. 1. If instead of making y- 2 k7Γ n 1 we had assumed ny = 2for + r, our second formula, a;3" — 2 cos. ny. xn + 1 =0, Would have been reduced to xqx + 2xn + Ϊ —0; or (#η + i)®=0, (because cos. (2for + it) — — 1), having foi its general factor the formula „ (2k + i)it X' — 2 cos.--------x +1=0, n Which is tlxe Other branch of the Cotesidn theorem. Cor. 2. From the theory of equations it follows, that 2 cos. — is equal to the sum of the two roots n of the equation 27Γ 2 COs. χ·+ 1=0, n which are also two of the roots of the proposed equation, and it is obvioiisly the same with all the other formulae j and hence it is manifest, that the 2 l4152- Analytical and Geometrical division of the circle depends upon the solution or the equation Χη-ί=0> !2tf and, conversely, by knowing the value of 2 cos. ·—-■» the roots of the proposed equation may be de- termined by the solution of a quadratic# PROP. II. 218. All the imaginary roots of the equation xn- 1 = 0, w being a primre number, are different powers of* the same imaginary quantity, and all different from each other. Before demonstrating this property of the roots of the proposed equation, it will be proper to short that, when n is a prime number, the roots of this equation cannot be the roots of any other equation xm- 1 = <>, m being supposed prime to n. For, if this be pos- sible, let r represent the common root, so that R* =1 and r“ =1; then, also, r“" = 1 and Rl“ = 1, whatever integral values are given to a and h; and, therefore, r“" = Rbm, or, dividing by r6”, we have })U^ sjnce a aiid b are here indetermi- nate*, and n and m prime to each other, such values of a and k may be found, that will make k4“; &c., Rf-'J*· which foots are all different from each other. For if any two of them be equal, let them be repre- sented by r/’-, and kqa, or Rz>e = R4'!, where p and q are each < n. And, since p and q are not equal,, let p>q·, then, dividing by Rqa, we have l; but since a is prime to n, and p — q < n, their pro- duct, (P~q)a> rs also prime to n; and, therefore, is impossible; for otherwise r“ would be a root of this last equation, and also a< root of the equation Rn=i, which we have shown to be impossible in the former part of the proposition, because (p — q)a 2 12484 Analytical and Geometrical and n arc prime to one another: therefore the fore- going series of roots, which belong to the equation are different powers' of the same imaginary quantity, and all different from each other. Cor. Since r* = 1, it is obvious, that Rn+t = R,- nn+2 — r\ and, universally, rp £> g\ &c., gn'\ when divided by n, leave different remainders, the same roots may he otherwise represented by 2 3 4 «-1 R*, Rg, r*, Rg, &c., ng } under which latter form it will be coUtenient for us to consider them in the following propo- sition, because this latter series will have the property of returning upon itself, if it be produced. it-1 beyond the term rs ; for gn'! — I is divisible by n (art. 87), or therefore, gn,^a'n + g·, and, consequently, r* = j/ =R°"+?: hence it is manifest,. that in this series of roots it is indifferent which of them is considered as the first. But if r be one root will be another, pro-Division of the Circle. 485 riding a be not equal to O, nor to re, nor to any multiple of»: therefore the same roots may be re- presented by *?*, RBe\ RS\ &C., pr by a*, nag, n“% &c., a"8" *j n-1 7i-l because gn~1 ztzna + 1, or r* = n, and r"s = Ra &c. The above periods of roots have, as we have seen, the property of returning upon themselves, if produced; and, since re is a prime number, η — 1 is a composite number; making therefore η — 1 = mky these periods of re — 1, or mk terms, may be decom- posed into k periods of m terms each, which shall have the same property; viz. by being produced, the same roots will recur in the same order as at first, as appears from the following proposition, prop. in. 219· To decompose the re — 1 imaginary roots qf the equation into k periods of ret terms such, that each, by being produced, shall recur in the same order as at first, m and k being supposed the factors of re — 1, or 7i— \=mk. The whole period of roots being 2 3 4 m(k-1) r, a*, Rs , a».,· - ... - af , the decomposition wilt stand thus:486 Analytical and (dremietrkal 1st period, k . 2 k 3k · ' . (m- ** i-8 > Ψ > ^ , ¥ k+1 2 A:-j-1 3A-4-1 (m - -1)Ac-f-: 2d period, vJ, ■ ψ 5 Ri , R? , /f + 2 2fc4-2 3k+2 (m - 3d period, R? , R? 4 1 -Rf k-i zk -1 3fc- 1 4A'-1 mk ~ ■ 1 k period, u* r/ j R? . Which are such,, that, being produced, they will give over again the same periods of roots; for the following terms in these periods will be, mk η — 1 rs· — Rs· = because gn~,-ct-.na +1; m/c+l n ~ ' ns 5= r£ i= r^, because gn m na' +g; &C. &C. &C.; and it is exactly the same with all the other periods. The method of separation is here ex- tremely obvious^ it being only necessary to write the first k terms of the general series., in the first vertical column, the second k terms in the next column, and so on. Cor. 1. The foregoing period of roots may be re- presented more simply in the following manner; viz. •by making gkz=zfh with which substitution they become 3 m -1 R, Rft it71 ? n4, r* ; h 2 3 m- 1 It^ 5 R'^% R*4 ; 2 2 € 2 · 2 3 C ! R 3 r!T * . 2-i 3-1 4-1 w-1 R4'1, R4 , R4 , R4 . R4 .Division of the Circle. 487 Hence again it follows, that any of these periods |nay be represented generally by m-1 ah r°, nah, if, r*4, κ“Λ, &c·, r“' Cor. 2. If the ηιιιηΐκ'ΐ· of terms, in this series m, be a composite number, as m — m'k', then may sjeach of the above periods be decomposed into If periods of m' terms each, as follows: ok’ R", H $ &ah „ &c.; R"Y Ic' + l Rah , 27c'--fl R"4 , &c,; a r", k’+it ft®4 , ak'+a Rah , &c.; k'- 1 . W-l Kah > 5 3k’- 1 Ra,‘ , &c. Which, by making = r** ==&*', becomes s, s'", s'"' , s'", &C.5 •2 3 c,a' c,o7/ fe , s , gaVi' ^ j &c.; a 2 a 2 2 3 ntt' ' ςχΟ?· hf X a 5 c< h' oft" hf δ 9 a y &c. y a similar result to the former: and in the same way the decomposition may be carried; on till the number of terms, in each period is a prime num- ber, after which no farther decomposition can be effected; and all these periods will have the pro- perty of returning upon themsel ves when produced, as may be readily shown as above. Remark. This decomposition of the imaginary (roots of an equation of the form 4f—1=0,488 Analytical and Geometrical n being a prime number, into periods, and on which the solution of the equation depends, is, in prap? tice, extremely simple, as will appear from the following examples; the foregoing complex ap- pearance arising sqlely out of.the generality that was necessary, in order to have a complete demon- stration of the proposition. The only difficulty is in ascertaining the quantity g, so that it may be a primitive root of the indeterminate equation a?*-' — l ==m(//); for which we might have given a rule, either in this place or in chapter v.; but as one, at least, of these roots is found among the first num- bers, it seems equally expeditious, or more so, to find them by trying the small numbers 2, 3, &c., till we arrive at it, than by any direct general rule for that purpose: it may not, however, be amiss to observe, that, if gm>n, and gm — 1 be not di- visible by n, m being a factor of η — 1, then will g be a primitive root (art. 202), where, likewise, it is demonstrated, that there are always several such roots, except in the case n = 3, in which there is only one. Ex. 1. It is required to decompose the four ima- ginary roots of the equation .r' -1 = 0, into two periods of twro terms each, which, by being produced, shall recur in the same order. Here, 2 is a primitive root of the equation a?5"1 — 1 = m(5), because 2' — 1 is not divisible by a; the first period of roots is, therefore,489 Division of the Circle, r, R2, R‘, r8; or R, R9, R4, r3; by rejecting the multiples of 5 5 and, therefore, the required periods are, 1st period, R, R4; 2d period, R2, R5. Ex. 2. Decompose the six imaginary roots of the equation a?7 - 1 = 0, into three periods of two terms each. Here 6 = 2x3, and neither 32 — 1, nor 3s — 1, is divisible by 7> therefore 3 is a primitive root of the equation a?7 —1 =h(7), and the powers of the roots of the proposed equa~ fion will be 1, 3, 32, 3*, 34, 3'; or, rejecting the multiples of ’J, the roots are e, r3, R2, R6, B4, r5; and, therefore, the periods sought will be, 1 st period, R, Rs; 2d period, R3, it4; 3d period, R2, R\ Ex. 3. It is required to separate the twelve imaginary roots of the equation ; vr,3_l=0, into three periods of four terms each; and these again into two periods of two terms each, which shall have the property of recurring always in the same order. Here, 2 being a primitive root of the equation, xn — 1 = m(13),#ge Analytical and Geometrical the powers of the roots of the proposed equation will be 1, 2, 2\ 2\ 2\ 2\ 2\ 2\ 2\ 2% 2W, 211; : and, therefore, rejecting ail the 'multiples of 13* the series of roots becomes 2 4 8 3 6 12 1Ϊ h K> .7, M, R , R a R , R , R , R , R , 13 li , R * fuid the first decomposition will therefore be, 1st period, R, p*, r!2, r&; 2d period, *% B% s”, R10j 3d period, r4, a6, n9, r7. And each of these will be divided into two pe* pods, as follows: }st period, j ^ J».* * ( rg 2d period, < 9: 3d period, f r4> »% . 1 R7· These examples are quite sufficient for rendering the decomposition of the roots, of any equation per- fectly familiar. PROP. iv. 22Q, The »—1 imaginary roots of theequatioq ^-1=0, being decomposed into periods, as in the foregoing proposition, then will the product of the sums of any two, or more, of these periods, or any powers of those sums, be equal to the sums of similar Let K°.+Ra*+R0* +R°* - - - Rafc , 2 3 7Π-1 Rt + Rte + RW + R.** - RM‘ ,Division of the Circle. represent any two periods of roots, of which the product is required, then, since we have seen that these periods being produced, give again the same foots, and in the same order, in multiplying these quantities together, we may arrange the products in the following manner; that is, by multiplying the whole of the upper series by each term in the lower one, only beginning always with that term of the upper line, that is over the term by winch we multiply, and produce the series of the upper line accordingly; thus, ,α+δ _j_ j^flft+δ -j- R04 +4 + Ra't+i -Γ &c. ip" +δ y 2 3 in — 1 (a+b)k Q(ah+b)b | H-b)h _j_ .^(αΛ. +b)h -r &C. R{"A 3 t 2 ■2 2 3 2 m- 1 % ,(a+b)h j j-^(ah-j-b)h +b)h _j_ ^ (ah + b)h + &c. R(a4 *f δ)7ί m- i m- i 2 7n - J |^(a+b)h . -^(αΛ+δ)δ -f'R' (ah +b)h + See·, And now, taking the sums of the different vertical columns, and writing a', a!', a!", ci\ &c., for a + δ, ak + b, ah? + b, &c., we have Ra' + Ra'4 + Ra'A + R"'4 + &C. . R‘ a'h 2 3 .7/1-1 ' Ra" + R"''4 + Η“"Λ + Κ“”Λ +&C. $?h , 2 3 1 7)1-1 Ra” + Ra'"'1 + Ra'"h + Ra'"lt + &C. V?'"k , &C. &c. &c. Each of which new periods consists of m terms, and they are similar periods to those from the mul- tiplication of which they were produced; because, if r“ be any imaginary root of the equation ^af—i=o,4(j2 Analytical and Geometrical ft®' another, providing a' be not divisible by nj and if a' be equal to n, or to any multiple of n, then, because r’“'=? I, the sum of the roots in that period will be equal to as many units as the period has terms, as is evident. The proposition, therefore, having been demon- strated for the product of two periods, it must ne- cessarily be true for any number of periods; and since there is nothing in the foregoing operation to prevent us from supposing a to be equal b, or, in other words, that the two periods that we have multiplied together are equal, the same is evidently^ trne of any powers of those periods; namely, that they may always he represented by the sums of pimple periods similar to themselves.. Cor. We are thus furnished with the following easy method of obtaining those products, qr powers; viz. Let /(r"), /(id), represent the sums of any periods, as /(r“) = r“ + r“’+ Ra"+ Ra", f (r*) = R* + R4' + R6" + R4"'. Under which form we shall have ./(R“) =/(f) =/(R“") s that is, the sum of the periods is the same, which·, ever is the leading term, because the periods are recurring ones, and the same have place with all other periods: then will /(R*)x /(R4) =/(R0+4)+/(Ra'+S): + /(Re''+i} + &C, which formula will be of particular use in the so- lution of the following examples, the principle of •\vhich is to subdivide the original period of roots.Division of the Circle. . 433- into two or more periods, as in the preceding pro- position; then, if there are only two, the sums of these separate periods may be ascertained, by knowing the collective sum of the two, which is always given, and the product-of the same two, which is obtained from the above formula; and having thus the sum of two quantities, and their product, the quantities themselves are easily found by a quadratic; if there are three periods, then, beside the suni of these three, we must know the sum of the products of every two of them, and the product of all three, whence the separate sums are found by means of a cubic equation, and so oh. 221. We may now proceed to the solution of a few examples, to illustrate what has been demon- strated in the foregoing articles. 3 βθ° Ex. 1. Required ttie cos. —-— — ?2°, and the imaginary roots of the equation Xs—1—0. The imaginary roots of this equation being de- composed into two periods, hy means of their* powers (as in ex. 1, art. 219), and representing the sums of these periods hyp, p'; that is, /{s.x)—p , = R + R4, /’(r4) —p' = R4 + Rs: it will only be necessary to find the values of p and p'·, that is, the sum of the two imaginary roots a + r4, or R4 + R5, which is readily obtained by means of the foregoing proposition, and the known property of equations; viz. that the sum of the roots of any equation is equal to the coefficient ofAnalytical ήήά Geometrical the second term, which* in the present case* is zef&j so that 1 + r + r2 + r3 + r4=o, or p+ // +1 - = 0, or p + p' - - =—1* Again, by cor.* art. 220, βκι) x /(rs) */(r1 +/(κ·'Ϊ=Ρ*·+Ρ j therefore, ρ+ρ' = — 1, and pp' — — 1; and, conse- quently, the equation which has for its roots the quantities p, p', will be +p—l=Q. Whence we obtain -1- A/r> - 1 + ya o—> which expressions represent 2 cos. 2.36o° 360° 2 cos, ·: therefore. cos. f2°=- 1+ vS •3090170* n - I - V5 COS. 144 =————- _ · 8090170. 4 The first of which values alone is obviously suf- ficient for the division of the circle into fiye equal parts. And, having thus determined the values of these cosines, we have, for finding the imaginary mots of the proposed equation, the two following quadratics:bivision of the Circle. 4$3 which, witli the real root 1, completes the solution of the equation. Ex. 2; Find the cosine of 360° : 7 ’ and the imagi- nary roots of the equation 3?— 1 = O. Having decomposed the powers of the imaginary foots of this equation* or* which is the same, the roots of the indeterminate equation a;6 — 1 =m(7)* into three periods of two terms each (ex. % art. 219), arid representing the sums of these pc— iiods by p3 p', p"; that is, p = R + R®* p' =R3 + R4* p" — R2 + R5; the object of inquiry will be, to ascertain the values of p + p' +p", of pp' +pp" +p'p", and of pp'p"; for then the cubic equation, having these quantities for its coefficients, will evidently have for its roots? pi p', and p". First, then, 7>+7>'+7>"=- 1, from the known theory of equations, and by cor., art. 220, pp' =// +p", p'p"=p"+p, PP"=P' +PI therefore, pp' + p'p" +ρρ" -2(th-/+ p") — — 2. Again, multiplying the first of the above pro- ducts by p" gives ρρ'ρ” +ρ”ρ"·496 Analytical and Geometrical Now p'p" = p"+pt and, by cor., art. 220, and p"p"=p' + 2; that is, =/(4)+/(7), the last of which^ viz./(r7) = 2, hy the same article j therefore, ρριρ"—ρ+ρ'+ρ” + 2=+ί. Hence the cubic equation that has p, p', and p"j for its roots, is J? +p*— 2p — Ϊ ==G; and, therefore, conversely, the roots of this cubic will be the values p, p’ and p", whence we find p = 1 · 2469796V p' = -1-8019376’, p"=- -4450420. And hence again, 360° 1·2469796 cos. · = · 6X34898; 7 2 _ which is sufficient for the division of the circle info seven equal parts. And by means of the above quantities p, p', p", we shall have the three following quadratics; viz,· X3 — p X + 1 = 0, x*—p'x+ 1=0, x* —p"x +1 = 0, which contain in them all the different imaginary roots of the proposed equation. ■ *' ' 360° Ex. 3. Find the cosine of ■ „ and the roots ■13.J of the equation 1 =*0.Division of the Circle. AQ'J The solution of this problem will be effected by means of a cubic and two quadratics, or a cubic and one quadratic, so far as it relates to the di- vision of the circle into thirteen equal’parts; ob- serving, that we must finish with the quadratic, in order that we may know the sum of two of its roots: the period of roots must therefore be first divided into three periods of four terms each, as in ex. 3, aft. 219. Whence, p =r‘ + r8 + r13 + r’, p' = R3 + R3 + Ru + R*°j y' = R4Tf Re + R9 +R7. And here, as in the foregoing example, it will be necessary to find the values of p+p' +p", of pp' + p'p" + jψ", and of pp'p", in order to ascertain the cubic equation that has these quantities for its roots. Now, p+p' +p" = -1, by the theory of equations, and by coi\, art. 219? pp' =p' Pp' +p +p"—p +2p' +p"y p'p" = p" + p" +p'+p —p' + 2p" + p, pp"=zp -\-p pp' pp" —p' p Qp pp". Whence, pp' +p'p" +ρρ" = 4(ppp' Pp") = — 4. Again, pp'p" =pp" + 2p'p" p p"p'\ But p p" = p' + 2p p p", 2p' p" = 2p' + 4p" -h 2p, p"p" = 2p'p P + 4 = /(R3) +2/(Rs)-f/(R1S). 2 K498 . Analytical and Geometrical Hence, pp'p" = 5 (p + p' +p") + 4= — 5 + 4 = — 1. Wherefore the cubic equation, having p, p', p", for its roots, is ps +pz— 4p+ 1 =05 and hence these values become known, each of which is the sum of four roots. And now, in order to get the sum of each pair of roots, the foregoing periods must be again subdivided into periods of two, the sums of which, for distinction sake, are re- presented by q, q', &c., as follows; viz. Period p, into j ? ΐ Period p, into j r3 + rK’ Periodp", into { Now, q + q'=p, ivhich is known from the preceding equation, and qq'=f + qv=p", which is also known. Whence, the equation containing the roots q, qr, is determined, being f-pq+p" = O; and the value of q in this will be equal to . 2 cos. ; and, by means of this, the roots of the proposed equation will be determined. It does not appear, from what has been done, which of these periods ought to represent theDivision of the Circle. m cos. ——; but this becomes immediately evident from considering, that the cos. of this angle will be positive, and greater than the cos. of any of the other angles, 2.36o° 13 3.360° 13 ’ &c. ; and therefore, that period q must be assumed into which p enters positively, and, in the same way, we readily discover the particular period which represents the cos. of any other angles, as, likewise, the negative from the positive, &c., by which means the apparent ambiguity is avoided.—These obser- vations must be particularly attended to in the fol- lowing example. Ex. 4. Find the cosine of 360° ~W* and the roots of the equation Xxi -1 = 0. Since If is a prime number of the form 2m+ 1, or 17 = 24 + 1, the roots of the above equation may be obtained by four quadratic equations, and the * 36o° , , . . cosine or by three quadratic equations; in order to which, the imaginary roots of the equation xlJ- 1 =0 must be decomposed, first, into two periods of eight terms each, then these into two periods of four, and these again into two periods of two terms each. Now 3 being a primitive, root of the equation x,6-i=m(i7), 2 k 2500 Analytical and Geometrical the whole period of powers will be, h 3, 32, 33, 34, 35, Q6 ° y 37, 38, 39, 0 10 ° y 0 11 012* O y O j ό 13 v y 314, 3 15; or 1, 3, 9, 10, 13, 5, 15, 11, 16, 14, 8, 7, 4f 12, 2, 6; by rejecting the multiples of 17· Whence (by art. 219) the first two periods will be P — K1 + R8 + R13 + R15 + R16 + R8 + R4 + R®, p' = R3 + r'° + R5 + r" -f Ru + R7 + R1® + R°. Now p+p'=-l; and pp' - p + p' + p + p + p + p'+p' + p' — 4 (p + p) - - 4, Hence the quadratic equation containing the roots p, p', will be p* + p — 4 = 0. Whence, p= + V17? and j»' = — 4- — τ λ/1 7* Again, the periods of roots p, p', must be now de- composed into the four following periods, the sums of which are, for distinction sake, represented by q, q'·, viz. Period p, ^ Period p', ·£ q — r1 + R13 + r10 + r4, q' = r9 + r15 + R8 + Rs. q" = R3 + R5 + R14 + Rla, q'" sc R10 + RU + R7 + R6. And here we'have qcf = q’" + q" + q'+ q =*p+p'~ l,'Division of the Circle, 501 Whence the quadratic equation containing the roots q, q', is q'-M--1*0: consequently, 5 = T? + i λ/(4 +/), and q' = \p-·^ v{4+p*). In the same way, f’—W + τ ν'(4 + pn), and q"e = \p' - \ ^(4 +pn). Again, the above periods of q, q', q", &c., are each decomposed into two periods of two terms each, which new periods are represented by t, t', t", &c,; viz, Period q, into j Period q', into +J Period into { £ ϊ£ + £> Period q , into j ^ _ r1, + R6’ Now t + f=q=xp + 4. v(4 + f)t ~ and tt' = th + f = q"=^p' + ^ v(4 +p'a). Therefore the quadratic equation containing the roots t, t', is f — qt + q" — 0, Whence, f t + i v{f-4q"), \t' = ^q-^ J{q*-4q"). The first of these is the greatest positive root,502 Analytical and Geometrical and isj therefore, the value of 2 cos. 36oe 17 ! which, by substituting for q and q", their respective values, in terms of p and p', becomes 2 cos. 36°° { 17 iP + * V(4 + f) } + + j{\p + * v/(4 + p*) }' + v(4 + p,s) }. Again, reestablishing the values of p,p', we have, s6o° m numbers, 2 cos. { 17 •M*(—i-+Wi7) + *vi(i7- vi7)} + ■W{*(-*+-Wi7) + W-Ki7-W7)1*- 4{i(—j— Wi7) + W*(i7 + a/i7)}5 s6o° which is the true numeral value of 2 cos. whence it is manifest, that, by the construction of three quadratic equations., a 17 sided polygon may be inscribed geometrically in a circle. We might have added here the solution of the equations* X ' 1 = 0*