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Theft, mutilation, and underlining of books are reasons for disciplinary action and may result in dismissal from the University University of Illinois Library NR V 41 a 291976 »-: vONPYy qd, MAR 97 VEC 12 wee MAR 40 (970 Ber | 1 MAR _9 per poi RECO é JAN 3 ieee tk Ly waa ; as SIDEC 18 i97q FEB 3 1 A Rep , DEC 4 7 RECUMAR 2 1 997 | MOV 13 ni he D (84 iov 19 an ~QEC 15 OCT 1 t 1974 _ APR 19 1989 ft OCT ~4 cud RELY sg “ bin’ 6 | f NOVe Of ae get 2 BF ey W2 RCD f DEC 2 GRECDVAN 6 my| SUN 02 208 WAY 6 75) O° afi | & | ECD APRI GA a t ete Fs ment a *t oe ' NOV 43 pred L161— O-1096 THHORY OF NUMBERS PATE Ic Cambridge PRINTED BY C. J. CLAY, M.A. AND SONS, AT THE UNIVERSITY PRESS. PE HOR Y OR NUMBERS PART LI. BY G’ By MATHEWS, M.A. FELLOW OF ST JOHN’S COLLEGE, CAMBRIDGE ; PROFESSOR OF MATHEMATICS IN THE UNIVERSITY COLLEGE OF NORTH WALES. CAMBRIDGE: DEIGHTON, BELL AND CO. LONDON: GEORGE BELL AND SONS. 1892 t/t 5 | aN a] Wt < "> Yy '< \3 I4ah31 WA PREFACE. HIS treatise is intended to provide the English student with an intelligible outline of the Theory of Numbers which may serve as an introduction to the detailed study of the subject at first hand. No single work of reasonable size could possibly do justice to every part of the theory; and in the choice of material it is not easy to adopt any plan which is likely to approve itself to everyone. I have been guided principally by a desire to give a fairly complete account of the theories of congruences and of arithmetical forms, so far as they have been developed hitherto ; to this I hope to be able to add a sketch of the different complex and ideal theories. Diophantine analysis proper, and questions of pure “tactic,” have been omitted, except in so far as they have been subsidiary to the general scheme. ) The range of this first volume is sufficiently indicated by the “table of contents. It is hardly necessary to say that I have "derived continual assistance from the works of Gauss and Dirichlet, vand from H. J. S. Smith's invaluable Report on the Theory of Numbers. I am also greatly indebted to Professor Dedekind for > permission to make free use of his edition of Dirichlet’s Vor- Slesungen tiber Zahlentheorie. So far as this present volume is concerned, the account of Dirichlet’s researches has been taken | _ primarily from his original memoirs; at the same time, I owe much to the study of the Vorleswngen, and I hope that I may “eventually avail myself of Prof. Dedekind’s kindness more directly, A by giving some account of his theory of ideals. M. b v1 PREFACE. In the references at the ends of the chapters and elsewhere I have done my best to indicate fully the sources from which I have derived my information. No attempt has been made to give an exhaustive bibliography; even if I had been equal to the task of compiling it, the result would probably be merely embarrassing to the beginner, whose attention should be directed in the first place to the works of the great masters of the science. Several friends, among whom I may mention Mr H. F. Baker, of St John’s College, Cambridge, Mr R. W. Hogg, of Christ’s Hospital, and Mr A. G. Greenhill, have kindly allowed me to send them proof-sheets; Mr J. Hammond has been good enough to revise my account of Professor Sylvester’s. researches on the distribution of primes; and I am indebted to my colleague, Mr A. Gray, for advice and assistance in seeing the book through the press. To all of these my best thanks are due; and I may add that I shall be grateful for any criticisms or corrections that may be sent to me by any of my readers. G. B. MATHEWS. UNIVERSITY CoLLEGE oF N. WALES, BANGOR. ART. CONTENTS. CHAPTER I. DIVISIBILITY OF NUMBERS. ELEMENTARY THEORY OF CONGRUENCES. 1. Laws of operation . 2. 3. Composite and prime maibaes Relative primes Unique resolution of a number into prime factors 4,5,6. Elementary theorems on divisibility (f 8. 1, 20. Kuler’s function oe ) Proof that 2¢ (d)= : ; Definition and Pats of Bevin, Residues EHiementary theory of congruences ; Roots of a congruence. Lagrange’s preeror thee a congruence ete have more roots than is expressed by its degree; the modulus being prime Linear congruences with one Syarinble Determination of a number having prescribed Seance with repent to given moduli . Greatest common measure of a a of ernpers Se reecall as a Tee function of them : Simultaneous linear congruences . Fermat’s Theorem, with Euler’s extension of it . Wilson’s Theorem ‘ : : : ; : ; : : Residues of powers. Exponent to which a number appertains. New proof of Fermat’s Theorem : Case of a prime modulus. Primitive ante Method of finding primitive roots 21, 22. Indices 23. 24. Residue of the product of the pancye are of a prime . Sum of the primitive roots of a prime. Congruence satisfied the primitive roots 25—29. Case of a composite ae tine} Authorities tg ea Q i NN DFP EP WH DOF V1 CONTENTS. CHAPTER II. QUADRATIC CONGRUENCES. ART. 30. Quadratic residues and non-residues. Legendre’s symbol 31. Quadratic character of a product of residues 32. Indices of quadratic residues and non-residues odd and even respec- tively . 4 33. Solutions of #?7= aimed pr) derived orn ‘aes of men ned 4 34. Modulus a power of 2. 35. Number of solutions of «7=a (mod 7) 36. Statement of problems to be discussed 37. Quadratic character of —1 38. Quadratic character of 2 39. Legendre’s Law of Reciprocity. Canes? S third roore 40. Hisenstein’s geometrical lemma : 41. Hxample 42, Jacobi’s extension of Terence rae 43—45. Gauss’s first proof by induction (after Dirichlet) 46. Determination of (D|n) when D is given 47. Solution of the congruence #2=a (mod p) by on Authorities CHAPTER IIL. BINARY QUADRATIC FORMS; ANALYTICAL THEORY. 48, 49. Algebraical forms 50. Representation of numbers, memntige or Peet 51. Determinant of a quadratic form. Definite and indefinite ime 52. Roots of a form ; , : : 53. Primitive and derived forms. Principal form 54. Transformation and equivalence . 55. Equivalent forms have the same sn eaeted vars oat “ef Pree 56. Distinction of proper and improper equivalence . 57. Composition of linear substitutions 58. Application to quadratic forms 59. Reduction of the problem of coronene unr to that of aivalenes 60. Summary of results to be obtained. Classes. Reduced forms . 61. Equivalent forms belong to the same order 62. Opposite, adjacent, and ambiguous forms defined 63. Reduced definite forms. Criteria of reduction 64. The number of reduced forms is finite. Limits of the ee of re coefficients of a reduced form . 65. Equivalent pairs of reduced forms 66. Set of representative reduced forms. eee 67. Reduction of indefinite forms. Gauss’s definition of a Poin oe PAGE 32 33 34 34 35 36 36 37 37 38 41 42 42 45 50 53 55 CONTENTS. ART, 68. Construction of a complete set of reduced forms. Example 69. Every form is equivalent to at least one reduced form : 70—73. Periods of equivalent reduced forms. Chain-fraction te ae of the root of a reduced form 74. Associated forms and periods. A period mene is its own Seat t contains two pga forms : : : : ‘ 75. The case when (4, B, —A’) and (-A, B, 4") belong to the same period : : . : : 76. Notation for continued Tenens Heme pontiined fractions . 77, 78. Auxiliary theorems on continued fractions 79. Two properly equivalent reduced forms belong to the same eerind 80. Algorithm for deciding whether two given forms are equivalent. Example . 81. Simplest representative Hore 82, 83. Automorphic substitutions 84—87. The Pellian equation 88, 89. Complete system of artemoroie : : : : ; 90. Representation of numbers resumed. Groups of representations; maximum number of distinct groups 91. Examples : 92, 93. Improper eqrvalente CHAPTER IV. BINARY QUADRATIC FORMS; GEOMETRICAL THEORY. 94—96. Elementary theorems on complex quantities 97—100. Geometrical interpretation of the linear eatioreinion ‘of a complex variable. Arithmetical group of proper unitary sub- stitutions . 101. Critical points 102. Equivalent points . : 103. The fundamental triangle Vv 104—107. Reduced points 108, 109. Every point above the axis of x is o@ihaatan to one fl onl one reduced point 110. Group of equivalent triangles 111. The same projected on to a hemisphere 112,113. Comparison with the analytical method of Pedic definite forms. Elementary substitutions 114. Indefinite forms. New criteria of reduction 115—119. Critical forms. Periods of principal reduced fore 120. Geometrical interpretation of the automorphs 121—124. Method of nets 125, 126. Extension to indefinite come Authorities 1X PAGE 74 75 76 79 80 81 81 84 86 87 88 90 95 97 98 100 103 105 110 110 111 111 115 116 117 117 119 119 123 124 128 130 > CONTENTS. CHAPTER V. GENERIC CHARACTERS OF BINARY QUADRATICS. ART, PAGE 127. Every primitive form (a, b, c) of determinant D, for which dv (a, 2b, c)=«o, is capable of ae a number on, where n is prime to D ; ‘ : i .) Leo 128, 129. Particular and total aubrtyern of a AaMeer form ? : ., 188 130. Dirichlet’s table of assignable characters. ; ; . "185 131, 1382. Determination of the characters of a form ty tnapechian ; .- 136 133. Examples. : eS: : : : : we eae 134, Principal form, class, nd genus enna A : : ; . 138 135. Proof that half the assignable characters are natore ne : : i BS Authorities : : : : ‘ : : : : : - hy Lae CHAPTER VI. COMPOSITION OF FORMS. 136. Bilinear transformation. Gauss’s nine equations , : . > 140 137. Composition of two forms . : ' : . 142 138. Determination of a form compounded of feo given niforiid : . 144 139. All forms compounded of bs jf’ in a prescribed manner are properly equivalent . : : 146 140. Reduction of the general case as that in aHGn nell fort is fale directly . : ; : : . : : : ; , . 147 141. Composition of classes. Duplication . : : . : : . 148 142. Arndt’s method of composition . : ; ‘ ; : ‘ . 149 143, 144. Successive composition. Proof that the result of compounding any number of forms is independent of the order in which they are taken . : 152 145. Theorems relating to the soniposiacn of (ofute of the same fei 156 146. Numbers of classes in the different orders compared . ; : » . 158 147—152. Lipschitz’s application of the theory of transformation ‘ Zed 153. Number of improperly primitive classes ; : : ; : wi LOT 154. Composition of genera . : . . 170 155. Each genus of the same order tniaine the same terri of keer: Ses We 156, 157. Number of properly primitive ambiguous classes . : fed at gee ae 158. Number of improperly primitive ambiguous classes . : : Ce 159. Limit to the number of actually existing genera . : : : Seedy bt 160. Gauss’s second proof of the law of reciprocity : : edi G 161—166. Periods of classes, Regular and irregular setae aure Ex- ponents of irregularity . : : : : : , Penn bi. Authorities . : : : f : : ; : ; : a LOS ART, CONTENTS. CHAPTER VII. CYCLOTOMY. 167—169. Properties of the roots of unity 170. 171. 172. 173. The cyclotomic equation X,=0 Proof that Xp is irreducible . Reduction of rational functions of the oct of xX, to a ora nn Outline of Gauss’s theory 174—177. Definition and properties of the mend: Ni 178. 179. 180. 181. 182. 183. 184. Method of finding the equation of the periods Proof that the equation of the periods is Abelian . Equation satisfied by the roots of X, the aggregate of iden Hirth up a period The solution of X,=0 ages to icner upon a series of cavalry: equations, the degrees of which are the prime factors of (p—1) Moduli for which the congruence X,»=0 has all its roots real. Analogy with the algebraic theory : Quadratic equation of the periods . Kronecker’s evaluation of Gauss’s sums 185—187. Dirichlet’s method 188—190. Gauss’s original investigation 191. 192. New proof of the law of reciprocity Gauss’s generalisation of the law . 193—195. The identity 4X=Y?+pZ? . 196. 197. 198. 199. Cubic equation of periods when p= on+ vi Kummer’s discrimination of the roots . Connexion of the theory of cyclotomy with other bars of the chore of numbers Authorities CHAPTER VIII. DETERMINATION OF THE NUMBER OF PROPERLY PRIMITIVE CLASSES FOR A GIVEN DETERMINANT. Principle of Gauss’s method . : } 1 200—205. Expression of the class-number in terms of the series a. (D|n) . 206. 207. 208. 209. 210. Explicit value of h for a negative determinant The same for a positive determinant Note on an alternative method (Dirichlet’s) . Kronecker’s formule. Fundamental discriminants Trigonometrical solution of the Pellian equation . x1 PAGE 184 186 186 189 190 191 195 196 196 198 199 200 201 205 209 212 214 215 219 223 228 229 230 231 239 247 252 252 253 X11 CONTENTS. CHAPTER IX. APPLICATIONS OF THE THEORY OF QUADRATIC FORMS. ART. PAGE 211, 212. Solution of the general Diophantine equation of the second degree 257 213—216. Determination of the factors of large numbers . : : eos Authorities : ; : : ; : : ; : : S278 CHAPTER X. THE DISTRIBUTION OF PRIMES. 217. Statement of the different aspects of the problem ; : = eee 218. Meissel’s method of enumeration . a Re : : : ; « Zhe 219—223. Tchébicheff’s asymptotic formule : 1: otherwise a would be a multiple of p, and at the © same time less than p. Now p being prime is not divisible by b: hence we may write p= mb-+ b’ where m, b’ are positive integers and b’a’: then 7 ee Deel 0o Cry, that is b*cY... is divisible by a, contrary to Cor. 2. Similarly if a >a; therefore a’ =a, and bPot.. = bP ev): A repetition of the argument gives 8 =f’, y =7/,... successively: so that the two resolutions are identical. |—2 4 DIVISIBILITY OF NUMBERS. It should be observed that any even number of the factors of A may have their sign changed without altering the product: thus, for instance, ' 60 = 2?.3.5=(— 2)7.3.5 =2?. (—3).(—5) =ete. but these resolutions are considered essentially the same, and this convention is understood in the statement of the theorem. Similarly any negative composite number may be reduced to the form (— 1) a*b®c’... where a, b, c... are different positive primes, and all other resolutions of the number into prime factors are essentially equivalent to this. 4. Ifa, b,c... are all prime to k, their product is also prime to k. For no prime factor contained in a or D orc... is contained in k: therefore the product abc... contains no prime factor of k& and is consequently prime to hk. | 5. Ifa, b, c... are prime to each other, and each divides hk, then their product divides k. For if any power of a prime, say p", occurs in the product abc..., it must occur in one of the factors, in a, suppose: therefore k, which is a multiple of a, must have p7 for a factor: and similarly for any other power of a prime contained in abe.... Hence kis divisible by the product. 6. If a is prime to 8, and ak is divisible by b, then & is a multiple of b. For ak is divisible by b, and also by a: hence, by Art. 5, ak is divisible by ab, that is, ue or f is an integer. ab ob The function ¢ (n). 7. Let n be any positive integer, and let (mn) denote the | number of positive integers, 1 included, which are prime to n and not greater than n. By definition ¢(1)=1. Also ifn is a prime number b(n) =n—1. Next suppose » composite, and let p, g, 7, s... be the different primes which divide n., ere THE FUNCTION ¢ (7). 5 Consider the series of integers 1, 2, 3...n. Of these the following are multiples of p: n Pp, 2p, Bn ot (n/p in all). - Write these down with the sign +. Similarly write down all the multiples of qg, 7, s... each with the sign +. In the same series there are a multiples of pg. Write these all down with the sign —: and do the same with all the multiples of pr, ps, qr... (taking all the products of p, g, 7, 8... two at a time). Next write down all the multiples of the ternary products pq’, pgs..., each with the sign +, and so on: until at last we come to the multiples of pqrs... with the sign (—)*~+, & being the number of different primes. Now take any number @ which is not greater than n and not prime to it. It will involve in its composition a certain number (A, say) of the different primes p, qg, 7..... How many times will it occur among the multiples already written down ? Evidently (taking its appearances in the order of the groups) X times with the sign +, then x Be Hy) times with the sign —, then ae) times with the sign +, and so on. If then we take the algebraic sum of all the groups, we have 6 occurring with a coefficient MUN Lue NOOO? ie _ Thus the algebraic sum in question is the sum of all positive integers not greater than n and not prime to it. Now the number of these integers is equal to the excess of the number of positive terms in the whole sum, as originally written, above the number of negative terms : that is, it 1s ni(Grg +i4..)- (See tit..) (State). a re 1 ONETIS EEN ta Ee OTE: 1 —1)-1 ..+(—1) at. Subtracting this from n, we have finally san(t-)(0-D(0-3)- 6 DIVISIBILITY OF NUMBERS. ] Corollary 1. Op ee (1 — 2 = p*! (» — 1). Corollary 2. If m is prime to m’, (mm!) = $m). $ (mn) For let p, g, 7... be the different primes which divide m, and p,q, 7... those which divide m’. Then dh (m). (m') =m (1 ~5) (1 ~ ) (1 _ ") sf 4 = mm’ (1 — ~) € _ A ae (1 _ *) (1 _ ae ip v P fl = d(mm’): observing that none of the primes p,q, 7... can be found in the Sey Oh eee It is not difficult to prove Corollaries 1 and 2 independently, and thence to deduce the main proposition. This is the method adopted by Gauss (Disquisitiones Arithmetice, Art. 38). 8. Ifd,d’,d”... are all the divisors of m (1 and n inclusive) then (d) +62) +6 (d')+...=0. Resolve n into its prime factors, so that n =ab8c’.... Then any divisor of n is of the form dS bee"... (at{rAtO, etc.) @§ q so that Sh (d) = Th (Wr...) = Uh (W) ob (L4). b (c?)... ={1l+¢(@)+o(a)+...+¢(a%)} x {1+6(0)+6(0)+...+¢ (6F)} x {L+ P(e) + O()+...4+h6(%)} x... ={1+(a—1)+a(a—1)+...+a77(a—-I1)} x11 (bb 6 ol) eee ee = (7, DPC) eet (For another proof, see Gauss, D, A. Art. 39.) _ 7) ‘ 4 ip Kr) par ) . UY FG LEAST RESIDUES. = Clavet ashore 2 =-U Congruences. 9. If the difference of two integers b and ¢ is divisible by m, b and ¢ are said to be congruent (or congruous) with respect to the modulus m, and this is expressed in writing by b=c (mod m). This is clearly the same thing as c=b (mod m). Each of the numbers 0, ¢ is said to be a residue (mod m) of the other. With respect to a given modulus, every number 0D has an infinite number of residues which are included in the expression b+ Am, » being any integer. In all that follows the modulus is supposed to be positive. Any number is congruent (mod m) to one, and one only, of the numbers 0, 1, 2...(m—1): or again to one, and one only, of the series 0, —1, —2,... —(m-—1). These may be called a complete series of least residues, positive and negative respectively. For a given number there will, generally speaking, be one and only one residue numerically less than a? this is called the absolutely least residue of the number. If m is even there will be a possible residue > which is equivalent to —>: the complete system of absolutely least residues may be taken 7 be ata sie Ze oe while if m is odd, the absolutely least residues are given by eae e = m—1l : lis SoU Ste Need ons 10. The following propositions are fundamental in the theory of congruences: most of them are so obvious as not to require a formal proof. I. Ifa=b (mod m), and a=c (mod m), then b=c (mod m). x Dope lf aie 0 =, c = G,, ete. (mod 7), oo then atbtct...=a tte +... (mod m). Ill. Ifa=qad (mod m), then ka = ka’ (mod m). of gfe] te privet, 2 pn) ipsa, opel Ni line Que Na 8 DIVISIBILITY OF NUMBERS. IV. Ifa@=q and b=D’' (mod m), then ab =ab' (mod m). For by III. ab=ab=av’. V. Ifa=a,b=0',c=Cc’, etc. (mod m), then abe... = ab'c'... (mod m). Proved by repeated application of IV. Hence if a=a (mod m), ak = a* (mod m), k being a positive integer. Finally, if a=a’, b=0’, c=c’,... (mod m) and @¢ denote a rational integral function, } (a, 0, c...)=6(a, 0, c,.,-) (mod m). All the above propositions are precisely analogous to the corresponding theorems for ordinary equations: but there is one case not yet considered where the analogy ceases to hold good. Namely from the equation ka =ka’ we infer that, if k is neither zero nor infinite, a= a’: but from the congruence ka = ka’ (mod m) we cannot infer that a= a’ (mod m) unless k is prime to m. The legitimate inference is contained in er ena theorem. VIL. Ifka=kb (mod m), ay" ae 3) then a = b (mod m/d), 118 ES Cad §) where d is the greatest common measure of kandm. (+2 For suppose k= k'd, m= md, where k’ is prime to m’. Then (ka — kb)/m =k'd (a — b)/m'd =k’ (a — b)/m’: and since k’ is prime to m’,a—b must be a multiple of m’: that 1s, a=b (mod m’), or, which is the same thing, a = b (mod m/d). 11. Consider the congruence ac” + be" + ...+1=0 (mod m), where a, b, c...J are given numbers and # is undetermined. Any integral value of « which satisfies the congruence may be called a root of the congruence. The coefficients a, b, c...l may be replaced by any other co- efficients which are congruent to them, and in particular by their least residues, without affecting the meaning of the congruence. If € is any value of w which satisfies the congruence, any one ROOTS OF A CONGRUENCE. 4) of its residues (mod m) will also satisfy it. It 1s convenient to say that one solution is given by «= & (mod m), or that the congruence has one root «= & (mod m). If » is a prime, the congruence f(a) = aa" + ba" +... +1=0 (mod p) cannot have more than n incongruent roots. For if a be any numerical quantity whatever, F(a) = (@— a) fr @)+F@ identically: f,(a) being a polynomial of degree (n — 1). Now suppose a an integer such that I (a) = 90 (mod p). Then f («) = (w@—4) f(a) (mod p), independently of a. ~Let 8 be another root of the congruence: then putting w= B, we get " (B— a4) fi (8) =f (8) = 0 (mod p), and therefore, since 8 is supposed incongruent to a, Ji (8) = 0 (mod p). It follows as before that a polynomial /, (wv) can be found, such that ° fi (#) = (@— B) fa (@) (mod p), independently of «: and so on. If then the congruence has n in- congruent roots a, 8, y...A, f(@)=a(a@—a) (a — PB)... (@—2) (mod p), independently of #: Le. this is an “identical” or “indeterminate ” congruence, Now let @ be any integer not congruent to any of the numbers a, 8,y...r: then f(@) =a(@—a)(@—B)...(8—r) (mod p). None of | the factors on the right is a multiple of p: therefore, since p 1s prime, their product is prime to p: consequently f(@) is prime to p, and f(«) = 0 cannot have any roots distinct from a, 8, ¥...X. Observe that it is not proved that the congruence actually has nm roots: in fact, this will not generally be the case. If a congruence of the nth degree is satisfied by more than n incongruent values of the variable, it must be an identical con- gruence: the modulus being supposed prime, as above. 10 DIVISIBILITY OF NUMBERS. Linear Congruences. 12. Every linear congruence with one unknown quantity can be reduced to the form 4 ax =b (mod m). Suppose in the first place that a is prime to m. Then if the numbers 0, 1, 2...(m—1) are each multiplied by a, the resulting products are all incongruent (mod m). For if two of them were congruent, say ae=af, then a(e—/f)=0; whence e—f= 0, since a is prime to m: but this cannot be, because e — fis less than m and different from zero. Hence the least positive residues of the pro- ducts will be the numbers 0, 1, 2...(m — 1), of course in a different order. For example, suppose m=12,a=5: then when (ALD, SAgsay DAlh.o,) aL an’= 0/5, 10-8, 8, 1,6, 11; 4,9, 2,7 Since the products form a complete system of residues, one and only one of them is congruent to b, hence the congruence aw = b has one and only one root. For instance, to solve 52 + 11 =2 (mod 12). This gives 5a=—9=3: and the preceding table shews that the solution of this is 2 = 38 (mod 12). When the modulus is large, this method becomes laborious, and it 1s necessary to find a more convenient practical method. In the first place, the solution of av =b may be deduced from that of aw=+1. Namely if =€ be the solution of the latter, then w = + b€ is the solution of the former. Now the congruence az=+1 (mod m) is equivalent to the indeterminate equation aw#—my=+1: and this can always be solved by reducing m/a to a continued fraction : namely, if p/q be the convergent immediately preceding m/a, ap —mq = + 1: so that the solution of aw=+1 1s c= +~>p (where, of course, the signs do not necessarily correspond). For example, to solve 365a = 11 (mod 1887) the work may be arranged as follows : 5 365 1887 5 7 55 62 1 6 6 7 i 1 LINEAR CONGRUENCES. ind The successive partial quotients being Pye oP sa hel Bey the numerators of the convergents are 5B 26 81 243 274 1887. Since 274 comes in an odd place 365.274 =— 1 (mod 1887) and therefore the solution of the proposed congruence is x=—11.274 =— 3014 = 760 (mod 1887). Next suppose that a is not prime to m: and let d be the greatest common divisor of m and a. Then in order that the congruence may be possible, 6 must be a multiple of d: so that the given congruence 1s equivalent to a = 5 (moa Aer O “). This may be solved as already explained: and supposing that the solution is given by w= & (mod m/d), the original congruence has d roots “=E+ oe (mod m). (e=0;, 1,2... (¢—1)). In the case of a composite modulus, it is sometimes convenient to proceed as follows. Let the modulus m = pq, and the proposed congruence ax = b(mod m), where a may be supposed prime to m, and therefore to p and gq. Let the solution of ax= b (mod p) be «= & (mod p). Substitute &+ yp for « in the given congruence: thus a& + ayp = b (mod pq), or ayp =b—a& (mod pq). Now b—a€ is a multiple of p,=b'p say: therefore ay=b' (mod q). Suppose the solution of this is y =n (mod q): then that of the original congruence is 7 =£&+ py (mod 7). | By repeated application of this process the solution of a congruence with a composite modulus may be made to depend on the solution of a set of congruences each with a prime modulus. For example, consider the congruence already solved, — 365x = 11 (mod 1887). 12 DIVISIBILITY OF NUMBERS. Here 1887 =38.17.37: and starting with 365x = 11 (mod pe which reduces to — 5a = 11 (mod 37), we find “= — ————— = - 17 (mod 387). Put «= 37y—17: then 37 .365y = 17.365 + 11 = 6216 (mod 1887), 365y = 168 (mod 51) whence 8y = 15 (mod 51), y =0 (mod 8) or y = 32 say, where 8z= 5 (mod 17), z=7 (mod 17), and thence successively y= 21 (mod 51), 2 = 37.21 —17 (mod 1887) = 760 (mod 1887). 13. v. Then a“~’=1 or a'=1 say, where ¢ is positive and less than m. Suppose f is the least positive exponent for which af=1 (mod m): then f is said to be the exponent to which a appertains (mod ™m). The numbers 1, a, a?,...a7— are all incongruent. For if any two of them were congruent, say a? =a’, where both p and q are less than f, it would follow that a?~4=1 where p~q 1s a positive integer less than f: this contradicts the definition of / Hence in the series e076. dd Uf the residues of the successive terms recur periodically, there being f residues in each period: and, generally, a’ = a* (mod m) if h=k (mod f), and conversely. For instance, suppose the least positive residue of 51 (mod 31) be required. We have 5°= 125 = 1 (mod 31): and since 1000=1 (mod 8), 5° = 5 (mod 31). It has already been proved that a*™ =1 (mod m): hence ¢ (m)= 0 (mod /), that is, the exponent to which a appertains is a divisor of ¢ (7m). M. 2 18 RESIDUES OF POWERS. This important result may be proved independently. It is clear in the first place that f cannot exceed $(m), because there are only @(m) numbers less than m and prime to it, and the numbers 1, a, a?,...a/—, are all incongruent and prime to m. If f is less than &(m), there will be at least one number 6 less than m and prime to it, and not congruent to any of the numbers 1, a, a?,...a‘, Consider the series DDG: 0G 00) =: These are all incongruent (mod m): moreover none of them can be congruent to any of the former series. For if ba’ = a*, then b=a' or =attt*, according as k& is greater or less than h: in each case, b is congruent to one of the first series, contrary to hypothesis. If all the numbers less than m and prime to it are congruent to some or other of the 2f numbers thus obtained, then 2f=¢(m): if not, let c be one of those that remain, and form the series ¢, ca, ca?,... cay: these are all incongruent to each other and to the first series. They are also incongruent to all of the second series: for if ca’ = ba*, c= ba*™ or = ba’** according as k is greater or less than /: in either case c is congruent to a number belonging to the second system, and this is contrary to hypothesis. If the least positive residues of the 3f numbers now obtained do not exhaust all the ¢ (m) numbers less than m and prime to it, take d, one of those that remain, and form the least positive residues of d, da, da?,...da‘; and so on. It is clear that in this way the complete set of ¢(m) residues must at last be exhausted, and since we get additional residues in sets of f at a time, @ (m) must be a multiple of f. _Fermat’s Theorem may be immediately deduced from this: for, putting ¢(m)= ee om) = (a)? = 1¢= 1 (mod m). 19. Suppose now that the modulus is a prime number p. It has been proved that the exponent f to which any number apper- tains (mod p) is a divisor of (p—1). The question arises: having given d any divisor of (p —1), are there any numbers to which the exponent d belongs, and if so, how many such numbers are there ? Let y (d) denote the number of integers, positive and less than p, to which the exponent d appertains. Suppose there is at least one such integer, a. Then all the numbers a, a’, a®... a% are incongruent (mod p) and they are all roots of the congruence PRIMITIVE ROOTS. 19 #*=1(mod p). Hence there are no other distinct roots of this congruence; otherwise it would have more than d incongruent roots. Now if we take a®, where k involves a factor of d,—éd suppose,—we have (a*)4* = (q%)#/> = 1, where the exponent d/6 < d: consequently d is not the exponent to which a* appertains. On the other hand, if & is prime to d, and f is the exponent to which a* appertains, (a*)/= 1 (mod p), therefore kf= 0 (mod d), whence f=9 (mod d), since & is prime to d. The smallest admissible value of f is therefore d: moreover (a*)4=(a%*=1 (mod p): so that in fact d is the exponent to which a’ appertains. We thus obtain ¢(d) numbers appertaining to the exponent d: but as it has not yet been proved that any such number as a actually exists, all that can be inferred at present is that yy (d) = ¢ (d) or else ar (d)= 0. But since every one of the numbers 1, 2, 3... (p —1) appertains to some exponent or other, and each exponent is a divisor of (p —1), it follows that PD+y(dy+y(d+..=p-1 where d, d’, d”... are the different divisors of (p—1). But it has already been proved that ?(d)+¢(d)+o(a")+...=p-—1, and therefore y(d) can never be zero, but must always be equal to @(d). Thus there are exactly ¢(d) numbers positive and less than p which appertain to the exponent d. In particular there are @(p—1) such numbers which apper- tain to the exponent (p—1). These numbers are called primitive roots of /p. 20. On account of the great importance of primitive roots, it is desirable to give a practicable method by which they may be found. When one has been discovered, the others may be found, if required, without difficulty: namely, if g is any one primitive root, then the whole system of primitive roots consists of the least positive residues of g, g*, g°,--. g*, where 1, a, 8,...»% are the ¢ (p — 1) numbers less than (» —1) and prime to it. The principle of the following method, which is due to Gauss (D. A. Art. 73), is to find a succession of integers appertaining to higher and higher exponents: it is clear that if this can be done a primitive root must at last be obtained. Take any number prime to p (in practice 2 is the most con- venient, as being the smallest): let this be a, and form the period 2—2 20 PRIMITIVE ROOTS. of least positive residues of its powers’. If there are (p —1) terms in the period, a is a primitive root. If not, suppose there are f terms in the period. Take any other number b not congruent to any power of a, and calculate its period. Suppose there are g terms in this period, where g < p—1: (otherwise b is a primitive root, and we need not continue). There are two cases to consider : elther g is, or 1s not a multiple of f. Take the latter case first, and let m be the least common multiple of fand g. Then we may always put m= f’g’, where /’, g’ are prime to each other, and such that f’ is a divisor of f, and g’ a divisor of g. For let q be a prime divisor of m, and g* the highest power of g contained-in m. Then g* must divide one or both of f and g: if it divides f but not g, take it as a factor off: if it divides g but not f, take it asa factor of g’: if it divides both f and g, take it as a factor of / or g’ (it does not matter which): and sey for any other power of a prime contained in m. Then evidently a/” appertains to the exponent /’ and 69 to the exponent g’: and therefore a/”, b9/7 appertains to the exponent fg, that is, m. For suppose % to be the exponent to which af, bv7 appertains: then a//7, b\/‘ =1 (mod p): this requires that % = 0 (mod /’) and A= 0 (mod g’), and since /” is prime to 9’ it follows that %=0 (mod /’9’), so that the smallest admissible value of X is fg’ or m. Secondly, g may be a multiple of f Then g>/f, so that in this case, as in the other, we have succeeded in finding a number appertaining to a higher exponent than that to which a appertains. The process may be continued, and since we get a higher exponent every time, we must at last arrive at a primitive root. For example, to find a primitive root of 97. Forming the period of the powers of 2, the least positive residues are 2 4 8) 16 ..327.04 (31) $620.2 ias 11-22, 44) 88° 79 617 225=50 5 one 12 24 48' 96 95 93 89 81 65 33 66,35 70° 43° 86 75 53" 9 187 436 72 47 94 91 85 73 49 1 f= 48. 1 Of course, in doing this, multiples of p are rejected whenever an opportunity occurs. sf INDICES. 21 (Observe that since 96=—1 (mod 97) the second half of the residues is obtained by subtracting each of the first half from 97.) The smallest number not contained among the preceding residues is 5, and on forming its period, we find it to be a primitive root. Again suppose p= 73 (Gauss’s example). Let a=2: the series of power-residues is Zea Oo L6is B28) G40 O52 3t © IL fo: Put b=3: the power-residues are 3.9 27 8 24 72 70 64 46 65 49 1 eg 12: Thus m=36, f=9, g =4, afl’ $9 =2,33 = 54, so that 54 appertains to the exponent 36. Forming its period, and taking the number 5, not Poecranteeh in it, this is found to be a primitive root. 21. Suppose that g is a primitive root of p: then since the least positive residues of 1, g, g?...g? are the numbers 1, 2, 3...(p —1) im a certain order, any number a which is prime to p must be congruent (mod p) to some power of g. If g*=a (mod p), a is called the index of a to the base g, and may be denoted by ind,a. Evidently, to a given base, a has an infinite number of indices all congruent (mod p—1): these are not considered to be distinct. It is sometimes convenient to consider the least pesitive value of a as the index of a (to the base g) par excellence. These indices possess properties analogous to those of loga- rithms: it is obvious from the definition that ind (ab) = ind a + ind b | ind a” = minda ind “ =inda—indb b ind 1l=0 the mdices in each congruence being supposed to refer to the same base. (mod p—1), 2 PRODUCT OF PRIMITIVE ROOTS. It should be noticed that the mdex of any number depends upon the particular primitive root which is taken for a base. Suppose that g, h are different primitive roots of p, and let ind, h = so that h=g*(mod p): then if m is any number prime to p, and pw its index to base h, m=h*=g™(mod p): so that ind, m = Ap = ind, h. ind, m (mod p—1). In particular, putting m =g, ind, h.ind, g =1 (mod p—1). A complete table of indices may be used to obtain the solution of the binomial congruence aa#”=b(mod p): namely this gives inda+n.inds#=indb(mod p—1), a linear congruence to find ind#; inda and indb being given by the table. Then inda being known, another reference to the table gives a. This method is of no direct theoretical interest, because a complete table of indices in fact contains a record of all the solutions of #’*=a(mod p); so that we are really only looking out a result already obtained by trial. The result, however, is valuable, indirectly, in connection with the further theory of binomial and other congruences. 22. Let f be the exponent to which a appertains: then at=1=g?" (mod p), g denoting (as usual) a primitive root of p. Hence f.inda=0(mod p—1): so that inda =; (p — 1), where & is some integer. Now & is always prime to /: for if not, suppose k = mk’, f=mf’, then = Gal) =" (p —1), and hence ~ at? =g* ?) =1 (mod p), where f’ ), and therefore the product of all the primitive roots =1 (mod p). The only exception occurs when k= p —1—k, or k=4(p—1) and is at the same time prime to (p—1): but this can only occur when 4(p—1)=1, that is when p =3. 24, Suppose p—1=a7b8cy.../4, where a, b, c...1 are different primes. Let A be any number which appertains to the exponent a*. Then 1+A+A2+...+A”+=0 (mod p). Afar Nee, Also Le ete rea = 0 (mod p). Now the sum of all the numbers, positive and less than p, which appertain to the exponent a* is congruent (mod p) to (l+A+A2+...4A"7)—(14+A%+ A+... 4 Ae"), and is therefore a multiple of p. It is here supposed that a>1. Ifa=1, the sum in question is congruent to A + A?+ A?+...4A%*=-—1 (mod p). Thus the sum of all the numbers, positive and less than p, which appertain to the exponent a* is congruent to 0 or —1 (mod p) according as a>1 or a=1. For the sake of brevity write ¢(a*) =a’, 6(08)=0,.. dA =: let A,A,...Ay be the a numbers which appertain to the exponent a*,and soon. Then any number of the form ABGY LE will appertain to the exponent a*b®cy...2, that is, to (p—1): it is therefore congruent to a primitive root of p. If we expand the product (A,4+-A,+...+Av)(B, 4+ B,+...4+ By)...(L,+L.4+...+ Ly), we get a sum consisting of ¢(p—1) terms, each of which is congruent to a primitive root of p. No two of these terms are congruent to each other: for suppose, if possible, ABC. Da AB C.),.£4(mod.p). 24 CONGRUENCE SATISFIED BY Raise both sides to the power b8cy...2: thus since BY =1 etce., Avre’... = Aver... (mod p), therefore b8cy...ind A = b8c...ind A’ (mod p—1), and hence ind A = ind A’ (mod a). Moreover since A and A’ appertain to the exponent a’, ind A = ind A’=0 (mod B8c7...), by Art. 22; therefore finally, since a* is prime to b8c7,..1, ind A =ind A’ (mod p — 1), whence | A = A’ (mod p). Similarly we could conclude that B=B, C=C,..L=L1/ (mod p); but any two terms of the expanded expression must have at least one pair of corresponding factors such as A, A’ which are incongruent (mod p). Hence no two of the terms are congruent: so that the expression is congruent (mod p) to the sum of all the primitive roots of p. Now if any one of the exponents a, 8, y...X is greater than 1, that is, 1f any square number can be found which divides (p — 1), one of the factors of the expression (A, + A,4+...)(B,4+8,4+...)..(2,+L.+...)=0 (mod p), so that in this case the sum of the primitive roots = 0 (mod p). If, however, each exponent is 1, so that p—1=abc...1, each factor =—1(mod p) and the sum of the primitive roots = (— 1) (mod p), where w is the number of different prime factors of (p — 1). This theorem may be proved in a different manner as follows. Let 91, 92... be the different primitive roots of p. For convenience write p—1=gq, and let a, b, c... be the different primes which divide g. Then the same argument by which ¢(m) was determined (Art. 7) shews that the primitive roots of p are the roots of the congruence (wt — 1) IT (a — 1). TT (waited — J), TI (av — 1), IT (w/e — 1+) I (wiaeete — 1), where IT (a9/* —1) = (2 — 1) (av — 1)... and similarly for the rest. The expression on the left-hand side of the congruence is a rational integral function of x of the degree @(q). This may be proved with the help of de Moivre’s theorem : for any linear factor of the denominator is of the form a — e%*“/4, = 0 (mod p), PRIMITIVE ROOTS. 25 where 6 is a number less than g and not prime to it: and it follows, just as in Art. 7, that this factor occurs precisely as often in the numerator as in the denominator. Now suppose that g has no square divisor, so that ¢ = abcd...: then it has to be shewn that the aforesaid polynomial is of the form 2? —(—1)*x?%— +... where w is the number of different primes a, b, c.... This is easily proved by induction. Namely beginning with «= 1, we have e*—] fy bai AE A Sr ea Pe FY bi next when p = 2 (2 —1)(a-1) (a@)*-1 at -1 (a*—1)(@®-1) a -1 “a1 Sa es ee pe A AIL AEN (ret eee EY == gh (ad) _ gpbilab\—1 4 This is seen by performing the first two steps of the actual division and observing that since b > 1 ab —2b< ab—b-—1. When p» = 3, the function is (asthe — 1) (a — 1) (a? — 1) (@* — 1) (a — 1) (a — 1) (a — 1) (x@—-1) Re CG iter alae ate tigger) {@P - YU i@y- 1 at -D@ 1)’ and by the preceding case this is (a7°4 (2) — wh (ab) —e oe .) . (acd (ab) — yb (ab) —1 fe, es) = yh (abe) di ee (abe) —1 ape the argument being as before. It is clear that the reasoning is quite general, so that in all cases the polynomial is go + (—1)H go + , Since this is identically congruent to @ — gi), the coefficients of #*— in the two expressions are congruent: that is D3 ool ed De oy =(— 1)" (mod p). If gq involves a square factor, we shall have g=m.abc..., and everything is as before except that the polynomial is of the form or? D + (—1y?. gems... where the term in 2?%— is absent: so that in this case gi = 0 (mod p). 26 PRIMITIVE ROOTS. It may be observed also that Lg’? = i=... = 2g” = 0 (mod p), xg” = (— 1)". m (mod p). For example, the primitive roots of 61 are 2, 6,7, 10,17, 18, 26, 30, 31, 35, 48, 44, 51, 54, 55, 59. Here g=60= 2?.3.5, so that «= 3, m = 2, and we ought to have Xg=0, 2g? =— 2 (mod 61); this may be easily verified. | 25. It has been proved that if a is any number prime to the composite modulus m, a?™ =1 (mod m). If ¢(m) 1s the exponent to which a appertains (mod ™), a is said to be a primitive root of m. It is only in a comparatively small number of cases that such primitive roots exist. For suppose m= pqg"r’... where p,q, 7... are different primes. Any number, a, which is prime to m is also prime to p', g", r’...: let fg, h... be the exponents to which @ appertains with respect to the moduli p*, q", 7’, ete.: so that af=1 (mod p'), a7 = 1 (mod 9"), ete. Then if ¢ is the least common multiple of f, g, h... a‘ = 1 (mod m). Now the greatest possible values of f, g, h... are d(p*), 6(q"), p(7”)...: hence ¢ is not greater than the L.c.mM. of $(p*), 6(g"), etc., and if it is less than this L.c.M. it must be a divisor thereof (since fis a factor of ¢ (p*), etc. by Art. 18). Again $ (p") =(p— 1) p*™, which is an even number, except when p=2 and X=1. Hence, generally speaking, the L.c.M. of ¢ (p*), ¢ (g"), ete. will be less than db (p*). P(g") ..., Le. less than $(m), and a fortiori t will be less than $(m), so that a cannot be a primitive root. The only exceptions are when m is a power of a prime, or twice a power of an uneven prime. Further the case of m=2* has to be rejected if X>2. For any odd number can be expressed in the form a=1+4 2k, whence @=14+4(+kh)=14+ 4k (k+1) = 1 (mod 8), and therefore, successively a*=1 (mod 16), a’=1 (mod 32), COMPOSITE MODULUS. 27 and generally a2 = 1 (mod 2°), Now UNG ee Sb (2M) so that 2* has no primitive roots if \ > 2. 26. The only cases which have to be considered are therefore when m = p* or 2p* where p is an odd prime, and the exceptional case of m = 4. The last case is easily disposed of: it 1s evident that there is one primitive root, viz. 3. Next let m=p*. Suppose a is any number prime to m and therefore to p, and let f be the exponent to which a appertains (mod p). Then at = 1 + kp. Hence Ge le cp ye =l+kp.p+k*p’. = 1 (mod p’), (as felons and similarly, av* = (a)? = 1 (mod 7’), af = 1 (mod #*). If, then, a is to be a primitive root of p* we must have fp a multiple of ¢ (p*), Le. of (p — 1) p*: hence f is a multiple of (p—1). But fis also a divisor of (p —1) by Art. 18: hence Abs ory that is, @ is a primitive root of p. Suppose, then, we put w=g, a primitive root of p; then Fo Lt kp, p ' being the highest power of p which divides g? — 1, so that k is prime to p. Raising each side to the power p we infer that gi? = 14 kp' p+ Pho" » (kp'y +. =1+kp*++ higher powers of p =1+kp**! (mod p**?), and similarly gi? eee ek kp't? (mod pts), get ph = tae k pith (mod pithts), 28 PRIMITIVE ROOTS. Putting h =r» —1, we have gi? = 1 (mod p*), so that g is a primitive root of p* if, and only if 7=1; that is, if (g? — 1)/p is prime to p. 27. It remains to find the number of distinct primitive roots of p*: 1e. the number of such roots which are positive and less than p*. Any primitive root of this kind may be written g =a-+ kp, where a is a primitive root of p which is positive and less than p. Hence g?> —1=(a?1—1)+ kp (p— 1) a”? (mod p?) =(a?1— 1)—kpa?~ (mod p?). There are two cases to consider : I. Suppose a?1— 1=0 (mod p’): then g? 1 —1l=-—kpa? (mod p’), “. (g?1 —1)/p = — ka? (mod p), and this will be prime to » if k is so. Now since g 2, requires separate considera- tion. Any odd number of the series 1, 3, 5... 2\—1, may be ex- pressed in the form a=2k +1, where & is odd, and n is at least equal to 2. Hence Of = On ie 1 = 2th +1, where fk, is odd: and similarly a — 2 iL a? = Int, + 1, where k,, k, ... k; are all odd. The only number which appertains to the exponent 1 is 1. If a?= 1 (mod 2%), it follows from the preceding expression for a that n+1 1: then n+t 3, the numbers appertaining to the ex- ponent 2’? (the highest possible, since n > 1) are 4+], 3.441, 5.441,...(27%-1)441. These may be written | i 441, 84+(441) 2.84+(441).... from which it is evident that the series comprises all the numbers less than 2* which are of the form 8n+3 or 8n + 5. AUTHORITIES. This chapter is substantially a paraphrase of the first three sections of the Disquisitiones Arithmetice. The invention of the symbol = by Gauss affords a striking example of the advantages which may be derived from an appro- priate notation, and marks an epoch in the development of the science of. arithmetic. References to the work of Gauss’s predecessors are given by himself (D. A. Arts. 28, 38, 44, 50, 56, 76, 93), and also by H. J. S. Smith in Arts. 1—14 of his Report on the Theory of Numbers (Report of British Ass. 1859). The most important of these, arranged according to the subjects treated, are the following : Continued Fractions and Linear Congruences. EvuLer: Solutio pro- blematis arithmetict... (Comment. Petropol. vii. p. 46 (1740), or Commentationes Arithmetice i. p. 11). Lagrange: Sur la Solution des Problémes Indéter- minés du second degré (Histoire de l Acad. de Berlin 1767, p. 165), and in the additions to the French translation of Euler’s Algebra. See also Smiru, H. J. S.: On Systems of Linear Indeterminate Equations and Congruences (Phil. Trans, cli. (1861) p. 293). The Function ¢(v). EuLer: Theoremata arithmetica nova methodo demonstrata (Comment. Nov. Petrop. viii. 74 (1760), or Comm. Arith. i. 274). Sylvester writes r(n) for @(n), and calls it the totient of n. See two papers by him, Phil. Mag., April 1882, p. 251, and Sept. 1883, p. 230, containing tables of @(z) and Sp(n) up to n=1000. Residues of Powers. EuLer: Theoremata circa residua ex divisione potestatum relicta (Novi Comm. Petr, vii. (1758) p. 49) ; Demonstrationes circa residua ex divisione potestatum per numeros primos resultantia (ibid. xviii. (1773) p. 85); Disquisitio accuratior circa residua ex divisione quadratorum altiorumque potestatum per numeros primos relicta (Opuscula analytica i. (1772) p. 121). These three memoirs may also be found in the Comm. Arith. i. pp. 260, 516, 487 respectively. It should be observed that Gauss first proved the existence of primitive roots for every prime modulus. Theorems of Fermat and Wilson. Fermat’s own statement of his theorem is contained in his mathematical correspondence (see Varia Opera AUTHORITIES. ol Mathematica D. Petri de Fermat (1679) p. 163). As usual, he gives no proof ; the first published demonstration is that of Euler: Theorematum quorundam ad numeros primos spectantium demonstratio (Comm. Petr. viii. (1736) p. 141, or Comm. Arith. i. 21). This practically amounts to showing that if p is prime, the expression (a+1)?—a?—1, is identically divisible by p, so that (a+1)”—(a+1)=a”-—a (mod p), identically: the theorem then follows by induction. A second proof, identical with that of Art. 18, will be found in the Novi Comm. Petr. vii. 49 (see title above). The proof adopted in Art. 16 is after Dirichlet: Démonstrations nouvelles de quelques théorémes relatifs aux nombres (Crelle iii. (1828) p. 390). Sir John Wilson’s theorem is stated by Waring, with a reference to its author, in his Meditationes Algebraice (1770) ; the first published proof is by Piacstan se (Nouveaux Mémoires de l’Acad. de Berlin, 1771, p. 125). The important theorem of Art. 11 is due to Lagrange: Nowvelle Méthode pour résoudre les Probléemes Indéterminés en Nombres entiers (Hist. de ?Acad. de Berlin, 1768, p. 192). Arithmetical Tables. The factor-tables of Burckhardt and Dase have been completed and extended by J. W. L. Glaisher as far as the 9th million. The Canon Arithmeticus, edited by Jacobi, gives a primitive root, and a table of numbers and indices, for all primes less than 1000. Gauss’s Tafel zur Verwandlung gemeiner Briiche in Decimalbriiche (Werke ii. 412) may be used, among other applications, to supply the place of the Canon Arithmeticus, at least as far as p=463. (Cf D. A. Arts. 312—318.) Another table, of less extent, but very compact, is given by Bellavitis: Sulla risoluzione delle congruenze numeriche... (Reale Acc. dei Lincei, 3rd series, t. 1. (1877)). Crelle published in his Journal (xlii. (1851) p. 299) a table of the least positive values of x, and x, which satisfy a,7,=a,%,+1 for values of a, up to 120 and all values of a, less than a, and prime to it. CHAPTER IL. Quadratic Congruences. 30. Ifthe congruence «= a (mod m) is possible, a is said to be a quadratic residue of m, or simply a residue of m, and this may be indicated by writing ahm: if otherwise, a is said to be a non-residue of m, and this is expressed by am. It is convenient to begin by supposing that the modulus is an odd prime, p say. If we form the squares of 1, 2, 3...4(p—1), the resulting numbers are all incongruent (mod p). For if two of them were - congruent, say 7?=s?, this would give (r+s)(r—s)=0, that is, r+s=0, or r—s=0, both of which are impossible, since 7 and s are different and each less than $p. Again since (p—k)? =k? (mod p), it follows that the squares of p+l pt+3 aD SD o>. 4 in the reverse order. Hence the series 1, 2,3...(—1) comprises saat 2 .. (p —1) are congruent to the other series of squares residues of p, and the same number of non-residues. For example if p = 11, PEl@=al, GEBVad PSVeo, CPE7=5, PSG=ss, so that 1, 3, 4, 5,9 are residues of 11, and 2, 6, 7, 8, 10 are non- residues. If aRp, so that a= a (mod p), IE = = 1 (mod p). Thus the quadratic residues of p are the roots of the con- gruence p-1 zx * —1=0 (mod p). QUADRATIC RESIDUES. 33 Every number prime to p satisfies the congruence ge41—]1=0, ee. ea that 1s (2? —1)(@#? +1)=0, hence the non-residues of p are the roots of the congruence p-1 e* +1=0. This affords a method of determining whether a given number a is a residue or non-residue of p, namely by calculating the least p-l residue of a 2 : but this becomes impracticable when p is large. Asan illustration 5°=5.25°=5.3?=45=1 (mod 11), so that 5 is a residue of 11. | The symbol Ee) or (a|p) is used to denote +1 or —1 ac- cording as @ ls, or is not a quadratic residue of p. It is the p-1 absolutely least residue (mod p) of a ? . 31. The product of two residues or of two non-residues is a residue: that of a residue and a non-residue is a non-residue. I. Let a, a’ be two residues: then two integers a, a can be found such that a = @, a’ = a?, whence aa’ = (aa’)’, that is, aa’ is a residue. Ii. Ifq@isa residue, the numbers a, 2a, 3a...(p—1) a are all incongruent, and their least residues (mod p) include all the quadratic residues of p and all the non-residues; but each product of a by a residue is a residue by I.: hence each product of a by a non-residue is a non-residue. III. Let 6 be a non-residue: then as before the series b, 2b, 3b,...(p—1)b is a complete system of residues (mod p): each product of b by a residue is a non-residue by II.: hence each product of b by a non-residue is a residue. The same thing may be proved as follows: we have pri p-l p-1 p= pal a* =(alp),b? =(b|p), (ab|p)= (ab) * =a? 6? =(alp) (|p). Hence (ab|p) =+1 if (a|p) and (b|p) agree in sign, that is if a, b are both residues or both non-residues: while if one is a residue and the other not, (ab|p) =— 1, that is, ab is a non-residue. M. 3 34 QUADRATIC CONGRUENCES. It is clear that (abe|p) =(ab|p) (c|p) = (a|p) (|p) (el p), and so on for any number of factors. 32. Taking any primitive root of p for a base, the indices of the quadratic residues are even, and those of the non-residues are odd. p-l1 = f(p=1) For let indja=/: then. a=g/ (mod p) anda? =g ? : if pr-1l then a 2 =1 (mod p), $f (p—1)= 0 (mod p — 1), that is $f is an pl integer: conversely, if f is even and =2f', a? =gf@V=1 (mod ). This gives another simple proof of the theorem of last article : for ind (ab) = ind a + ind b, and this is even if inda and indb are both even or both odd. 33. Consider the congruence a? = a (mod p*), where p is an odd prime, and a is prime to ». Then if this is possible, so also is x? = a(mod p). Let «=a be a solution of this last congruence, so that a? —a=hp. Put L=a+yp: then x —Aa=a—a+ 2ayp+ yp =(h+ 2ay) p+ yp’. Now determine y so that h+2ay=0 (mod p): then 2—a=0 (mod p*): that is, a solution of 2?=a (mod p?) can be deduced from that of «?= a (mod p). More generally, from a solution of a=a (mod p*) we can deduce a solution of a= a (mod p**1), For suppose v=a is a — solution of #®=a (mod p*),so that e&—a=hp. Write v=a+ yp»: then 2? — a=(h+ 2ay) p+ y°p* = (h + 2ay) p* (mod p*4): and hence if 2ay+h=0 (mod p), c=a+yp* is a solution of =a (mod p**), Thus from any solution of #=a (mod ») can be deduced a | solution of a?= a (mod p*). Moreover if #?= a (mod p) is possible, there are two distinct solutions of the form w=a and a=—a (mod p): so that there will be just two solutions of 2?= a (mod p’), COMPOSITE MODULUS. oD For example, to solve a = 2 (mod 343). Here 2=9 (mod 7): so that the roots of # = 2 (mod 7) are w= + 3 (mod 7). Write Gad 7y; then a —2=T7 (1+ 6y)+ 49’. The congruence 1+6y=0 (mod 7) gives y=1 (mod 7): so that «=10 is a solution of #=2 (mod 49). Finally, putting x©=10+49y, 2 —2=49 (2 + 20y) + 492y?; 24+ 20y=0 (mod 7) gives y = 2 (mod 7), and hence c=10+2.49 =108 is a solution of the given congruence, the complete solution being «= +108 (mod 343), 34. The case next to be considered is when the modulus is a power of 2. The square of any odd number 2n + 1 is 4n? ++ 4n+1=4n(n+1)+1=1 (mod 8) since either n or n+ 1 is even. Hence in the first place the congruence «=a (mod 4) is possible if, and only if a=1 (mod 4): and if this condition is satisfied, there are two distinct solutions, given by #= + 1 (mod 4). Similarly the congruence 2?=a (mod 8) is possible only if a=1 (mod 8): and in this case there are four distinct solutions given by v= 1, 3, 5, 7 (mod 8). Next consider the congruence #?=a (mod 2*) where \>3. This involves 2?= a (mod 8): and hence a= 1 (mod 8): conversely if this condition is satisfied, it may be shewn by an inductive process simuar to that of last article that the proposed congruence has always four distinct roots. | Namely, suppose « is a root of 2?=a (mod 2*)so that a—a=2*h: then putting «=a+ 2".y we get e—a=2(h+ay) + 2% °y? where since X>3, 2A—2>A+1: so that x will be a root of «a —a=0 (mod 2'*1) if h+ay=0 (mod 2): this always gives a suitable value of y, since a is odd. If # = & is any one of the roots, it is easily seen that the four distinct solutions are given by wa+é& w= +(E4+ 2%”) (mod 2?). 36 QUADRATIC CONGRUENCES. 35. We are now able to discuss the congruence «= a (mod m) where m is any modulus whatever, and a any number prime to it. Let TESA OO where , g, 7... denote different odd primes. Then if the proposed congruence is possible, so also must be the following : x= a (mod 2*), 22= a (mod p*), 2 =a (mod gq") etc. Conversely if these are possible so is the given congruence. For suppose &, 7, €... ete. to be any values of w which satisfy the ae wae : =a (mod 2*), # =a (mod p), 2 =a (mod ¢g*),.. ees Then since 2*, p*, gq"... are relative primes, a number can be found so as to satisfy simultaneously the congruences x = & (mod 2"), c= ny (mod p'), x =€ (mod ¢*“)... ete and all such numbers are congruent (mod m) (see Art. 13). Each set of solutions of the auxiliary congruences furnishes therefore one distinct solution of the proposed congruence. Now the conditions to be satisfied in order that the auxiliary congruences may be possible were determined in Arts. 33, 34. If p is any odd prime factor of m, a must be a quadratic residue of p: and if this is so, the congruence a?= a (mod p*) has two distinct soe If «= 0 or 1, there is no further condition: the congruence =a (mod 2) has one root =1 (mod 2): if «=2 it is necessary ie a=1 (mod 4), and then «=a (mod 4) has two roots; and if | k >3 we must have a=1 (mod 8), and then # =a (mod 2*) has four incongruent roots. If then ¢ denote the number of different odd primes which | divide m, the number of distinct solutions of the given congruence, when it is soluble, is Qt Qt gtts according as «<2, « = 2, « > 2 respectively. 36. In order to complete the theory of quadratic congruences, two problems have still to be solved. They are, first, to determine practically whether the congruence 2?=a (mod p) is possible, p being an odd prime, or, in other words, to find the value of (a|p): and secondly to find the roots of the congruence when it has been shewn to be possible, VALUES OF (—1|p), (2|p). 37 It follows from Art. 31 that the determination of (a@ p) may be made to depend upon that of the symbols (— 1) p), (2)p), and (q\p), where q is a positive odd prime. These three cases will now be considered in order. 97. By Art. 30, (— 1p) =(-) ®: this is + 1 if $(p—1) is even. Putting $(p—1)=2n, this gives p=4n+1. On the other hand if p is of the form 4n +3, $(p—1) = 2n +1, and (-1|p)=(- == 1. Hence —1 is a quadratic residue or non-residue of p according as p is of the form 4n+1 or 4n +3. 38. It is found by trial that 2 is a quadratic residue of 7, 17, 93, 31, 41, 47 and a non-residue of 3, 5, 11, 13, 19, 29, 37, 43. The first set are all of the form 82 +1, and the second set are all of the form 8n +3. It may be shewn that this law is general: namely that 2 is a residue of all primes of the form 8n +1 anda non-residue of all primes of the form 8n + 3. We begin with the latter part of the proposition, viz. that 2 is a non-residue of all primes of the form 8n +3. If this is not true, let p be the least prime of this form for which the congruence a= 2 (mod p) is possible. Suppose #=e is a solution of the congruence: then we may take e to be positive, less than p, and odd: for if the congruence is possible there are two suitable values of « which are positive and less than p, and their sum =p which is odd, so that one of the values of « must be odd. Hence e=1 (mod 8), and e? =2 + pf where f is positive and less than p. Now pf=e@ —2=—-1 (mod 8): and since p=+3 (mod 8), f=F3 (mod 8). Consequently f must have at least one prime divisor q of the form 8n + 3: for if all the factors were of the form 8n +], their product would also be of this form. Since f

p, and consequently if a is a residue of sq, and £ of tg, sg+tq=0 (mod p), and hence, since ¢ is prime, s+t=0 (mod p), which is impossible, since s and ¢ are both less than dp. Consequently the series (p — %), (p — 2) noe Yom ea) feed chae eee must consist of the numbers 1, 2, 3... p’ only in a different order ; and therefore io 2) =(p—a,)(p—a,)...(p—a,). 8, 8,.-. Bx (mod p) = (— 1)". 0, Os: ay 9) P2 -=- Oa (mod p). But Cedars; Smet 20 od 2.0 DO age Ly Ze ote). and therefore (—1)".¢@?.1.2.3...p°=1.2.3...p° (mod p). Dividing by 1.2.3...’ which is prime to p, we get (— 1)". q@” =1 (mod p) or gq? =(—1) (mod p). But | g” = (q|p) (mod p) by Art. 30: therefore finally (¢|p)=(—1)".. It remains to calculate the value of this expression. It is convenient in what follows to denote by [x] the greatest positive integer contained in the real positive quantity w; so that “ =|#]+ 0 where @ is a positive proper fraction. Making use of this notation we may write q=pPla/pl+r 2q =p [2q/p] +s PO=PlPaPlt+7y, 40 LAW OF RECIPROCITY. where 7,, 7, ... 7p are all positive and less than p: hence by addi- tion Pa" q=Mp+A+B (1), where M=[q/p) + [2¢/p] +... + [p'a/p] A=,+4,+... +O, B= B+ Bet... +B). Now (p —a,)+(p—%)+...+(p — ay) +2, + Bot... +O, =142434...4- 7, that is yp-A+B=P— Combining this with (1) we have 2-1 PF (q-1)=(M-p)p +24. Now & a : is an integer, since p is odd; also (q— 1) is even: therefore (IM — w) p= 0 (mod 2) and hence » = M (mod 2), so that (qip)=(-— 1 —(-1). In exactly the same way (plq) =(— 1" where N=[p/q] + [2p/q] + --- + [¢'p/4). Of the two numbers p, g one must be less than the other: suppose q iS (mod 2). Il. With the same notation The proof is very simple : ere R=(14+r—-DQ4r—-)(4+r’-1)... =1+3(r—-1) (mod 4), Similarly R=(14+r—1)(14+r?-1)... =1+> - —1) (mod 16), and therefore (mod 2). therefore aa 4. P, as before, being an odd positive number, pal (=) (ieee P-1 (Q|P)=(-1) ©. For by definition | (-1UP)=-lp)Clpiyelp)... 3 P=} = (— ie 2 =(-1) it 3 Oke Lemma I. pe pees 1 andigimilariy) » <(@)02) =a) ae nT by Lemma IT. 5. Ifthe positive odd numbers P, @ are prime to each other, CO) OA re | Namely, if P = ppp”... and Q=q¢q"... it follows from theorems (1) and (2) that (P| Q) = II (pl), where the product extends over all the different combinations of a factor p with a factor gq. & GAUSS’S FIRST PROOF. 45 Similarly (Q|P) = II (q|p), and therefore (P|Q)(Q|P) = (pig) |p) =I] (— Lye a1) a (— 1) 23 (p—1)(q-1), Now since the summation extends over every combination (, 7) —1 —l 2t(p—-l@- = Soe x ee Citra <0 = i dita (mod 2), by Lemma I. P-1 Q-1 1 Hence (P|Q)(Q P)=(-1) = It is clear that by means of Jacobi’s extension of the law of reciprocity the algorithm for finding (q|p) may often be abbreviated. For instance, in the latter part of the example of Art. 41 we may proceed thus: (39|47) = — (4739) = — (8|39) = —(2|39) =-1: the resolution of 39 into its prime factors being avoided. 2 = (— 1) L-VE@-D, 43. Like many other important theorems, the law of quadratic reciprocity was first proved by an inductive method. Gauss’s original proof (D. A. Arts. 125—144) has been considerably simplified, without altering its character, by Dirichlet (Crelle, t. 47, p. 139). As it is an admirable example of matixematical induction, it seems proper to reproduce it here. It is assumed that the formula (p|q) (q\p) =(— 1) *2-V@-9, where p and g are positive odd primes, has been proved for all such primes which are less than a particular prime, say q: it is then shewn to be true for every combination of q with a smaller prime: the theorem being true in the case of the two smallest primes 3, 5 it is thus seen to be true universally. We observe in the first place that if Legendre’s formula is true for every combination of two primes less than q, Jacobi'’s genera- lised formula (P|Q) (Q|P) = (— 1)?@-Y@-» is also true, provided P,Q are positive odd numbers, every prime divisor of which is less than g. This is obvious from Art. 42. 46 LAW OF RECIPROCITY. It will further be supposed that the results of Arts. 37, 38 with respect to the values of (—1|p) and (+2|p) have been proved: from these follow the values of the generalised symbols (— 1|P), and (+ 2|P) by Art. 42. The complete proof requires the discussion of three separate cases. I. First let pRq: then an integer e can be found such that Cie aay Further, we may suppose e less than g and even: for there are two suitable values of e which are less than qg, and their sum is q¢ which is odd: hence one of the values of e must be even. Hence f is odd, positive, and less than gq. Now if fis not divisible by p, it follows from the above equa- tion that (p| ff) =1, and (¢f|p)=1, that is, (¢|p) (f|p) =1. Hence by multiplication, (qip) F |p) lf) = 1, or @p=(Slp) lf) =C 1) by the extended law of reciprocity. Now by Lemma I. of Art. 42 ae nae a (mod 2) 2 2 Sal =° (PE) (mod 2) =—P** (moa 2) since é is even, pb meheg iy eich) coals ee tee wpa tot aanoP ED as 4 (mod 2) = 0 (mod 2), (q| pp) =(— 1Tke a = (— 1a which agrees with the law of reciprocity, since (p|q) = 1. If f is divisible by p, so also is e, and we may write e=e'p, f=f'p, so that ep Eat =qf" where eé’ is even, and /’ is odd and less than gq. Hence e?p = 1 (mod /’) and therefore (p| /’) =1. Also Cee that is, (1)? Gip)(f|p)=1- GAUSS’S FIRST PROOF. 47 multiplying, etc. as in the other case, we infer that oe Cpe = (— Lyee4 aah 1 = a ty el} Since gf =—1 (mod 4), one of the numbers q, / must be of the form 4n + 1 and the other of the form 4n +3, whence - aly and therefore (q|\p)=(— 129 as before. II. Next suppose pNq and g=38 (mod 4): then —pfq, and we may write e+p=af where, as before, ¢ is even, and f odd, and less than gq. f-1 Hence 1=(-p|f)=(-1) 2 (p|f) and 1=(qf|p)=(q|P) (Ff |p): ee therefore (q\p)=(-1) ® (pf) CFP) = (— 1)Hepty yy, Now since gf = p (mod 4) and q = 3 (mod 4), f= 1 or 3 (mod 4) according as p= 8 or 1 (mod 4): therefore p—f= 2 (mod 4), and consequently £(p+1)(f-)-4(p-1)(f +1) =3(f- p) = 1 (mod 2) therefore (q| p) = —(— 12) 49), As in Lemma L (q—-1)+(f+1) = af+ 1 (mod 4) =p+1 (mod 4); Peal) (pol) ih |) pig L {mod 3), = 0 (mod 8), 4(p—1)(q—-1)+4(p-1) (Ft 1) =0 (mod 2), and therefore finally (q|p) =— (— 1)? which agrees with the law of reciprocity, since (p|q) = — 1. If p divides f, we put, as before, e = e'p, f=f p, ep+l—af, where f’ is odd and = 8 (mod 4). 48 LAW OF RECIPROCITY. Hence (plf)=-1 l=(@f |p)=(@\p) (F'|p) and (91) ao) 2) ee oak) a =—(- 1 #29 (@-) since F =¢ (mod 4). 44. The case which remains to be discussed is that in which pNq.and q=1 (mod 4). It has to be shewn that in this case qNp. The proof is effected with the help of the following lemma :— There exists a prime number p’ less than q such that qNp’. When g=5 (mod 8), this is easily proved: namely q—2=38 (mod 8), and therefore g—2 involves at least one prime factor of the form 8n+3 or 8n+5: if p’ be any such factor, g—2=0 (mod p’), and therefore (q|p) = 2|p) =— 1. Next suppose g= 1 (mod 8). Let m be the greatest integer which is less than /q, and suppose, if possible, that g is a quadratic residue of all odd primes which do not exceed 2m+1. Then if M=(2m+1)! the congruence a®=q (mod M) is possible (cf. Art. 35), and we may therefore find a positive integer k such that k°=q (mod J). Hence (q— 12) (q — 2°)...(¢ — m’) = (# — 1°) (? — 2°)...(k? — m?) (mod M). Now k (k? — 1°) (k? — 2?)...(k? — m?) =(k+m)(k+m—1)(k4+m-—2)...4(k-1)...(&—m), and since this is the product of (2m-+1) consecutive integers it is divisible by (2m+1)! 1e. by MW, so that k (q — 1?) (q —2°)...(¢ — m?) = 0 (mod M), and therefore, since k is prime to I, (g—V)@q— 2)...¢ — m)/M is an integer. But this quotient may be written in the form 1 g—l q— 2? g—m m+1'(m+1P—2 (m+ 1? — 22° (m+ 1P—m?? where each factor is a proper fraction: so that 1t cannot be an integer. GAUSS’S FIRST PROOF. 49 Thus there must be at least one prime less than 2m + 1, of which q¢ is a non-residue: and 2m+1 < 2/¢+1* (mod 2): also (pp —1) = (p—-1) + 4 (y'— L) (mod 2): so that the index d is congruent (mod 2) to —-1 p-l (Pot PS) (PE) +t DW D, that is to 4(p?-1)4+4(p4+1)(7'- 1), which is evidently even. Hence qipp)=+ 1, and therefore as before (q\p)=—1. The case in which f is divisible by p but not by p’ may be similarly treated. III. Finally suppose / is divisible by pp’. Putting e=epp’, f=f pp’ we get e"*pp —1l=aqf', whence (ppl yp) = 1, (— af" | pp) = @l pp) (S| pp) = 1, and therefore (q| pp) =(—S"| pp’) (pe P) =(— 1) (pp —ayty 7A) Now f"'=—1 (mod 4) so that I ee is even, and therefore (q| pp’) =+1, and thence as before (q|p)=—1. The law of reciprocity has therefore been completely proved. 46. It is now easy to find the forms of the prime numbers of | which a given number JD is a quadratic residue: more generally, we can determine all the positive numbers n which are prime to . 2D, and such that (D|n) = + 1*. Evidently we may suppose that D is not divisible by any square: for if D= aD, (D\n) = (a?\n) (D!|n) = (D'In). If then P denotes the positive product of all the odd primes which divide D, we have D=+P, or D=+2P. 1 Cf. Dirichlet, Zahlentheorie, p. 121, DIRICHLET’S TABLE FoR (Din). 51 There are four cases to consider: 1s D=+P=1 (mod 4). By the generalised law of reciprocity (D\n) = (n|P). Now (n'|P)=(n|P) if n =n (mod P): moreover n is to be odd: so that it is sufficient to consider the ¢ (P) odd numbers which are less than 2P and prime to P. Let a be any one of these numbers for which (a|P)=+1: then (D|n)=+1, and n is odd, if | n=a (mod 2P). Similarly (D|n) = —1, and n is odd, if n= b (mod 2P), where b is any one of the numbers less than 2P and prime to it, for which (b|P) = — 1. For instance if D=138, the odd residues of 13 which are less than 26 are Le Sil e238 25 so that (13|n)=1 if n=1, 3, 9, 17, 23, 25 (mod 26), and (13}%) si at mid, Geli 15; 19, 21 (mod 26). iap D=+4 P=3 (mod 4). n-1 Here (D\n) =(—1) 2 (n|P), so that . (D[n)=+1 if (n|P)=+1, n=1 (mod 4), = or (n|P)=—1, n=3 (mod 4): while (D|n) =—1 if (n|P)=+1, n=3 (mod 4), or (n|P)=—1, n=1 (mod 4). Til. D=+2P=2 (mod 8). In this case (D\n) =(— 18, (n|P): (Din)=+1 if (n|P)=+1, n=+1 (mod 8) or (n|P)=—1, n=+8 (mod 8), (D\n)=—1 if (n|\P)=-1, n=+1 (mod 8) or (n|P)=+1, n=+3 (mod 8). 4A—2 III 52 QUADRATIC CONGRUENCES. IV. D=+42P=6 (mod 8). (D\n) = (—1) 3H, (n| P) (D\n)=+1 if (n|P)=+1 n=1, 3 (mod 8) or (n|\P)=—-1 n= 5, 7 (mod 8) (D\n)=-1 if (n|P)=—-1 n=1, 3 (mod 8) or (n|P)=+1 n= 5, 7 (mod 8). In each of the cases [I-IV the values of n group themselves into arithmetical progressions with a common difference 4D instead of 2D as in the first case. Example. Let D=— 80: this is a case of III, with P=15. It will be found that (n|15)=+1 if n=1, 2, 4, 8 (mod 15) and that (n|15)=— 1 if n=7, 11, 18, 14 (mod 15). The conditions (n!15)=+ 1, n= +1 (mod 8) are simultaneously satisfied if N= VAT, Zool Aig Ad, 79, 113 (mod 120); and (n|15)=— 1, n= + 3 (mod 8) simultaneously if n=11, 13, 29, 37, 43, 59, 67, 101 (mod 120). Thus (— 30|n) = + 1 if n= 1, 11,13, 17, 23, 29, 31, 37, 43, 47, 49, 59, 67, 79, 101, 113 (mod 120). It should be carefully remembered that if n is a composite odd number the condition (D|n)=+1 is necessary but not sufficient in order that D may be a quadratic residue of n. The necessary — and sufficient conditions are that (D|p)=+1 for every odd prime p which is contained in n (ef. Arts. 33—-35). For instance (2\15) = 41, because (2/15) = (2/3) (2/5) =(—1)(—1) = 41; but 2 is not a residue of 15. The integers of which a given number D is a quadratic residue are sometimes referred to as the divisors of the form a?—JD, or of the form «— Dy*?: the meaning of the expression being that integral values of # and y, prime to each other, can be found so as to make 2?— Dy? a multiple of the divisor in question. Thus the problem of the present article has been completely discussed by Legendre under the head of “finding the linear forms which belong to the divisors of # + cw?” (Théorie des Nombres, 2°™ Partie, GAUSSS METHOD OF EXCLUSION. 53 §§ x—x11): and he gives at the end of his work (Tables III—VI1) lists of the odd lnear divisors of a large number of forms ¢ + cu’. A comparison of Legendre’s method with that here given will shew that both agree in principle, but that a good deal of clearness and brevity is gained by the use of the extended law of reciprocity. 47. The practical solution of the congruence «=a (mod p), when its possibility has been established, is by no means easy when p isa large prime. A direct solution may be given when p is of the form 4n +3 or 8n+5. Namely if p=4n+4+3 and aRp ati = 1 (mod p); whence a=a"* (mod p), and the roots of x = a (mod p) are therefore x=+a"" (mod p). Similarly when p = 8n+5 and aRp a+? — 1] = 0 (mod p); whence either Ties A| (mod p) or att = — | (mod p). In the former case a”#=a and the solution is given by e=+a"" In the latter case, since —1=(8n+4)! (by Wilson’s Theorem) = ]?. 27.37... (4n+2) = M? (say), anti = iM? Now determine y so that ay=1 (mod p): then multiplying the last congruence by y”, asy™ M?, and therefore x=+ My”. When p=1(mod 8) an indirect method has generally to be adopted: and in fact, as Gauss has remarked, indirect methods are always preferable to the preceding, as being less laborious. Gauss has given a “method of exclusion,’ which may be applied with advantage when p is moderately large. The solution of the congruence a? = a (mod p) is identical with that of the indeterminate equation | v= pyta. 54 QUADRATIC CONGRUENCES. We may suppose that « is positive and less than 4p: hence a* ) 5) unitary substitution ( of determinant e: and let ,, o, be the roots of (a, b, c), @,', w, those of (a’, b’, c’). Then if w’ is ag root of (a’, b’, c’) corresponding to @,,- aw + 8 yo +8’ QQ; = and therefore "Rael do, —8 el 6(—b+/D)— Ba —=yo,+a —y(—b4+V7D)+aa _ (aat by + yD} {— a8 — 08 + 8D} (aa + by)? — Dry?” a {aaB + b(ad + By) + oy8} + a (ad — By) VD a (aa? + 2bary + cy’) ~~ oteVD, = 7 5 a consequently w’ = @,' or @,/ according as e=+1 or —1; that is, according as the forms are properly or improperly equivalent. This way of expressing the distinction between proper and improper equivalence is due to Dirichlet*. 1 Berlin Abhandal., 1854, p. 99 ; Liouville (2nd series), vol. ii., p. 353. COMPOSITION OF SUBSTITUTIONS. 63 57. Suppose that a form f= aw#?+ 2bey +cy* is transformed into f’=a'v? + 2D'a'y'+ cy” by the substitution S= ie 5 OL determinant ¢; and that f’ is transformed into fl= a” a2 4 Ob" aly! +y"”? by the substitution S’= is s , of determinant é’. Y> Then since x= ax’ + By’ = a(a'a" + fy") + B(x! +8y’) = (aa + Byes (08 + Bo’) y" and similarly = -y = (ya’ + 89’) @” + (v8 + 88) y”, it is clear that f is transformed into f” by the substitution a + Bry’, a8’ + a) ; ya’ + by’, y+ 60'/° This substitution is said to be compounded of S and S’, and is denoted by SS’. It is to be carefully observed that yge he + By, va B+ a) yat dy, 7B+56 is, in general, different from SS’. In forming the elements of SS" we proceed in the same way a, B| |o’, | y, 6 a m0: multiplication being performed according to rows of the first matrix and columns of the second. The determinant of SS" is faa! + Br’) (yB" + 68’) — (a8 + 88’) (ya + Sy’) = (a6 — By) (a0 — By’) / =€€. as in the multiplication of the determinants , the 2 This is positive or negative according as e, e’ agree or differ in sign; therefore SS’ is a proper substitution if S, S’ are both proper or both improper; otherwise it is improper. We may compound SS’ with any other substitution S” and thus obtain (SS’)S”. It may easily be verified and is, in fact, obvious that this is the same as S(S’S”), so that it may be expressed without ambiguity by SS’S”; and, in general, the _ result of compounding any number of substitutions §,, S,,...S,, in this order, may be written S,S,...S,, and called the product 64 BINARY QUADRATIC FORMS. of the substitutions (in this order). In particular, a substitution may be compounded with itself any number of times and the result written in the form S?. The substitution & a leaves the variables unaltered; it is called the identical substitution, and may be denoted by 1. Let S= & B be a unitary substitution, so that e& =1. We y, 6 have (qj B ( d/e, — ah & (se = ie eC Y> 6 7 y/€, a/e 0 ’ (ad ag By)/e Nee = ( ; +) =af The substitution S’= ( o/e, — 8} ) is said to be inverse to S, —y/e, ale and is denoted by St. Evidently (S7)7=S, and S7S=SS7=1. We may also write S~ for (S)", where n is a positive integer ; and it is clear that S~"=(S”)—; for instance, Deo We SA ino rie > AO ane Seale and so in general. Thus for all positive and negative integral indices, S™S” =S™*" ; and after the admission of negative indices it still remains true that in the composition. of substitutions the associative law of algebraical multiplication is valid, but not the commutative. 58. Applying these results to the theory of quadratic forms, we draw the following conclusions :— I. Ifft, f....f, be any number of forms each of which contains the next following, f, will contain /,; and a substitution which transforms f, into f, may be obtained by compounding the sub- stitutions which convert f, into fi, f, into f,... fr. into f,. The resulting substitution is proper or improper according as the number of its improper components is even or odd. Il. If fi, fi, fs are any three forms, such that f, is equivalent to fy, and f, to fs, then f, is equivalent to f;. | In particular, if fiw f,, and fiw fs, then fiw f. For the present only proper equivalence will be considered, and ‘equivalent’ will be used in the sense of ‘properly equivalent.’ Similarly ‘unitary substitution’ will mean ‘proper unitary sub- stitution, unless the contrary is stated REPRESENTATION AND EQUIVALENCE. 65 Reduction of the problem of representation to that of equivalence. 59. It has already been observed (see above, Art. 50) that in discussing the representation of numbers it is sufficient to con- sider primitive representations. Suppose, now, that m = aa? + 2bay + cy? is a primitive representation of an integer m by the form (a, b, c), so that a, y are integers prime to each other. Let integers £, 6 be chosen such that ad—®By=1. Then the substitution bs 5) converts (a, b, c) into an equivalent form (qa’, b’, c’), where a =ae+ 2bay + cy? =m, b’ = aaB +b (ad+ By) + cyd =n, say, c = a8? + 2b86 +c& = l. Conversely, if we can find a form (m, n, l) equivalent to (a, b, c) and if (5 7% 6 into (m,n, 1), then m may be represented in the form (a, b, eX, y? by putting «=a, y =¥. ) is any unitary substitution which converts (a, b, c) Since n? —la=D, it follows that n? = D (mod m), and therefore D is a quadratic residue of m. Hence no number of which D-is a non-residue can be represented by any form of which the determinant is D. On the other hand, if m is any number of which D is a residue, and n a root of the congruence n?= D (mod m), then (n?— D)/m is an integer, / suppose, and the form (m,n, is of determinant D, and one by which m can be represented. If tn, tn’, +n”, etc. are all the incongruous solutions of n?=D (mod m), and J, I’, 1’, etc. the corresponding values of (n* — D)/m, every form (a, b, c) of determinant D by which m can be represented must be equivalent to one of the forms (m,n, l), (m, —n, Ll), (m, n’, U’), (m, —n’, UV’) ete. Conversely m can be represented by any form which is equivalent to one of these. When a, y are given, the general values of 8, 6 are of the form B=B,+ha, §=68,+ hy, M. 4) 66 BINARY QUADRATIC FORMS. k being any integer, and ,, 6, two particular values of 8, 8 so that ao, — Boy=1. Hence the general value of n is n=(aa+ by) B+ (ba+ cy) 6 = (aa + by) By + (ba + cy) 6 + k (ae? + 2bay + cry’) =n, + km =n (mod m), m) being a particular value of n. The representation m = aa? + 2bay + cy? is said to appertain to the root m of the congruence n?= D(mod m). Representations appertaining to the same root of the congruence are said to belong to the same set. We shall return to the problem of representation later on: meanwhile, enough has been said to shew its dependence upon the theory of equivalence, which will now be considered in detail. 60. Before doing so, however, it may be well to give an out- line of the principal results which will be obtained. Forms which are properly equivalent are said to belong to the same class. Each class contains an infinite number of forms, any one of which may be taken as a representative of the class; all the other forms of the class being derivable from it by means of unitary substitutions. For every determinant the number of distinct classes is finite. The proof of this fundamental proposition consists in shewing that for any given determinant there exists a limited number of forms, called reduced forms, the coefficients of which satisfy certain conditions of inequality, and that each class contains at least one reduced form. The criteria of a reduced form are quite — distinct for definite and indefinite forms; but in each case we are able to construct a complete system of reduced forms and to ar- © range them into sets of equivalent forms. The number of sets is equal to the number of classes for the determinant considered. A method is devised by which any proposed form may be transformed into an equivalent reduced form; so that in order to find out whether two given forms of the same determinant are equivalent or not, it is sufficient to find reduced forms equivalent to them, and then decide whether these reduced forms belong to the same class. This can be done in every case; and moreover, if the two given forms are equivalent, the process by which this fact ORDERS AND CLASSES DEFINED. 67 is established also enables us to find a unitary substitution which will convert one of the forms into the other. Finally, by a method which is applicable to all quadratic forms, we are able to find all the substitutions by which a given form may be converted into another given form which is equivalent to it. 61. Let the form f= (a, b, c) be transformed into the equiva- lent form f” = (a’, b’, c’) by the proper unitary substitution @ ‘s : then (Art. 54) a = ae? + 2bary + cy? b’ =aaB +b (ad + By) + cys c = a8? + 2b88+ c& From this it is evident that every common divisor of a, b, c is also a common divisor of a’, b’, c’. Now since f’ is converted into > f by the inverse substitution e ; he as prtegral linear functions of a’, b’, c’; therefore every common divisor of a’, b’, c’ is also a common divisor of a, b,c. Therefore dv (a’, b’, ce’) =dv (a, }, c) and similarly du(a., 2b’, c’) =dv (a, 20, c). That is to say, f and /’ are either both properly primitive, or both improperly primitive, or both derived forms; and moreover in the last case they are derived from equivalent mince forms. j. a, b,c can be expressed Classes, therefore, like forms, may be distributed into primitive and derived classes. Any two forms (a, b, c) (a’, b’, ce’) of the same determinant are said to belong to the same order if dv (a’, b’, c’) =dv (a, b, c) and dv (a’, 2b’, c’) = du (a, 2b, c). There is a corresponding arrangement of classes; so that we may have, in the most general case, orders of ayes primitive classes, of improperly primitive classes, and of classes derived from properly and improperly primitive classes respectively. Inthe theory of equivalence it is sufficient to consider primitive forms ; and in like manner with regard to the representation of Le For it follows from the results of this article that every form equivalent to (wa, wb, wc) must be of the type (wa’, wb’, uc’), 9—2 68 BINARY QUADRATIC FORMS. and from the equivalence (ua’, wb’, uc’) ~ (ua, wb, wc) we infer (a’, b’, c’) ~ (a, b,c). Similarly if m is representable by a derived form (ua, wb, wc), m/u must be an integer, and to every representa- tion of m by (ua, ub, wc) corresponds a representation of m/w by (a, 6, c) and conversely. In what follows, therefore, unless the contrary is stated, it will be supposed that the forms considered are primitive. 62. ) ; and by compounding these we 70 BINARY QUADRATIC FORMS. obtain ( rf io ai by means of which (10, 17, 29) is transformed into (1, 0, 1). Or, again, let the form be (29, 51, 90): then the series of forms 1s (29, 51, 90), (90, 39, 17), (17, — 5, 2),:(2, +1, 5), where the forms (2, 1, 5) and (2, — 1, 5) are both reduced. If w,, @, are the roots of Aw?+2Bo+C=0, the conditions that (A, B, C) may be a reduced form are equivalent to |w, + @,|>} 1, |w, a.|¢ 1. Observe that since @,, @, are conjugate complex quantities, |@,|=|@,)|, so that the second condition may be replaced by |,|¢1. The significance of this will be seen later on. 64. For a given negative determinant, the number of reduced forms ts fumte. . Let (A, B, C) be a reduced positive form for the determinant D=—A; then A= AC—B* The conditions of reduction being 2|B\>A, and C+ A, we have AC ¢ A, and B?+4A?; hence A= AC— B+ 8A, or A V/4A. Hence also |B) $V1A, and AC=A +B? +4A, so that C$4A. The values of A, B, C being all limited, it follows that the number of reduced positive forms is finite. From each positive reduced form (A, Bb, C) we obtain a cor- responding negative reduced form (— A, B, — C) by changing the sign of the extreme coefficients. | In order to construct a complete set of reduced forms for the determinant — A, we take B=0, +1, +2,...+%, where X is the greatest integer contained in V4A; then, attributing to B any one of these values, we break up A+ 6? in all possible ways into the product of two integral factors A, C: finally, we reject those com- binations for which the conditions | C|<¢|A|{2)B| are not satis- fied. The remaining sets (A, B, C) give reduced forms, Example 1. Suppose A= 40. Here V40/3 is between 3 and 4, so that 7 =3, and the calculation is as follows :— B=0 AC =4F 407220 24t 10 mos, tel a 1.41%, +2 1. 44") 2 22" 4, 1 hd 140 eee REDUCTION OF DEFINITE FORMS. 71 The decompositions marked with an asterisk have to be re- jected: so that the positive reduced forms are (1, 0, 40), (2, 0, 20), (4, 0, 10), (5, 0, 8), (4, £2, 11) (7,54 3, 7 )5 eight in all. It is to be observed that this process gives the derived as well as the primitive forms; thus, in the above example, (2, 0, 20) and (4, 0, 10) are derived forms. Example 2. A939, MS 3; Be 0 A Ot ee aoe Liye hel LAO" 20204 .10,75..8, +2 1. 43%, ue 3 Lie4S 2 S245 63, 1684 128658: The reduced positive forms are: (1) properly primitive, (1, 0, 39), (3, 0, 13), (5, + 1, 8): (ii) improperly primitive, (2, + 1, 20), (4, +1, 10), (6, +3, 8). 65. We have now to inquire whether two positive reduced forms (a, b, c), (a’, 0’, c’) of the determinant — A can be properly equivalent. Suppose that the substitution 6 ae converts (a, b, c) into (a’, b’,c’). Then (aa! = 0 (ae + 2bay + cy?) =(aa + by) + Ar? ¢ Ar’. Now an > 4A, and vy is an integer: therefore y?=0 or I. Similarly, by considering the inverse substitution (a ; ae) which converts (a’, b’, c’) into (a, b, c), we conclude that 6?=0 or 1. If 8=y=0, a=d6=+ 1 and the forms are identical. Next suppose y=0, B=+1; then a=d=4+1,a@=a, 0 =dta. The conditions of reduction 2|b|+a, 2|b’| +a’, cannot both be satisfied unless b=+4a, b’=+F4a. The forms (a, $a, c), (a, — 4a, c) are in fact equivalent, the first being converted into the second by vate 1; —1 the substitution G A : i2 BINARY QUADRATIC FORMS. Suppose y=+1, @=0. This is merely the inverse of the preceding, and leads to the same result. Next, let y=+1, B=F1; then ad=0. If a=0, Ca 0; b’ =—b +06, c =a 2b6 + co. The conditions 2\b| +a, 2|6'|+a’ cannot both be satisfied, except in the two following cases :— (i) 8=0. The forms in this case are (a, b, a), (a, —6, a), which are obviously equivalent, the transforming substitution being (_ 1’ 9): Gi) S6=+1, Here 2|b'| >c, except when c=a, and bo) =—b=+ha. The forms are (a, 4a, a) and (a, —4a, a), of which the first ; . 1,- may be converted into the second by the transformations ( : _ 0 Sret and cs i - . The case where 8 = + 1, y=0 leads to the same result. Finally, let GB=y=+1; then ad=2, so that either a= + 2, 6=41, or a=4+1, 6=+2. Suppose a=2, 6=1. Then by the equations of transformation, a =4a+4b+6¢, ) Ca Oat 20 ic. therefore a —c =8a + 2b, which is positive; hence a >c’, and the form (a’, b’, c’) cannot be reduced. The other cases may be treated in the same way: and the conclusion is that the only possible pairs of equivalent reduced forms are (a, $a, c), (a, —4a, c) and (a, b, a), (a, —b, a); with (a, ga, a), (a, — ha, a) belonging, as wt were, to both cases. 66. From every such pair of equivalent reduced forms we reject that form of which the middle coefficient is negative. . The REDUCTION OF INDEFINITE FORMS. 70 remaining forms are all non-equivalent, and may be taken as representatives of the different classes for the given determinant. Thus, for instance, when A = 39, we reject one of each of the pairs (2, + 1, 20) and (6, + 3, 8): there remain eight representative forms, four properly primitive, ai 0, 39), (3, 0, 13), (5, ih eH); (5, a 1, 8), and four improperly primitive, (2, 1, 20), (6, 3, 8), (4, 1, 10), (4, —1, 10). Similarly when A=40 the equivalent pairs are (4, +2, 11) and (7, + 3, 7), and we may take as representative forms (1 0840 \o5 (5.0) Sn (405 11) (%) 32-7) (2, 0, 20), (4, 0, 10). In order to discover whether two given definite forms are equivalent, we find, by the process of Art. 63, a reduced form equivalent to each. The necessary and sufficient condition for the equivalence of the proposed forms is that these reduced -forms should either be identical, or form one of the special pairs of equivalent reduced forms. Reduction of Indefinite Forms. 67. When the form (a, b,c) is indefinite, the roots of the equation aw? + 2bm + ¢=0 are both real. The process of reduction is quite different from that employed for definite forms, and is closely connected with the expansion of a root of the equation in the form of a periodic chain-fraction. As a first step, we observe that for a given positive determinant D there is only a limited number of forms (a, b, ¢) in which a, ¢ have opposite signs. For if (+a, b, #c¢) be such a_ form, b?+ac=D, and hence |6||C|>/D-—B: that is, |A|, |C]| satisfy the same conditions of imequality. III. The roots of dw? +2Bme+ C=0 are —B+/D oO, =— + ‘ , a proper fraction, and —B-WV/D . @, = ——,—— , an_ Improper fraction ; moreover @,@,= C/A, which is negative, so that w,, @, differ in sign. Conversely, if ,, w, are of opposite signs, and | @,| <1, | »,|> 1, the form is reduced; for these conditions require that — B+ /D and —B—,/D should have opposite signs, and that —B—».W/D should be the greater numerically; hence B is positive and less than /D. Also since |,|<1, ~D—B<|A|, and since | ,| > 1, VD+ B>| A). IV. Since | Aj |C\>|Al>/D-B. Hence B is positive and < /D, and moreover | (D+ B>|Aj\>VD-B: that is, (A, B, C) is reduced. Let 6, 0’ be chosen so that b+b =— da, VD—|a'\|a”|, the process can be repeated; and it follows, as in Art. 63, that after a finite number of operations we must arrive at a form (A, B, C) for which ali the mequalities are satisfied. For example, let the form be (76, 29,11). Here D=5,rX=2, and the process of reduction leads to the series of forms (76, 29, 11), (11, —7, 4), (4, -1, -1), (1, 1, 4), of which the last is reduced. The reducing substitutions are 0, i) 0, i) oe . es yh tex Diei | Ae tes 2200 Fi af, 9 which (76, 29, 11) is converted into (—1, 1, 4). and by compounding these we obtain ( by means of 70. The next thing to be done is to find out which of the reduced forms are equivalent. To this end, we require the following proposition :— Tf (a, b, a’) ws a reduced form, there exists one, and only one adjacent reduced form (a’, b’, a”) which is equivalent to it, and has its first coefficient equal to a’. Suppose, in the first place, that a is positive: and let us write —a’ instead of a’, since the third coefficient must be negative (Art. 67). Let 6, b’ be chosen so that b+b'=da, VD-a a, PERIODS OF REDUCED FORMS. 77 aes i, ; and therefore the substitution i 5 :) converts (a, b, —a’) into a form (a’, b’, a”) which is reduced. In a similar way, from a reduced form (—a, b, a’) of which the first coefficient is negative, we may derive an adjacent reduced form (a’, b’', —a@’). Namely, if we determine 6, b’ by 6+)’ =— 6a’, VD—-a a’, so that the substitution ( x 4 converts (—a, 6, a’) into a reduced form XT +5 (a’, b’, #5 a” 71. Starting, now, with any reduced form (a, 6, a’), we deduce in this way a series of reduced forms (@’, b’, a”), (a”, b”, a” etc., each of which is equivalent to the one before it. The total number of reduced forms being finite, we must at last arrive at a form identical with (a, 6, a’). In this way we obtain a cycle or period of reduced forms CASO UAC 0 Cae (eet Dat oY, If these forms are supposed to be arranged in a ring, each form will be connected with two adjacent forms, one following, and one preceding it; thus (a, b, a’) is followed by (a’, b’", a’) and preceded by (a, 64, a). The number of forms in a period is necessarily even, because the final coefficients a’, a”, a”... are alternately positive and negative, and the last of these, namely a” =a, has a sign opposite to that of a’. | After completing one period, we may take any one of the reduced forms which are left, and form its period; and so on. Finally, all the reduced forms will be arranged in a finite number of periods, each containing an even number of forms. Thus for D=13 the periods are I. ( 1,3,-4),(-4,1, 3),( 3, 2, —3), ee eA 3 ati) (Le 304). ( 4,1,-—38), (—3,2, 3),( 3,1, —4), (-4,3, 1). IL. (2, 3, — 2), (— 2, 3, 2). 78 BINARY QUADRATIC FORMS. 72. Each form of a period, such as (+ a, b, ¥ a’), is transformed into the next following, (F a’, b', + a”), by means of a substitution ( ; :| . If, w are two corresponding roots of the forms, at 1 8-—a Suppose, in particular, that » is the principal root of its form: —b+/D oe that is, let o =, ee then when the upper sign is taken for +a, & is positive, w is a positive proper fraction, and tree Weeeee | a Sym eel : — th) a ree : since oa which is negative. If we take the lower sign for +a, » is a negative proper fraction, 6 is negative, and we have Jo|--o=— 5 oe ers ray 1 }S|+]e'|’ since w’ 1s positive. In every case, therefore, 1 O|= rT 7- | | }d|+ | @ | Similarly, if (_{’ y) is the substitution which converts a (F a’, b', +a”) into the next form of the period, o'l=a7= B+ [a" and so on. Hence, if g;, go... Pom are the forms of a period, ( : rd rs red the substitution which converts ¢; into dgi, |O,|=dz, @, the principal root of ¢,, the absolute value of w, may be expanded as a pure recurring chain-fraction in the form ana 1 | =>— Fr Eee lw, | d,+d,+...+dom+.... * * In the same way, if w; 1s the principal root of ¢,, ib 1 1 Lier erase ane iil gs * * ASSOCIATED FORMS. 79 For example, let ¢,=(1, 3, —4); then the corresponding period contains ten forms (cf. Art. 68). The values of 6; are I —I, i, ae Me 6, aly ib —I, y sah and hence ’ 1 ae alk jal Saas gi pies igh A Os % * Here it will be observed that the period of ten forms gives rise to a recurring fraction with only five partial quotients in its period. The reason for this will appear later on. 73. It is now evident that the reduction of an indefinite form is precisely equivalent to the expansion of its principal root in the form of a recurrent fraction; in particular, we have found the necessary and sufficient conditions that a root of a quadratic equation should be expressible as a pure recurring fraction. Another curious point is that in the expansion of expressions such as (b+ /D)/a only a limited number of distinct periods of partial quotients can occur; thus for D=13 there are only two, namely (1, 1, 1, 1, 6) and (3). Here periods derived by cyclical permuta- tion, such as (1, 6, 1, 1, 1), are not considered to be distinct. 14. If (a, b, —a’) is a reduced form, so also is (— a’, b, a). These may be called associated forms. Two associated forms may occur either in different periods, or in the same _ period. In the first case, it 1s easily seen that each form of one period is associated with a corresponding form of the other; and the periods may be called associated. Suppose, on the other hand, that the forms occur in the same period. Then if (—a’, b’, a’) is the fornt of the period next after (a, b, —a’), its associate (a’, b’, —a’) is the form next before (—a’, b, a); and similarly, if (—a@’, b’, a”) is followed by (a’”, b”, —a@”), (—a”, 6’, a’) will precede (a”, b’, —a’). Proceeding in this way, forwards from (a, b, —a’) and backwards from (—a’, b, a), we come eventually to a pair of associated forms (A, B, — A’), (— A’, B, A) which are consecutive forms in the period. Since these are adjacent, B+B=0 (mod A’), that is, 2B=0 (mod A’); (—A’, B, A) is therefore an ambiguous form (cf. Art. 62). In the same manner, by going forwards from (— a’, b, a) and backwards from (a, b, —@’), we arrive at a pair of associated forms which are adjacent; one of these is therefore ambiguous. Thus every period which is its own associate contains two ambiguous forms; and conversely, if a 80 BINARY QUADRATIC FORMS. period contains one ambiguous form, it must be its own associate, and will contain a second ambiguous form. There cannot be more than two ambiguous forms in a period; for by proceeding forwards and backwards from an ambiguous form (A, B, — A’) and its predecessor (— A’, B, A), we obtain continually new pairs of associated forms (a, b, —a’), (—a@, b, a), and if either of these is ambiguous, it is adjacent to the other, and the period is completed. 75. If (A, B, —A’') is a reduced form, so also is (— A, B, A’). The principal roots of these forms are equal and opposite ; hence the recurrent expansions of their absolute values coincide. It may happen that (A, B, — A’) and (—A, B, A’) belong to the same period; suppose that this is the case, and that the period contains 2m forms. Then if, with the notation of Art. 72, we put (A, B, —A’)=q,, it is easily seen that (— A, B, A’) =i, and that 8inoi= — 91, Om+e = — O9,-+-O2m = — Sm. Consequently Cig ae Orie Og tat as ear = Oly and the series of recurring partial quotients in the expansion of the principal root of (A, B, — A’) contains only half as many terms as there are forms in the period. An example has already been given above (Art. 72) for D= 13. Here the forms (1, 3, —4) and (—1, 3, 4) belong to the same period of ten forms; and the periodic expansion of the absolute value of the principal root of either is (0; 1, el ele 6). All the forms of a period of this kind may be arranged in pairs such as (a, b, —a’), (—a, b, a’), with equal and opposite principal roots. a, B Y 5) is a substitution transforming (a, b, —a’) into (—a, 6, a’), then, being the principal root of the first form, If the forms (a, b, —a’), (—a, b, a’) are equivalent, and ( _a(-0) +8 y¥(— ao) oe or yo? —(a+6)o+PR=0. Comparing this with aw? + 2bw — a’ =0, and proceeding exact] y as in Art. 83 below, we have a=(t—bu)/c, y=au/e, B=-au/o, 6 =—(t+bu)/c, CONTINUED FRACTIONS. 81 where o = dv (a, 2b, c), and (#, uv) is an integral solution of — Dw? =— 0°. Conversely, if this equation admits of integral solutions, the forms of any period for the determinant D and divisor « may be arranged in pairs (a, 6, —a’), (—a, b, a’). If the equation is solvable for c=1, it is so for o=2; in this case, both the properly and the improperly primitive periods have the special character in question. If the equation has integral solutions only for o=2, the property will belong only to the Steal ‘primitive periods. 76. In order to complete the theory, it has to be shown that two equivalent reduced forms must belong to the same period. This is the most difficult part of the whole investigation, and requires the proof of some auxiliary propositions. The notation (4; #1, fe,---) will be used to express the continued fraction 1 1 Pothyin Tees and in the case of a recurrent fraction, the period will be indicated by asterisks: thus Mo + Leeeel Gla 2, 3, 4) means 145 5454.. and (0; 3) eins, © s Ze Os A continued fraction (with a finite or infinite number of partial quotients) is said to be regular, when it is of the form (+ fo; Pa, fe, fs---), Where all the quantities jy, Me, My... are positive and not zero. When this is not the case, it is irregular. 17. If a continued fraction contains only a limited number of negative or zero partial quotients, then rt is possible, by a finite number of operations, to convert it into a regular continued fraction. Let pw, be the last partial quotient which is not positive. There are the following cases to consider. I, [SpA We have identically 0+ ame Dae = x 82 : BINARY QUADRATIC FORMS. hence without altering the value of the continued fraction we may replace the four partial quotients Pray 9, Mrsiy rte by Pra = Myr+ip My+e; leaving all the rest as before. IE fy=—n, and n>1. We have identically iA k! x sf 1a pbe2d hp = x be (b—1)e#-1 ova ba—1 =a—1+ = Sh gen Ryan =a—1+ : 1 PUT ae eae | (b—2)+ =a-—1+ ze 1+ 7 012) ee Leah Hence we may replace Bray NN, My+i by pra ye 2 2 La le Of these only the first can be negative. If either »—2 or 441 — 1 is zero, we can apply the reduction of the previous case. III. fy =— 1. We have identically A ile yee Paeris a—1 av a—2 aie eos et =a—-2+ - 1 + —- CONTINUED FRACTIONS. 83 Therefore Leis Mare L a [lp may be replaced by Pra —2, 1, pry —'2. - If 4, = 2, the first reduction may be applied. If w,,,=1, the preceding process fails: but we have 1 = on ; Ly CERN ee 0 Hf ——f ii Le ater bs x =a—b—-24+ o : So emsih so that we may replace Mra, —1, 1, Brite, press by Pras fra, —2, 1, pros—. If +3 — 1 1s zero, the first reduction can be made. It will be seen that in every case the last irregularity is brought at least one place nearer to the beginning of the fraction. It is therefore possible, by repeated reductions, to bring it to the first place, and the fraction has then become regular. Moreover each reduction either leaves the total number of partial quotients unaltered, or else increases or diminishes them by two. If the process be applied to an infinite continued fraction (that is, one with an infinite number of partial quotients), only a limited number of partial quotients will be affected, and those which remain unaltered -will occupy odd or even places in the regular expansion according as their places were odd or even originally. 78. Another lemma which will be required is the following :— If the quantities x, y are connected by the relation pee Tue yo+6 where a, B, y, 5 are integers such that a5— By=1, wt is always possible to express y in the form y=(tp; Mi; Ma,+++Mor, +p, 2), Where fy, flo, +++ Moy are all positive. 84 BINARY QUADRATIC FORMS. Let + be determined so that 8/64 is a positive proper fraction: this proper fraction may be expanded into an ordinary continued fraction in the usual way, and if it happens that the number of partial quotients is odd, the last of them, .,, say, may be replaced by (#»-1—1)+-1/1, and then we may write fra for fy 4—1, and put p,=1. Thus in all cases B s a (+ Kb; Pa, Pass + »flor Suppose that a,/y, is the convergent immediately preceding B/S; then %6 —y7,8 =1l=ad—y8, and hence a=a,+ vf, ee) LE vo, where v is some integer. It follows from this that aly = (+ Bb; Mi) He,-++Mor, G v), and that GHEE Ss ya +6 Site (lee ae Ma,+++ Mor, Pe, 2). ie CON eae Lea Heample 1. Suppose G s) = as 9. eH Here — 6/5 =—2+44/5 =(—2; 1, 4); a=-—l, m= 1, a =11=—1+(-2)(— 6), y =—-9=1+(—2) (5), and therefore llz—6 Poi 25a apa elaaaealieilaie a, B\ ( 30, —67 Ee 74, ee S)s be 47, anal Bid=(—1502).1, 3, 4.5191); a, = — 37, 1 = 58, a=a,+67, y =y7,—105; therefore v= —1, and _ 80a- 67 ) =4725-105 =—(— 715 2, L 3, 4, i il. —1, 2). 79. We are now able to prove that two properly equivalent reduced forms must belong to the same period. Let the reduced EXAMPLE. 85 forms be ¢=(a, 6, —c) and ¢’=(a, b’, —c’): we may suppose that a, a are both positive, because if, for instance, a were negative we could take instead of @ one of the two adjacent forms of the same period. Let @, w’ be the principal roots of ¢ and ¢’; these will be positive proper fractions. If $’~¢@, there will be a proper sub- stitution tes RS) which will transform @¢’ into ¢, and therefore , to+68 oO = . yo+6 By the lemma just proved, we can express o’ in the form wo =(+p; Mi Ma,+++far, ray, @) . . =(+p; By, ba,seefeoy; Gr vy +d,), dz, Cire Cant d,), if (0; he en) is the recurrent expansion of o. If d,+v happens to be negative, the expression for w’ can be reduced to a regular expansion, so that we may write in every case On (Pye ly ola Gy. 3); where J,, J,, J;,... are all positive. Now ’ is a positive proper fraction; therefore w’=0. Again, w can be expressed in one way only as a pure recurring’ fraction ; hence J,, /,, Js... must form the recurring period of w’. But in the reduction of the first expression for w’ into a regular form, only a finite number of partial quotients after + v are affected. Therefore the series (J,, 1,, J;...) only differs from (d,, d., d;...) by beginning at a different place; in other words, the period of partial quotients in the expansion of ’ is only a cyclical permutation of the period belonging to w, and moreover a permutation equivalent to an even number of transpositions (cf. Art. 77 above). Therefore the forms ¢@ and ¢’ belong to the same period. The following example will illustrate this theorem, as well as the reduction of irregular continued fractions. The form ¢’=(6, 1, —7) is transformed into ¢=(9, 5, — 2) by ee 29, —5 the substitution ( _93 :) : 5 Now me os 13): and pees rae (29 11, 3, —6, w); WS =1930 -- 4 86 BINARY QUADRATIC FORMS. or, since (Un e tes, tel, 12), ci (a SPRINT TES UT ye SO) ENP Te Oy LS Sotto ot ges) my eLoeaih sake he SE 1, (2) 0 deere * xk ={07 etl oe oe deerme) the period of which is derived from that of w by cyclical permuta- tion, each element being removed four places backwards. 80. In order to discover whether two given forms, f, /’, of the same determinant, are equivalent, we find, by the process of Art. 69, two reduced forms ¢, ¢’ equivalent to f, f’ respectively. The necessary and sufficient condition for the equivalence of f and fis that ¢ and ¢’ should belong to the same period of reduced forms. Suppose that this is so, and that i B ) : & im ) : & | y, O/” \y, pl’ \y, 8 respectively transform / into ¢, ¢ into dq’, and /’ into ¢’. Then 5 / / will transform f into /”. Example. D = 48, f=(58, 72, 97), f’=(—19, 47, —114). We have GER P RYO ESS Oh ere ~( 6, 5, —8) o(—3, 4, I)=¢ and (—19, 47, —114)~(— 114, 67, — 39) oo (Ceo) ee 2) (2 5h 88) =o The period of (— 3, 4, 9) is (—3, 4, 9), (9, 5, —2), (—2, 5, 9), ete, hence dw ¢’, and therefore fr’. SIMPLEST REPRESENTATIVE FORMS. 87 Again f is transformed into ¢ by the substitution ea 1) ( 0, Sy DN pace hve hee re oh ela Le uti) el into ¢’ by (% i Joa = °), meee | Cae =( i, ety J’ into qd’ by (a ae ere Allo aera Therefore fis transformed into f’ by the substitution ( 5, aa) —1, ‘ a ee) 25 (~ 26, i — 4, og 1,-— 2/ ~\ 21, —59/° Simplest Representative Forms. 81. The most important result which has been obtained is that both for definite and for indefinite forms the number of classes for a given determinant is finite. To assign, a priort, the number of classes, without constructing a system of reduced forms, is a fundamental problem of which the interest is equalled by its difficulty, and all the solutions hitherto obtained depend upon the most abstruse analytical methods. An account of these investiga- tions will be given later on; meanwhile, it may be observed that, from this point of view, a system of reduced forms is merely a finite number of forms such that every form of the determinant considered is properly equivalent to at least one of them. Con- sequently the definition of a reduced form is to a certain extent arbitrary; and in like manner with regard to the choice of a complete system of representative forms. We may, in fact, take as the representative of a class any form which is contained in it ; however, it is convenient to fix upon a set of ‘simplest representa- tive forms, which are defined as follows. When J is negative, there cannot be more than two reduced forms in any class. When there is only one, that is chosen for the representative; when there are two, in which case they are opposite, that one is taken of which the middle coefficient is positive (cf. Art. 66). When D is positive, the period of reduced forms for any class will contain either two ambiguous forms or none (Art. 74). In the 88 BINARY QUADRATIC FORMS. former case, let the ambiguous forms be (a, b, c), (a’, 0’, c’). We can find two other forms equivalent to these, of which the middle coefficients shall be the absolutely least residues of b, b’ to the moduli a, a’ respectively. If in one only of these new forms the middle coefficient be zero, that form is chosen; if neither or both of the middle coefficients be zero, that form is taken of which the first coefficient is numerically least. If the first coefficients are numerically equal and of opposite signs, we choose the form of which the first coefficient is positive. When the period contains no ambiguous form, we choose that form (a, b, c) of which the first coefficient has the least numerical value (if there are two such coefficients, only differing in sign, we take the form of which the first coefficient is positive); then, as before, we deduce from this the form (a, b’, c’), where b’ is the absolutely least residue of b (mod a), and take this for the representative form. Thus for D=58, there is a period (2, 6, —11), (—11, 5, 3), (3, 7, -—3), (—3, 5, 11), (11, 6, —2), (—2, 6, 11), ete. The ambiguous forms are (2, 6, — 11) and (— 2, 6, 11), from the former - of which we derive the representative form (2, 0, —29). Or again for D=99 we have a period (5, 8, —7), (—7, 6, 9), (9, 3, — 10), (—10, 7, 5); here we choose (5, 8, —7) and deduce from it the representative form (5, — 2, — 19). Automorphic Substitutions. 82. Suppose that it has been discovered, by the methods . already explained, that two forms f and /’ are equivalent. The process by which the equivalence is established also furnishes a proper substitution by which / is transformed into /. The question arises; is this the only substitution by which the transformation can be effected ? and, if not, how can we find all the substitutions which transform f into /’ ? This problem may be at once reduced to that of finding all the substitutions which transform /f into itself. It is clear that if A is any substitution which transforms / into itself, and S any substitu- tion which transforms / into f’, then the substitution RS will also transform f into. f’. Moreover, if 8,, S, are any two different substitutions which transform f into /”, the substitution S,S,~ will transform f into itself. Hence, putting S,S,;17=A, we have S,= RS,: so that all the substitutions which convert / into /’ AUTOMORPHIC SUBSTITUTIONS. 89 may be obtained by compounding any one of them, such as S,, with all the substitutions which leave f unaltered. A proper substitution R which transforms / into itself is called an automorphic substitution (relatively to f), or simply an automorph of 7. 83. Let f=(a, b,c) be a primitive form of determinant JD, and let » be its principal root; then if (" 8 be an automorph, oa +68 yo +6 or yo? + (6—a)o—B=0. Comparing this with aw? + 2bw + ¢=0, it follows that y=au/o, 6 — a= 2bu/o, 8B =-— culo, where wu is an integer, and ¢ =dv(a, 2b, c): that is, ¢=1 or 2 according as f is properly or improperly primitive. Since (6 +a)+(d6—a) = 26, an even integer, we may evidently put 6+ a= 2t/c, where ¢ is an integer. Thus a=(t—bu)/e, B=—culo, (i) i Sia Se cE and substituting in a6 — By =1, we obtain GOL) Popemndts tiaetin ated resets es (11) a, 8 y, 6 given by (i), where ¢, u are integers which satisfy the indeterminate equation (11). Thus all the automorphs ( ) are expressible in the form Conversely, if t, uv are any integers such that #—Du*=o?, the values of a, 8, y, 6 given by (1) will all be integral, and @ ) will be an automorph. It may easily be verified that (i e ) is an algebraical automorph ; so that all we have to do is to prove that a, 8, y, 6 are integers. This is obvious when o=1; if o=2, D=1 (mod 4), and the equation # — Du? =4 shows that ¢, uw are both odd or both even. Also b is odd, otherwise f would not be primitive ; 90 BINARY QUADRATIC FORMS. therefore t— bu and ¢+ bu are both even, and consequently a, 6 are integers. Finally, a, ¢ are both even, and therefore 8, y are integers. 84. The equation (ii), although originally proposed for solution by Fermat, is usually known as the Pellian equation. Its character, from our present standpoint, is essentially different, according as D is positive or negative. First, let D=— A, a negative integer; then, in general, the only real integral solutions of. -+ Aw? =o? are t=+0, u=0. Ifo=1,A=1, they aret—+1;u=0, and?#=—0,u=+1. Ifo =2, A=3, they aret=+2, u=0, andt=+1,u=£1. In general, therefore, there are only two solutions; the only exceptional cases (for primitive forms) being A=1, c=1, when there are four solutions, and A = 3, o = 2, for which there are six. Since the two solutions (¢, w) (—#, —w) lead to the same substitu- tion, there is in general only one automorph, the identical substitution; for the exceptional cases we have two and three automorphs respectively. Thus the form (1, 0, 1) has the 1, 0 0,-—1 : automorphs & "4 and & a : while (2, 1, 2) has 1,0 Ly 0,-1 (0. i é 1, a ei 1): 85. On the other hand, if D is positive, the Pellian eqitation | admits of an infinite number of solutions. It can be shown, in the © first place, that there is at least one solution distinct from t=+a, u=0. For suppose (a, 0, ¢) is a primitive reduced form of determinant D for which a is positive, and dv (a, 2b, c)=o. Let — its principal root be expanded into the recurrent fraction 2 2 @ =O Ged Ose le), and suppose that Pom—1/qom—1» Pom/Gem are the (2m —1)th and 2mth convergents. Then since w=(0; dy, ds, ... dam, demt @), it follows that ie P2m-1 @ + Pom Pom—1 © 1 Jom ‘ hence (? sae aa a is an automorph, and therefore if we put dem—1> Yom Pm = (¢ i bu)/o, Pam = — culo, dem = aul CO, dom = (¢ =f bu)/ CG, t, w will be positive integers such that # — Dw = o°. THE PELLIAN EQUATION. 91 For instance, if D=17, we may take the reduced form (2, 3, —4), for which o=2. The expansion of its principal root 1s * * (Ofer is bee lal’ 3). whence p= 9=4( —3u), g,:=16=4, so that Pet nlipe ioe ly, Excluding the case for which t=o, w=0, there will be one positive integral solution of — Du? =o? for which t, wu have the smallest possible values. This will be denoted by (7, U) and called the fundamental solution. If nis a positive integer, the expression (7’+ U»/D)" may be reduced to the form P+ Q/D, where P, Q are integers, so that if we write (=* ey Ln + U, alee oO Oo T,, Un will be rational. Moreover since /D is irrational, we shall have ) (= ove a U,/D Oo Oo and hence, by multiplication, Le — DU? 4. (P—DUN® ome aia ad) Therefore (7, U,) 1s a moral solution of #— Dw=oc. It may be shown that 7',, U, are integers. This is evidently the ‘case when o=1; if o =2, it'may be proved by induction as ea Suppose the theorem true up to (7',, U,); then since = 1 (mod 4), the equation 7? — DU,? =4 shows that 7, U, as a as 7’, U are either both even or both odd. Now eed as Ces VD = TP + U /D T, + Gi, VD f 2 2 2 TT,+DUU, , Tr,U+2Un = } | - aoe 3 vD} hence Teas (LL, + DOU.) and Ona =4(T,U + TU) are both integral, and since they satisfy 7?,,,, -DU*,4, = 4, they are both odd or both even. The theorem being true for n = 1, it is true universally. 92 BINARY QUADRATIC FORMS. 86. Every positive integral solution is of the form (T,, Un) where Laan De (“+ U ey Oo Oo and n is a positive integer. For suppose, if possible, that (¢, wu) is any other positive solution. Then t¢+u/D>Z74+ UA/D, and since 71+ UV/D>a, gives 0< t'—w'/D T—UAJ/D: consequently (¢+u /D)—(t —w J/D) <(T+UV/D)-(L-U A/D), and hence w’< U. Therefore also t 1, and U,4> Un. We may, if we like, assign negative values to n, and put Pin+ UnvD _ (a: EEN Co Co : .. (Be Uy _T-UyD Oo oO Pe. Din + Un VD _ Tn = UnvD Oo oO so that ee eta If, in the notation of the hyperbolic functions, we put @ = cosh (T/c) = sinh (U/D/c), we shall have T7,=oacoshnd, Un»/D=csinh nd. The convenience of this is that the known formule of the hyper- bolic functions may be used to express the relations between the different values (7, U,). Thus from the formulee cosh 26 =2cosh?¢ —1, sinh 2¢ = 2 sinh ¢ cosh ¢, we deduce of, = 27,2 — 02, cUg, = 27,0»; and so in other cases. It may be specially observed that Paa = ae 5 Dn T Path bees aeons Un, a Bites oO these formule are very convenient for the calculation of the successive values of 7,,, U,. | 87. Let o=1 for the moment, and for the sake of uniformity rte = O07 — 7, U,—U. If pis'an odd prime; we have T, HUsND = (2, + Uy DY = 7? + U,? D®” (mod p) = 7,+ De-) U, /D. Hence Lo=L;, Up =(D |p) U;, 94 BINARY QUADRATIC FORMS. if we adopt the convention that (D|p)=0 when p is a factor of D. If we write e for (D|p), we have T,, = T., U,= U., supposing that p does not divide D. Also Teele pete) Gti =7,7T.+ DU, U.=%i.. Uti 0 = es and hence in general Dnp+h See O np th ay Ee 7 We conclude, therefore, that the least positive residues of (T,, Un) with respect to p form a recurrent series, the number of terms in each period being a factor of p—(D| p). If p divides D, 7; = + 1 (mod p), and the same reasoning shows that the residues recur with a period of p or 2p terms, according as T’=1 or —1 (mod p). With respect to the modulus 2, the system of residues may be (TO) or OL EL eO) Orn 12) )e (tesa). It is easy to see that a similar recurrence will exist if the residues are taken with respect to a composite modulus m. Thus if (7", U’) is the least positive solution of ¢ — m?Du?=0, (7", mU’) is a Solution of #@ — Dv? =1 for which u=mU’ =0 (mod m), and it is the first of the kind which occurs. Putting 7” = 7,, mU' = Uj, we have 7',?=1 (mod m); if 7,=1 (mod m), the period contains — » terms ;.if not, 7, =27,?-—1=1 (mod m), U., = 27, U, = 0 (mod m), and the period contains 2 terms. It does not seem easy to assign the number of terms in the - period of residues when m, D are given. By arguments similar to those employed in Chap. I. it is possible to prove the following | theorem, which may serve as an exercise for the reader. Suppose that m contains at least one odd prime factor which does not divide D: then if p,q, 7... are the different odd primes which divide m but not D, and if w is the Lom. of m/pgqr..., p—(D\p), ¢g—(D|Qq), ete., the number of terms in a period will certainly divide ». A fortiori it will divide sll (1-5 (DIp)). where the product applies to all odd primes which divide m but not D, and a is the number of such primes, FUNDAMENTAL AUTOMORPH. 95 If every prime factor of m divides D, all that can be concluded is that the number of terms in a period divides 2m. Eaamples. D=11, T,=10, U,=38,.m=%5. Here, since (11 |5)=+1, and (11|3)=—1, wis the L.c.m. of 5, 4, 4, that is, 20. The period of residues does in fact contain 20 terms, viz. CLOTS (4) 00) F000) 12), a.0ha30) (25, 3), (49, 30), (55, 72), (1, 60), (Aj) (49, 8O)e (40080296 CL, Lo): (55, 3), (49, 45), (25, 72), (1, 45), CLO WSs (40 ne LOCO sa )en (len), | If m=11, there are 22 terms in the period; namely (10, 3), era), (10,9), (1,10), ete. The number of terms in the period is often less than w: thus for D=11, m=9, pw is 12, but the period is (1, 3), (1, 6), (1, 0). 88. The complete system of automorphs for a primitive form (a, b, c) is given by a, B\ /((1,—6U,)jc, —cU, Jo « 6 ) ( aU,/o, (T+ ne) , where 7',, U, have the meaning already defined ; except that in this formula they may be taken positively or negatively. If, as we have tacitly done hitherto, we consider @ Cae the same as —y, - } be é it is sufficient to give to n all positive and negative integral values and take Z',, U, with the signs appropriate to them in each case, according to Art. 86. With this convention, let the substitution above written be denoted by S,. Then it is easily verified that if m, n are any integers, . Sin- Sh om Simin = 8,.Sm, and hence, if S denote (T—bU)/o, —cU/o ) ( aU/o, (L'+bU)/c/’ Sy = 8” } therefore all the automorphic substitutions may be expressed as powers of the fundamental substitution S. 96 BINARY QUADRATIC FORMS. It has already been shown (Art. 85) that from the period of a reduced form (a, b, —a’) an automorphic substitution may be derived. It is not difficult to prove that this is the fundamental automorph. For simplicity, suppose that a is positive, and let the derived automorph be ge (i —bu')/o, ao ) awa, (t' + bu')/o/’ where ¢' is positive. Dae: Since ¢? — bw? = aa’w? + 07, ¢ — bw’ and t'+ bw’ are positive. If S' is not the fundamental automorph, it must be a power of it, say S*- It follows from this that if aw’/(t’ + bu’) be expanded into a continued fraction, the series of partial quotients will consist of those belonging to the expansion of aU/(7’+bU), repeated k times. But in the series d,, d2,... ds», derived from the period of reduced forms, such a repetition can only occur once. The only possible supposition therefore is that S’ = S?, and that S is derived from the partial quotients d,, d,,...dm. But these partial quo- tients lead to a solution, not of # — Du? =o°, but of ? — Dw = — oa? (cf. Art. 75). Therefore S' is the fundamental automorph. The automorphs of a derived form are the same as those of the primitive from which it is derived. 89. li k= & B ) is any substitution which converts (a, b, c) _ into the equivalent form (a’, b’, c’), and if S is the fundamental automorph of (a, b, c), then all the substitutions which convert | (a, b, c) into (a’, b’, c’) are given by (ig eS _ oR Yn» Sn where n is any positive or negative integer. The values of a, Bn» Yn» On are an = {aT, —(ba + cy) Un}/o, Bn={B8Tn — (08 +6) U,}/o, Xn = (yTn + (aat by) Un}/o, 8, = {6T, + (a8 + 56) Un}/c. The relations between the original and the new variables may - also be expressed by writing Tn, + Un VD (ox ax +(b+/D)y= {a (aa + By’) + (b + VD) (ye’ + dy’) ; GROUPS OF REPRESENTATIONS. 97 and similarly, the automorphs of (a, b, c) may be derived from av+(b+/D)y= fn + ae oe fax’ + (b + /D)y'}-. Representation of Numbers resumed. 90. It has already been shown (Art. 59) that if a number m is properly representable by a form of determinant D, D must be a quadratic residue of m. Further, if n is any root of the congruence n? = D (mod m), and we put (n?—D)/m=l, then the form (m, n, 1) is of determinant D, and any primitive representation m= aa? + 2bay + cry? by the form (a, b, c) of determinant D leads to a proper substitution a, B a 6 In order, therefore, to determine whether m can be represented by a given form (a, b, c) we first find all the solutions of n?= D (mod m); taking any one of these, say n, we examine whether the forms (a, b, c), (m, n, 1) are properly equivalent. If they are, the process by which we discover the equivalence enables us to form ) by which (a, 6, ¢) is transformed into (m, n, 2). a substitution ty fe ) which converts (a, b, c) into (m, n, 1), and ve then =a, y=y gives a primitive representation of m by the form (a, b,c). This representation, moreover, appertains to the particular solution (n) of the congruence n?= D (mod m). If, for a-given value of n(mod m), a particular solution is w=a, y =v, then (by Art. 89) the most general solution apper- taining to this root of the congruence is | pa ea (bat ey)u pon Lad OL BOY) ES on Oo where o = dv (a, 2b, c) and (¢, u) is any integral solution of P — Du? =o. By giving to ¢, uw all suitable values we thus obtain a group of representations; there will be a finite or infinite number of representations in the group according as D is negative or positive. The maximum number of distinct groups will be equal to the number of the solutions of the congruence n?= D (mod m). M. 7 98 BINARY QUADRATIC FORMS. 91. Some illustrations of the general theory will now be given. Example 1. Suppose D=—1. There is only one positive class for this determinant: its representative is (1, 0, 1). Let m be a positive integer such that m or 4m is odd; then the congruence n?=—1(mod m) is solvable if every odd prime which divides m is of the form 4n+1. If this condition is satisfied, there will be a form (m, n, 1) of determinant —1, and this must be equivalent to the reduced form (1, 0, 1); therefore m can be expressed as the sum of two squares. A special case of this is the famous theorem first enunciated by Fermat (Observations on Diophantus, No. VIL.) ; | Every odd prime of the form 4n+1 can be expressed in one way, and one way only, as the sum of two squares. For if p=4n+1 be a prime the congruence n? = —1 (mod p) has two solutions, each of which (Art. 84) corresponds to four representations, so that there are eight representations altogether : but if e=t, y=u be any one of these, we get exactly eight by putting e=+t, y=+u, or e=+tu, y=Ht, so that if we neglect the order of the terms, p is expressed in one way only in the form p= +w. For instance, let p=89. The roots of n?=—1 (mod 89) are n= +34; taking the upper sign, and applying the usual process of reduction, we have (89, 34, 18)0(13, 5, 2)0(2, 1, 1)(1, 0, 1). Hence we derive the substitution ( 0, =f 0, 5) 0, 1\_ /+3, +2 —1, -—3/ \-1, -8 ee which converts (89, 34, 13) into (1, 0, 1); and the second form is converted into the first by the inverse substitution & a a This gives the representation 89=(—5)?+(+8)2 If we start with (89, —34, 18), we obtain in a similar way 89 =(+ 8)? + (—5)? = 64 4 25 as before. Eaample 2. To find all the representations of 85=5.17 as the sum of two squares. EXAMPLES. 99 The roots of n?=—1 (mod 85) are n= +13, +38. The forms (85, 13, 2), (85, 38, 17) are converted into (1, 0, 1) by the ee fi, ar iA) Git i ate substitutions Gs 7. 4 a and ti 2 9 respectively ; and hence 85 = 49 + 36 = 4+ 81. The forms (85, —18, 2) and (85, —38, 17) lead to the same results. Example 3. To find all the representations, primitive and derwed, of 81 in the form 3a? + 2ay — 12y’. For the primitive representations we solve n? = 37 (mod 81), whence n= +19 (mod 81). Then by successive reductions we find (81, 19, 4) (4, 5, —3)0(—3, 4, 7) (7, 3, —4) co (4 on (3 847) ~( 3, 1, —12). 73g un converts (3, 1, —12) into (81, 19, 4); and the corresponding representation is «=— 163, y= 69. From this is derived the substitution ( By Art. 89, all the representations belonging to the same set with this are expressed by 2=—168t+ 991u, y= 69t— 420u, where (¢, w) is any integral solution of ? — 37? =1. Putting ¢=—73=—-T7, w=—12=-— U, we obtain the simpler solution « =7, y =3, and hence we derive the general solution in the form a= Tt + 29u, y = 3t + 24u. The form (81, — 19, 4) is not equivalent to (3, 1, — 12), so that the above solution gives all the primitive representations. The derived representations may be of two kinds, according as dv (x, y)=3 or 9. The former set is deduced from the primitive representations of 9 by (3, 1, —12). Proceeding as before, we obtain the representations of 81 in the form o=— 9t + 45u, y= 8t— 24u, where, as before, #? — 37w? = 1. 100 BINARY QUADRATIC FORMS. There are no representations for which dv(a, y)=9. This is easily seen from the fact that 1 is representable by the form (1, 0, —87), which is not equivalent to (3, 1, —12). Improper Equivalence. 92. A few remarks may be added with regard to improper equivalence. Every form is improperly equivalent to its opposite — (Art. 62): hence if f is improperly equivalent to /’, it is properly | equivalent to the opposite of /’, and conversely. Moreover, to every proper substitution @ ss ) which converts ff into the opposite of /’ corresponds an improper substitution & a which changes f into /’, and vice versa. Thus the whole theory of improper equivalence is practically reduced to that of propa | equivalence. There is, however, one point of special interest; the forms f and f’ may be both properly and improperly equivalent. In~ this case every form of the class to which f and /’ belong is improperly as well as properly equivalent to itself. It may be shown that the necessary and sufficient condition for this is that the class should be ambiguous, that is to say, should | contain ambiguous forms. First, let 7, f” be definite: then if we find a reduced form ~ equivalent to them, it must be equivalent to its own opposite, — which is also a reduced form. By Art. 65 this is the case only — when the reduced form is of the type (a, 4a, c) or (a, b, a). The first of these is ambiguous; the second is converted by the substitution & x into the ambiguous form (2a + 2b, a+b, a). Secondly, suppose that fand /’ are indefinite. Let (a, b, —@’) be any reduced form in the period of f By the improper ~ substitution (7 5) it 1s converted into its associate (—a’, b, a): hence if f, f’ are improperly equivalent, their periods must be : associated; and if they are properly equivalent as well, their common period must be its own associate, and is therefore ambiguous (Art. 74). IMPROPER EQUIVALENCE. Oa Every ambiguous form is improperly as well as_ properly equivalent to itself!; for if (a, b,c) is ambiguous, and 2b = 8a, eS ) is an improper automorph. Hence for the double equivalence of two forms it is sufficient as well as necessary that they should belong to an ambiguous class. 93. The effect of an improper automorphic substitution is to interchange the roots of the form to which it is applied (Art. 56); hence if 6 B ) is an improper automorph of (a, ), c), y 6 —b-VD_a(—b+VD)+ fa a y(—b+/D) + éa’ whence, on multiplying up and equating the rational and irrational parts, a(a+6)=0, a (ba — a8) + b (by — ad) —yD = 0. Now a is not zero; hence a+6=0, and on substituting —« for 6 in the second equation we obtain 2ba —aB+cy=9. The condition a6 — By = — 1 leads to a+ By=1. Since aB — cy = 2ba, we may put 7 AaB + cy = 2«, where « is an integer; hence B =(« + ba)/a, y = (« — ba)/c, and substituting in a? + By =1, we obtain Ke? — Do? = ae. Thus every improper automorph of (a, 6, c) must be of the form (i ae yle, ‘s aes where (x, X) is an integral solution of «@—Dd=ac. Since 1 Hence the term ‘ ambiguous,’ which is an unsatisfactory rendering of Gauss’s anceps = two-headed ; ef, ‘sacer ancipiti mirandus imagine Janus’ (Ov. Fast. i. 95). 102 BINARY QUADRATIC FORMS. improper automorphs exist for forms belonging to ambiguous classes, and for such forms only, we are led incidentally to the result that if (a, b, c) belongs to an ambiguous class, there will be integral solutions of «?— Dd? =ac for which («+ 0bA)/a and («—bnr)/e are both integral. As an illustration, let (a, b,c) =(2, 5,7). Here D=11, and the values e=5, X=1 lead to the improper automorph € » , ; . from which all the rest may be derived. In like manner the | solution «=17, %=—5 leads to the improper automorph —5, —4). . . ( refi ee ve on the other hand, no improper automorphs can be derived from the solutions x =5, N=—1, and c=17, X=5. CHAPTER IV. Binary Quadratic Forms; Geometrical Theory. 94. THE theory of the reduction and equivalence of binary quadratic forms is made much more intelligible by the introduction of a complex variable. The discussion of complex quantities belongs to the elements of formal algebra, and will not be . reproduced here: for convenience, however, a few of the funda- mental results will be stated, and the notation to be used will be explained. | The general form of a complex quantity is Z=2+Y1, where 4, y are real, and ?=—1. The modulus or absolute value of «+iy means +V2?°+y. It is denoted“ by mod (w+yt), or by |#+ yi|; in what follows, the second notation will be exclusively used. The square of the modulus of a complex quantity z is called its norm and denoted by Nm (z); thus if z=a#+y, Nm (2) =a? +7". The argument of z= x+y, denoted by arg (2), is a quantity 0 Le le |2 ran mined only up to multiples of 27. such that cos @= —, sin @= It is many-valued, being deter- 95. If we take an origin O and rectangular axes X’OX, Y’OY, and mark the point P whose coordinates referred to these axes are a, y, the complex quantity z=#-+ yi may be considered 104 BINARY QUADRATIC FORMS. to be represented either by the point P, or by the vector OP drawn to P from the origin. In particular, 1 and 7 will be represented by points A, B on OX, OY at unit distance from 0. Considered as an operator, «+ yi will denote the operation by which the vector 0A =1 is converted into the vector OP=a+yi. The modulus and argument of # + yz are equal to the radius vector and vectorial angle of P, if OX be taken as the initial line, and the radian as the unit angle. Calling these 7 and 6, we have “o+yt=r (cos @+7sin 8) =re®, 96. If z=xv+yi, and 2 =a'+y't, the four fundamental operations are performed according to the formulze ete =(eta')tyty)iar/+z, e-2 =(w-a') + (y-y/)i 22 =(au' — yy’) + (ay'+a'y)i = 22, Py RAG Oe PACES: PA ey = Py seer Me, 1. Of these, the first and third may be taken as definitions; the second and fourth are derived from them by means of / (z¢—2)+2' =z, and = ee We shall also require the following results, which are given here for the sake of reference :— Je+2'|pl2|+]2'), |ze|=|2/|2'| |zle!|=|21/|2 arg (22’) = arg (z) + arg (2’), arg (2/2’) = arg (z) — arg (2’), log z= Log | z|+7¢ arg (z). In the last formula, Log | z| means the ordinary real logarithm of the real positive quantity | z|. The quantities « + yi and w— yi are said to be conjugate. It is sometimes convenient to write 2 for the conjugate of z; it will be observed that zz, = Nm (z) = Nm (z,). COMPLEX VARIABLE. EOD 97. Consider now the linear transformation of a complex variable defined by ; Sade 0 = —_, cz +d, where a, b, c, d are any constants, in general complex, such that ad — be + 0. This may be regarded as a transformation of the plane of reference by which the point 2 is made to correspond to the original point z. There are, in general, two points which are unaltered by the transformation ; for convenience they may be called the stationary points of the transformation. They correspond to the roots of the equation : yet Becz dl. or cz? + (d—a)z-b=0. If the roots are 2, and z, the equation of transformation may be written in the ‘normal form’ , zo —Z Z—-2 = =r.——. Z— 2, Z—2, It is easily seen that %=(a—cz,)/(a—cz,), and it may be verified without difficulty that » satisfies the equation 1 @+d' + 2be gi Aaa ad — bc This is a reciprocal quadratic, as might be expected, because the interchange of z, and z, converts X into 1/A. When the roots 2, and z, are distinguished, the value of X is determined, and conversely. It may happen that the stationary points coincide: this will be the case when (a—d) + 4bc = 0. Such a substitution is called parabolic. It is clear that the values of \ must also coincide, each being equal to 1; and, in fact, the equation for » reduces to 1 ny a x saan 2, whence (A — 1)?=0, as stated. 106 BINARY QUADRATIC FORMS. _ 2-2 Z- The equation ———= “= now becomes illusory: it may, —— vp) Boe 29 however, be replaced by 1 1 Z— 2, hee Zz where 2, =(a — d)/2¢, k = 2c/(a+ d). A special case occurs when c=0 and a—d=0: here the coincident stationary points are at infinity, and the transformation 1S +k, zZ=ztk, where k = D/a. Observe that k cannot be infinite in either of these last formule of transformation: for it will be found that k= leads in each case to ad—be=0, which was expressly excluded at starting. Those substitutions which are not parabolic may be arranged into three classes, according to the value of X; namely (i) elliptic substitutions, for which % is complex, and || = 1, (ii) hyperbolic » : » | Ais real,and + 1, (iii) lowodromic _,, - » Ais complex, and |r| +1. 98. It is a characteristic property of a linear substitution — that it transforms circles into circles. This important proposition may be proved as follows. Since the most general expression for a linear substitution _ involves three independent quantities, any three assigned values Z, 2,, 2; may be made to correspond to any other three 2%, 2, 2s. In fact, the linear transformation required for this purpose is / Fat f / / ye ea VE a | £3 — 2% Canon Z—Z: £5 —f, ee—-2 &—ho or, which is the same thing, / ' / / a3 — &o Z—h, 43 — & ie" 23 — 2, 2 — Ba Zg— By Shy —a /6— ayes It is convenient to write (aPy6) for = / B = and call it a cross-ratio: in this notation, the above substitution may be written (21222,2)} = (2120252). MOBIUS’S CIRCULAR RELATION. 107 Now if A,, Az, A;, P are the points corresponding to 2, 2, 25, Z respectively, | (2:20%22) | = AA, ALP. 1°23 3A, : PA, ? while ALY (2,20%,2) = 2A,A,A,— 2A,PA,, the angles being described as indicated by the arrows in the figure; that is, by rotating lines from A;A, to A;A,, and from PA, to PAs, in the positive sense. A 3 Ae Fig. 2. It follows from this that (2,2.232) is real if and only if Paisley, vals Wake are concyclic; for the condition of reality is that LAA, A,A,— 2 A,PA, should be a multiple of 7, and this is precisely the condition that the four points should be concyclic, the angles being measured as above explained. . LS ar eR Ae A When (2,22;2) is real, (22,232) 1s real also: therefore any point z on the circle 2,2,2, is transformed into a point 2 on the circle 2/22/25. It should be observed that, in this connexion, straight lines are to be considered as equivalent to circles of infinite radius; in fact a straight line is in general transformed into a circle, and a circle may be (exceptionally) transformed into a straight line. 99. In the arithmetical application, it will be sufficient, for the present, to consider the substitutions , az+B g =- yz+6 where a, B, y, 6 are real integers such that aé — By = 1. 108 - BINARY QUADRATIC FORMS. The equation for » is in this case (Art. 97) N+A1T= e+ &+ 2Bry =(a+ 6) —2. The discriminant of this quadratic 1s 4— {(a+6)— 2? =(a+ 8)? {4 -—(a4 8}*I. This is negative, and the substitution is hyperbolic, except in the following cases :— I. a+6=0, whence also By + a?+ 1 =0. Here the equation for X is (A+1)?=0, and the stationary points are the roots of ya? —2az — 6B =0, whence Z=(a +2)/y. The substitution may be written in the form g—(at+viy_ — 2—-(atr/y e—(a-—aiy @—(a—A)/ry’ from which we see that it is elliptic and of period 2. Tl (aoe It is enough to suppose that a+6=1, because if a+é6=—1, we can change the signs of 4, 8, y, 6 throughout, without altering the substitution. The equation for » is M+2A+1=0, and therefore, putting p =e" = maa aa 3 N=p or p’. The equation for the stationary points is ya? —(2a—1)z-B=0, eT Cee ICES) _2s 2 whence _ 2a-1+iV3 ay a+p a co on reduction HYPERBOLIC SUBSTITUTIONS. 109 If we take X =p, we must put (Art. 97) © ees Dowie ve tS P 1 =a b) 2 5) Z a and the normal form of the substitution is Ben Sart ea dorms e—(a+piy " 2—(a+p)/y This is elliptic, and of period 3. The other substitution of the same period is obtained from the preceding by changing p into p®. Til. a+6=2. | Here (A — 1)? = 0, and the equation for the stationary points is —2(a— [yee = = 0, The substitution is parabolic, and may be written 1 one Z=(a-Diy 2-(a-Viy* If it happens that y=0, we must have a=6=1, and the substitution becomes g=Zzt+Pp. All the remaining substitutions are hyperbolic, the stationary points being defined by ss ye —(a—8)z-B =0. 100. As in last chapter, it may be shown that to every quadratic equation with integral coefficients and real roots corre- sponds a group of hyperbolic substitutions which have. the roots of the equation for their stationary points. Namely, if the equation is reduced to the form az?+ 2bz2+c=0, where du (a, 6, ¢c)=1, and if we put as usual b?— ac = D, dv (a, 2b, c) =a, the substitutions in question will be given by i (‘+ al Z—-2, Z—2, oO "g— 2," where (¢, w) is any integral solution of #— Du?=o°, and 2,, 2 are the roots of az?+ 2bz+c=0. 110 BINARY QUADRATIC FORMS. (It is left as an exercise to the reader to verify that the normal form of the substitution here given is only a transfor- (t — bu) z—cu auz + (t+ bu) - squared factor ensures that % and A™ are both positive, as they ought to be, because \ + A = (a + 6)? — 2, which is positive.) mation of z= It is interesting to see that the 101. Points which can be stationary points for a substitution of the complete group may be called critical points; it appears that they fall into four sets, which may be termed respectively (i) rational points, for which z = p/g, (11) surd points De ee Ge, ie (iii) 7-points » » 2=(ptr/q, (iv) p-points » » 2=(ptp)/¢. Here p,q, 7 denote real integers; in (ili), p and g must be relative primes; in (iv), with the upper sign p and qg must be relative primes, and with the lower sign p+1 and q must be relative primes. 102. A complex quantity z=2+yi for which y>0 will be said to have a positive imaginary part; its representative point will be on the positive side of the axis of a. Two quantities z, 2 will be termed equivalent when real integers a, 8, y, 6 can be found such that , a+B = ’ yz+6 ao — By =1. The corresponding points will also be called equivalent. Equivalent points are on the same side of the axis of x. For suppose that 2’ = a8 cal yz + ) as above,.and let z, 2 be the conjugates of z and 2’: then , a%+ 8 4 = ’ YZ + ) SAGO By) ie 20) (yz + 6) (y2 + 8) C=) a ~ Nm (yz+ 8)’ and therefore Z — oe THE FUNDAMENTAL TRIANGLE. 111 If z=a+yi and 2 =2a'+y'%, this gives y/y’= Nm (yz+ 9), a positive quantity, so that y and y’ have the same sign. If y=0, y'=0 also; that is, a quantity equivalent to a real quantity is also real. 103. Suppose now that o=a#+y2i is any complex quantity with positive imaginary part, and let o, =#— yt be its conjugate. Consider the points w which satisfy the conditions lo+o,\<1, ww, > 1. Expressed in terms of # and y these are 2\e\<1, e+y>t. Now the points (a, y) for which 2|#|<1, #+y?>1, y>0 are all contained within the area which les above the axis of a, outside the unit circle 27+ y2—1=0 and between the parallel lines e=4 and e=—4. (See the annexed figure, in which the area in question is shaded.) Fig. 3. This region may be considered as a triangle enclosed by three circular arcs (two of which are accidentally straight), the angles of the triangle being 0, 7/3, 7/3. It will be referred to as the fundamental triangle, and denoted by V. 104. Zhe points on the boundary of V may be grouped into equivalent pairs. For it is evident that the transformation wo =S(o)=o+1 112 BINARY QUADRATIC FORMS. converts every point » on the line «=—¢# into a corresponding point w’ on w=4, and if one of these is on the boundary of V, so is the other. Also the transformation wo = T (wo) =— 1/a, or, which is the same thing, , ‘ = oh ae yr sak eer converts every point w on the circular arc extending from p to 7 into a point w’ on the arc from 7 to (1 + p). A point will be called a reduced point if it is either within V or on that half of the boundary of V which is on the negative side of the axis of y. The point ¢@ is included. The reduced points p,7 are critical. The third vertex of V, which is at infinity, may also be considered a critical point for the substitution S(@)=@ +1. This seems the proper place to observe that from our present point of view z=oo simply means that the absolute value of z is infinite, and we shall have no reason to distinguish between (say) z=, a real quantity, and z=ai,a pure imaginary. We shall therefore speak of the point «. Perhaps the reader may be assisted by considering the effect of inverting the plane of reference into a sphere: to every accessible point on the plane will correspond one point on the sphere, but all the infinite elements of the plane will be represented by a single point on the sphere, and this representation will be sufficient so long as z= can be used without distinction for any quantity of — which the norm is infinitely great. This is the case, for instance, in the theory of algebraic integrals and Abelian functions as expounded by Riemann. On the other hand, if we distinguish those complex quantities z=w+y7 for which Nm (z) is infinite, according to the ratio of w to y, our ‘infinite’ elements will form a linear multiplicity. This is what is done in plane projective geometry, and it is for this reason that we speak of ‘the straight line at infinity in a plane.’ A great deal of misunderstanding is avoided if it be remembered that the terms mfinity, infinite, zero, infinitesimal must be interpreted in connexion with their context, and admit of a variety of meanings according to the way in which they are defined. 105. There are no reduced critical points except 1, p, ©, the vertices of the fundamental triangle. For if we put w =(a+7)/y, the conditions that » should belong to the fundamental triangle are y>0, \$4, @4 te, and the only integral solution is a=0, y=1, whence w =7. REDUCED POINTS. TLS Similarly w = (a+ p)/y is reduced only if a=0, y=1 or a=1, -y=1; and of the two points p, 1 + p only p is reduced. - Obviously no rational or surd points can be reduced. 106. No two reduced points can be equivalent. Suppose, if possible, that or and ol se ie yo + ) are both reduced. We have (by Art. 102) Doe at see @ — W 2 sly at ~ Nm (yo + 8)’ now Nm (ya + 6) = yoo +76 (@ + @) + >y—yo+ >, with a possible exception when either y=0, 6=+1, or y=4#1, 5=0. Hence, with these possible exceptions, lw’ — a) |<|@—@)l. But since o= 20 a i8e =o + a and w’ is reduced, we have in the same way |a—a@)|<|o’—a@, |, except perhaps when either y=0, a=+1, ora=0,y=4+1. It is impossible that |’ — @,'|< | @—,| and |@ —a|>|o—@)| simultaneously. Hence it is sufficient to examine the cases v0: Oo =a and y= +e a=d=0, whence also Se ‘The first case gives o=o+ £, so that if (#, y) are the coordinates of the point , those of »’ will be (+8, y). Now B is at least equal to 1 numerically, and since the width of V is just equal to 1, it follows that and o’ cannot both be reduced. In the second case wo =—1/a, whence Nm (a). Nm (o’) = 1. M. | 8 114 BINARY QUADRATIC FORMS. If Nm(@)>1, Nm (o@’) <1 and o’ cannot be reduced: similarly if Nm(o’)>1, is not reduced. If Nm(w)=1, w can only be on the circular are going from p to 7, and then w’ =—1/m is on the arc going from 7 to p+1, and is consequently not reduced. (Cf, Art. 104.) 107. We shall now state and prove a proposition the object of which may not be very evident: it is, in fact, preparatory to the important theorem which follows it. Within the area, above the axis of x, which is enclosed by the parallel lines x= +4 and the line y=c, where c 1s a finite positive quantity, there can only be a finite number of points equivalent to a given point w, the ordinate of which exceeds c. Let o=&+ 1, where 9 >. aw+ 8 Then if o = yot+6 = a+ yi be an equivalent point, Lae n Y= 5p (Om Wa) = a Baas) + DOES OF Therefore if y >, P(E +7?) + 2ydE + & < y/o, that is, (Ey + 6) + 127? < n/c. Now since the expression on the left-hand side is the sum of two squares and &, 7» are real, it is clear that only a limited number of real integers y, 6 can be found to satisfy the inequality. Suppose that (y’, 6’) is any suitable pair of values, 9/, 6’ being relative primes. Then if a’, 9’ are any integers such that . a's’ Stet B'y’ os 5 the most general solution of a6’ — By’ =1 is a=a+my, B=R' +m’, , +8 awot+Pf’ OS ey fier Og yYot+o yor m being any integer. Hence Suppose, now, that we consider points on the boundary #=-— 4 as belonging to the area defined in the enunciation, but points on x=+%4 as being outside of it. Then it is possible in one way only to determine the integer m so that the conditions = / —4=an<} REDUCTION UNIQUE. ELD may be satisfied. Therefore for every combination (y’, 8) in which y’, & are relative primes, and (y’£ + oy + yn? < y/e, we obtain one and only one point o’, equivalent to , within the area defined by —$ =a <3, y’>c. The number of these points is therefore finite. 108. We are now able to prove the fundamental proposition, that Every point above the axis of x is equivalent to one and only one reduced point. Let o =&+ 1, where n> 0. The integer m may be uniquely determined so that —s af +m <5. Put o=o+mM; then if |’) >1, w’ is reduced: if not, let o” =—1/o; then the ordinate of w” =7/Nm (o’)> 7, except when Nm (’) = 1. If Nm (@’) = 1, either w’ or w” is reduced. If otherwise, let m’ be chosen so that i} A VEN oe =o 4+ =n" 4+y% satisfies the conditions SY 4 = gl! < 4; and if w’” isnot yet reduced put w'”=—1/’”. Then, as before, the ordinate of w'” exceeds that of wo” and a fortiori that of o. By proceeding in this way we must at last arrive at a reduced point, because the points o’, w”, etc., all lie within the area defined by —$=0, and a(a—b+c)<0; then the form (c, —0, a), derived from (a, b, c) by the substitution T= tj iy , , will be reduced. For it is clear that (a+b+c) and (a—b+c) have opposite signs: hence either c(¢+6+a) or c(c—b+a) is negative. The same conclusion follows if a(a+b+c)<0 and a(a—b+c)>0. Therefore if (a, b, c) is any reduced form there will be, in general, two adjacent reduced forms equivalent to it; if (a+b+c) and (a—b+c) have the same sign, these forms are derived from (a, b,c) by the substitutions S and S+; while if (a+6+c) and (a—b+c) differ in sign, the substitutions are 7 and either S or S™ according as a (4 +6+c) or a(a — b+ ¢) is negative. 116. It may happen that one of the quantities a(a+b+c) is negative, while the other vanishes; in this case (a, b, c) will be called a critical form. Its representative circle passes through one of the points p, p+1 and includes the other. We have | a’ +ab+b?=D,so that these critical forms will only occur when 2D can be represented by the form (2, 1, 2). Suppose that (a, b, c) is a critical form with a+b+c=0 and a(a—b+c)<0: then ab is positive. The substitution S changes (a, b, c) into (a, —c, b) which is not reduced, because a(a—c+b)=2a(a+ bd), which is positive. Similarly the substitution Z’ converts (a, b, c) into (c, — b, a), which is not reduced, because c(c—b+a) =2b(a+ b)=+4. On the other hand S™ converts (a, 6, c) into (a, b—a, — 3b) which is reduced, because a (a+b —a-—3b) = — 2ab=-, and is not critical except when a= 2b. EXAMPLE. La 1550 Wav (b, —a,c), which is also critical, and then S transforms this into (b, b—a, — 3a), which is reduced but not critical, except when be 2a, Now if a=2b, c=— 3b, and the form is (2b, b, — 3b): this is primitive only when b= +1; and similarly the primitive critical forms for which b= 2a,a+b+c=0 are (+1, +2, 43). Therefore with the single exception of D=7, we are able to associate each critical form with two other reduced forms, one of which is critical, and the other not. Moreover, the substitution S7'S =( ) converts (a, b, c) into Exactly similar reasoning applies to the other case of critical forms. The final conclusion is that the reduced forms may be arranged in periods such that each form is converted into the next following by one or other of the five substitutions Oe Lea at Ome, where the last two are only to be applied to critical forms. Observe that S77'S7 = a a 4 101 Sm) lee Dencesttenc put U for STS, the five substitutions are S, S~, 7, U, U™. An example will make this clearer. If D=37, the periods of reduced forms for the classes represented by (1, 0, —37), (3, 1, —12) are Pee nl) ele), e—30) nL 233 )) Las a 28), eee ols Gaee dy. (16. aly, (ade oD) —d 61) (6, — 1), = 5,12): (1, —1, 36). II. (3, 1, —12), (8, 4, —7)*, (4 —38, —7)*» (4 1, -9), eee ym 5.4) (8,2, 11), (3,41, 19), (—38, 4, 7)*, (—4, —3, 7)*, (—4, 1, 9), (—4, 5, 3), (3, —5, —4), (38, —2, 11). In the second period the critical form (3, 4, —7) is converted into (4, —3, —7) by SZS, and (—3, 4, 7) into (—4, —3, 7) by eid iNee [22 BINARY QUADRATIC FORMS. 117. In tabulating periods of reduced forms, we may omit those forms which are connected with both the adjacent ones by the substitutions S,S; the remaining forms may be called principal reduced forms. Thus, for the two periods given above, the principal reduced forms are Te (66,8 1 a 6) 16), BO ee ee II. ( 3,4,-7), ( 4,-8,-7), ( 4,5, —3), (-3,-—5, 4), (—3,4, 7), (-4,-3, 7), (-—4,5, 3), ( 3,—-5, —4). The characteristics of a principal reduced form (a, b, c) are that either a+b+c and a—b+c have opposite signs, or if one of them vanishes, the other has a sign opposite to a. Geo- metrically, the representative circle of a principal reduced form includes one of the points p, p +1, and either excludes or passes through the other. For the exceptional case D=7, there are two periods of principal reduced forms; of these one is (1, —2, —3), (1, 2, -3), (2, —1, —83), (2, 1, —3), and the other is obtained by changing the signs of the coefficients throughout. All the principal reduced forms are critical. 118. Every form is equivalent to at least one reduced form. | For suppose the representative circle of the form constructed: this will intersect an infinite number of triangles, and the substitution which transforms any one of these into the funda- mental triangle will convert the given form into an equivalent reduced form. 7 119. It may be proved by considerations similar to those adduced in Chap. III. that two equivalent reduced forms must belong to the same period. It does not seem worth while to give the proof in detail; it may be observed, however, that with regard to the automorphs derived from the periods, and the corresponding chain-fractions obtained from them, a distinc- tion has to be made between periods which contain critical forms and those which do not. Thus for D = 87 the non-critical period (1-6, 11), (= Gee 8) eer een SHEAFS. | AS} leads to an automorph of (1, 6, —1) in the form ee LZ 12 La ? Spe Ses B 7e : and exactly as in the Gaussian theory we obtain from this the expansion ) was * 737 —6 =(0; 12). On the other hand, the period of (3, 4, —7) leads to the automorph STS*7S+7TS> 7S = e 84 36 ii) , and the corresponding expansion gy * * ihe eae oe) eet yy 120. In conclusion, a word may be said about the geometrical meaning of the automorphic substitutions. The infinity of tri- angles which have a common vertex at a rational point may be said to form a sheaf of triangles. In particular, those which have a common vertex at infinity may be said to constitute the primary sheaf. Suppose now that >’ is a sheaf with its vertex at the rational point —6/y, where y, 6 may be taken as relative primes. Determine a, 8 so that ad—SBy=1; then the substitution ,_ a0 + 8 Ooo= yo+o6 will transfer the pomt — 6/y to infinity, and >’ will therefore be transformed into the primary sheaf >. The most general substitution by which this is effected is nibs (a+ my) w + (8 + mo) yo+6 =o +m, where m is any integer. Now the substitution o”’ =o +m simply produces a cyclical permutation of the triangles of the primary sheaf: hence we see that it is always possible, and in one way only, to transform any sheaf >’ into the primary sheaf so that any assigned triangle of >’ may become the fundamental triangle V. More generally, any sheaf =’ may be uniquely transformed into any other, =”, so that any assigned triangle of =’ may be converted into an assigned triangle of >”. 124 BINARY QUADRATIC FORMS. It is now evident that the representative circle of any form will intersect the same number of triangles in every sheaf which it crosses: this number being equal to the number of reduced forms in its complete period as defined in Art. 116. The effect of applying an automorphic substitution is to produce a cyclical permutation of the sheafs which are crossed by the representative circle of the form: that is to say, if we represent the series of sheafs in order by ory: DEN Eas Pann PD ee D Feos EDIE RAN PrP Sein they will be changed into es Die hat he Ee Lee ee ee where / is some integer. In particular, the fundamental automorph eee —cU/o aUl/o, (T4+bU)/o will convert each sheaf into a consecutive sheaf. The sheafs divide the upper half of the representative circle into an infinite number of equivalent arcs; and the effect of applying the fundamental automorph is to transform each of these into the next following. Method of Nets. 121. Another useful geometrical method is that of réseausx, or nets. Let the plane of reference be divided up into a system of equal and similar parallelograms by means of two sets of equi- distant parallel straight lines; such a system will be called a net, each of the parallelograms a mesh, and each point, where two lines cross, a node. Through any node O draw two rectangular axes OX, OY (fig. 5). Then if OPRQ is a mesh with one vertex at O, and if @,=p+q, Ge = 7 4S), are the complex quantities corresponding to P and Q, the net is completely defined by a, a, and may be denoted by (a, a). The system of nodes is given by Z=Mo,+ NB, where m, » have all possible integral values, zero included. NETS. B25 The quantity ps— qr measures the area of the parallelogram OPRQ: it is called the norm of the net, and written Nm (a, @,). Fig. 5. Let a, 8, y, 5 be any integers such that a’— By=1: then if wy =o, + Yo, By = Pa, + da», the net (a,’, a,’) has the same nodal system as (a, w,). Two such nets are said to be properly equivalent, and we may write (ay, By) © (@1, By). More generally, if ad — By =n, a real integer, all the nodes of (a, a) will belong to the nodal system of (#,, #,) but not con- versely ; in this case (a, a) is said to be a multiple of (a, a). It is easily verified that Nm (ay, o,/) = (ad — By). Nm (a, @), and hence, in particular, if (a, @,) (a, @), Nm (a, a) = Nm (a, a). 122. Suppose, now, that f=am?+2bmn-+ cn? is a definite form of determinant — A; then af =(am + bn) + An? = Nm {ma+n(b +A). This suggests that the form af may be represented by the net (a, b+%/A); and in the same way if f’ = a’m? + 2b’'mn + cn? is the 126 BINARY QUADRATIC FORMS. form into which / is converted by the substitution (e A we may say that af’ is represented by the net (a, a), where wm, =aa+y(b+1/A), @, = Bat 6 (b+ WA). If s,, 8, are the lengths of the sides, and d,, d, the lengths of the diagonals of a mesh of the net (@,, a), we have s2=(aa +yb)P?+7A=ad, s2= (Ba + db% + &A =a’, d,=(aa + yb+ Bat db? + (y+ oyPrA =a(a'+c’ + 2b’), d.2 = a(a'+ c’ — 2b’). (Cf. Art. 54.) 123. It is easily proved geometrically that every net 1s properly equivalent to at least one net of which the mesh is such that neither of its sides is greater than a diagonal. For suppose that O is any node of the given net; then there will be at least two other nodes which are at a minimum distance from 0. Let P be any one of these; then the line OP produced indefinitely both ways will contain an infinite number of nodes. Let this line be moved parallel to itself until it first passes through another set of nodes; and let Q be a node on the line in its new position which is at least as near to O as any other node on the new line. Then the parallelogram OPRQ, of which OP, OQ are adjacent sides, will be the mesh of a net which contains all the nodes of the given net: moreover, it follows from the way in which P, @ were chosen that OR ¢ OQ ¢ OP, while if RQ is produced to Rf’, so that QR'= RQ, Ff’ is a node, and PQ=OR’¢ OQ; hence OPRQ satisfies the geometrical conditions above stated. Such a net will be called a reduced net. With O as origin, let a, a, be the complex quantities as- sociated with the points P and @; the conditions of reduction are \oy| plo, + wo), > |r, + We oo If (@,, w,) is a reduced net, we obtain four associated reduced nets by variation of sign from (+@,, + @,). Of these (a, a) and REDUCED NETS. bea (@,, — @2) are improperly equivalent, so that one of this pair must be properly equivalent to the given net. The nets (a, a), (—#,, —@,) may be considered identical ; moreover, if (@,, @,) is reduced, so also is the equivalent net (@., — @), hence we may add the further condition |a,| +|a,]. The corresponding quadratic form af’ satisfies the conditions aw’ pa(a’+2b'4+c), ac pa(a + 2b'+¢’), aw pac’, or, which is the same thing, lo |a’| + 210", These are precisely Lagrange’s conditions of reduction ; so that the method of nets gives a complete geometrical interpretation of the theory of transformation and reduction as applied to definite forms. It may be observed that it immediately follows from the geometrical method that if (a, b, c) is a reduced form, a is the numerically least of all the numbers representable by forms of the class to which (a, b, c) belongs, and that if |c|>|a|,c is the next least numerically. This may, of course, be proved analytically ; and, in fact, it is upon the existence of minimum representable numbers that Hermite has based his general theory of the reduc- tion of definite quadratic forms. 124. It is easy to see that, in general, the geometrical reduc- tion of a net is unique ; there are, however, two exceptional cases. The first of these is when there are two nodes P, P’ nearer to O than any others, while there are fowr nodes at the next smallest distance. These nodes are the vertices of a rectangle with its centre at O; and if a, is the complex quantity corresponding to P, the quantities which define the vertices of the rectangle may be taken to be + o,, +(@,—a,). There will be two reduced nets properly equivalent to the given one, say (@,, @,) and (a,,-—@#,+ a); the corresponding reduced forms will be of the type (a, + 4a, c) with |a!<|c|. Secondly, there may be st# nodes, all at the same minimum distance from QO. They must evidently be the vertices of a 128 BINARY QUADRATIC FORMS. regular hexagon, and the corresponding reduced forms are of the type (a, +4a,a). Here there are three reduced nets, say (a, @.), (a, —@,+,),(@,, —@,+o,) with wo, =e7"%,a,, but there are only two reduced forms; the reason being that (#,, —@,+%,) 1s connected with (a,, a.) by the substitution G 3 i , which is an automorphic of (a, 4a, a). All this is in agreement with the results of Art. 65. 125. The method of nets may be extended so as to apply to the theory of indefinite forms. We may construct a perfectly consistent algebra of ‘hyperbolic’ complex quantities z= a+ yj upon the lines indicated by the formule lj=j.l=j, p=, (ety)t@+yp=@te)+yty)), (w+ yj) (@ + yp) = (aa! + yy’) + (ay! + #'y) J, Nm (@+ 9) = 2 — ¥, w arg (x te Y)) —= cosh Vea yp = sinh Be y C It is easily verified that Nm (27) ='Nm'(@) Nim (2%), arg (22) = arg 2+ arg 2’. Following the analogy of the ordinary theory, we represent the quantity «+ y) by a point whose rectangular coordinates are (vz, y): then the formule for addition and subtraction have a geometrical interpretation exactly the same as that for the ordinary theory; and the other formule express geometrical relations to the hyperbola a —y?=1. Thus if P be the point corresponding to «+ yj, and if OP meet the hyperbola in P’, Nm(#+y))=OP/OP’, and arg («+ yj) is twice the numerical measure of the area of the hyperbolic sector OAP’, where A is the vertex of the hyperbola on the same branch with P’. Now if f=am? + 2bmn + en? is any form with positive deter- minant D, we have | af =(am+ bn) — Dn =Nm {|ma+n(b+jy7D)}. METHOD OF POINCARE. 129 We represent the form af by the net (a, b+ 4/D), and it follows just as above (Art. 122) that if w,=aa +y(b+j/D), w= Ba+6(b+4/D), the net (w,, #,) may be taken to represent af’, when /f’ is derived Bom f by the substitution & a) 126. If we draw the lines x+y=0, «—y=0, it is geo- metrically evident that the net (a, a.) equivalent to (a, b +.4/D) can be determined so that the points a,, #, are on opposite sides of one asymptote and on the same side of the other. For suppose that OM is either asymptote: choose any two nodes P, Q, one on each side of OM, and on opposite sides of the other asymptote, and let PQ meet OM in R. Then if the triangle OPQ contains no nodes within it or upon its perimeter (except at O, P,Q), the paral- lelogram of which OP, OQ are adjacent sides may be taken to form a mesh of the required net: if otherwise, there will be a point P’ within or upon the perimeter of OPA and a point Q’ within or upon the perimeter of OQR such that OP’Q’ is a triangle without any nodes except at O, P’, Q’, and then if the quantities a’, a,’ correspond to P’, Q’, either of the nets (a@,’, a,'), (a,—a,) will have the property required; and one of these must be properly equivalent to the given net. If we call a net of this kind reduced, the analytical condition for a reduced net (@,, a) is that Nm (a@,) and Nm (#,) must have opposite signs. Now with the notation of last Article Nm (a,) =a’, Nm (a,) = ac’, so that in the form f’ = (a, b’, c’) the coefficients a’, c’ will have opposite signs. A form of this kind may be termed reduced; and it immediately follows (cf. Art. 67) that every form of determinant D is properly equivalent to at least one reduced form, and that the number of reduced forms is finite. It should be carefully observed, however, that although the number of reduced forms is finite, the number of reduced nets is infinite. In fact, if (@,, 7.) is a reduced net, it is geometrically obvious that either (a,,a,+ a.) or (a, + a2, a2) is also reduced; and similarly either (#,, 7,--a,) or (;—@»2, @) is reduced. Each reduced net is therefore connected with two adjacent reduced nets M. 9 PS 130 BINARY QUADRATIC FORMS. 14 Al AO by means of one of the substitutions i “a i 1) and one of the substitutions & is 3 Gi ‘ i) . In the same way every reduced form is connected with two adjacent reduced forms: for mstance (4, 3, —2) is converted into (8, 1, —2) and (4, —1, —4) by the 1,0 1,-—1\_ substitutions @ 4 and ( 0, 4] respectively. The infinity of reduced nets belonging to a given class may be arranged into a linear series by means of the above substitutions ; the corresponding reduced forms will form a recurring series, and everything proceeds as in the Gaussian theory. AUTHORITIES. THE analytical theory of binary quadratic forms contained in Chap. Im. is based upon that given by Gauss in Arts. 153—-222 of the Dzusquisitiones Arithmetice: much help has also been obtained from the 4th section of Dirichlet’s Zahlentheorie, and Smith’s feport (1861) Part m1. The memoirs of Dirichlet which relate more particularly to this part of the theory are entitled Démonstration nouvelle dune proposition relative ad la théorie des formes quadratiques (Liouville, 2nd series, 11. (1857) p. 273), and Simplification de la théorie des formes binaires du second degré a déterminant positif (ibid. p-. 353 : translated, with additions, from the Berlin memoirs for 1854). The most important researches anterior to Gauss are those of Lagrange contained in his Recherches d Arithmétique (Nouveaux Mém. de l’Acad. de Berlin 1773, 1775). Lagrange considers forms of the type dz?+ Bry+Cy?, and distinguishes them according to the sign of B?-—4AC: he introduces the ideas of transformation and equivalence, and shews that every form is equiva- lent to one of a limited number of reduced forms. The criteria of reduction are that neither | A | nor | C| is less than | B|: for definite forms, this leads to conclusions similar to those in the text, while if D= B?-44AC is positive, it follows that A, C have opposite signs, and | B| }»/1D. Correct classifications are given for a considerable number of special determinants, together with the corresponding forms of linear divisors ; in fact, the memoir abounds in valuable and suggestive matter, and is well worth careful study. Still, the improvements which Gauss effected are undeniable, and he justly claims credit (D. A. Art. 222) for the important distinction between Proper and improper equivalence. For the history of the Pellian Equation, see Smith’s Report (1861) Art. 96. The first rigorous theory of its solution was given by Lagrange: Solution dun Probleme @ Arithmétique (Miscell. Taurin. t. iv. (1766—9), or Oeuvres, t. i. p. 671). A list of fundamental solutions of 72— DU2=1 up to D=1000 will be AUTHORITIES. liait found in Legendre’s Théorie des Nombres. Degen’s Canon Pellianus gives also the partial quotients for the recurrent expansion of JD. Many of Euler’s memoirs relate to special problems of the theory of quadratic forms: the reader is referred to the analytical table of contents prefixed to the Comment. Arith. (t. i. p. 1x). The geometrical method employed in the earlier part of Chap. tv. first arose in connexion with the theory of elliptic modular functions. The best authority on this subject is Klein’s Vorlesungen tiber die Theorie der ellip- tischen Modulfunctionen, edited by R. Fricke, of which the first volume only has yet appeared (Leipzig, 1890). In this work full references are given to the numerous papers on the subject ; the part which relates more particularly to quadratic forms will be found on pp. 163—268. According to Klein (Le. p- 250, note) the first application of the ‘Modultheilung’ to the theory of indefinite quadratic forms was made by H. J. 8. Smith in his paper Sur les ¢quations modulaires, written in 1874 and published in the Atti dell’ Acc. Reale dei Lincei, t. i. (1877). However, there can be little doubt that the modern development of the modular-function theory dates from Dedekind’s letter to Borchardt (Crelle, lxxxiii. (1877), p. 265), and is principally due to the researches of Klein and his school. Special reference should be made to Hurwitz’s Grundlagen einer independenten Theorie der elliptischen Modulfune- teonen (Math. Ann. xviii. p. 528). On the method of nets, see Gauss’s review of Seeber’s work on ternary quadratics (Werke, ii. p. 188, or Gott. Anz. J uly 1831); Dirichlet, Ueber die Keduction der positiven quadratischen Formen mit drei unbestimmten ganzen Zahlen (Crelle, xl. (1850), p. 209); Poincaré, Sur wn mode nouveau de repré- sentation géometrique des formes quadratiques définies ou mdéfinies (Journ. de l’Ecole Polyt. cah. 47 (1880), p. 177). An interesting modification of Gaussian methods is given by Hurwitz : Ueber eine besondere Art der Kettenbruch-Entwickelung reeller Grissen (Acta Math. xii. (1889), p. 367). Hermite’s important memoirs on quadratic forms will have to be con- sidered later on- it may be mentioned here that the criteria for a principal reduced form (Art. 117) are enunciated by Hermite in his paper Sur Véntro- duction des variables continues dans la théorie des nombres (Crelle, xli. (1851), p. 207). | bo CHAPTER Vs | | Generic Characters of Binary Quadratics. | 127. As already explained, the classes of binary quadratic : forms belonging to the same determinant D may be arranged into — orders, according to the values of dv (a, b, c) and dv (a, 2b, c), where (a, b, c) is any form of the class considered. The classes belonging’ to each order may be further arranged into groups, each of which | constitutes a genus. The principle of this distribution depends | upon a few simple propositions, the proof of which will now be. given. | Let (a, b, c) be a primitive form for which dv (a, 2b, c)=a;, then tt is always possible to assign integers x, y, prime to each other, so that (ax*+ 2bey + cy?)/o =n may be prime to any prescribed | number m. | For suppose that p, p’, p’... are the different primes which | divide a/c, c/o and m simultaneously; q, q’, q’... those which divide a/c and m, but not c/o; 7, 7’, r”... those which divide e/e and m, but not a/c; finally, s, s’, s”... those which divide m, but not a/o or c/o. : Let P= ppp os | Q=497q"..- : R=rrr ... | SiS 55 Sse : Since (a, b, c) is primitive, 2b/c is prime to P. Now choose x, y so that both are prime to P, x is a multiple of Q, but prime to R, y a multiple of R, but prime to Q, zy a multiple of S, but not divisible by the square of any of the prime factors of S. This may be done in an infinite number of ways and still leave wz, y | ] PARTICULAR CHARACTERS. £33 prime to each other. The simplest way is to put z=QS’, y = RS”, where S = S'S” is any resolution of S into two factors. This particular choice of x, y gives a ax’? + 2bry+cy? _c Oo oC ‘ Ue (mod q); according as y or v=0 (mod s). Here p, g, 7, s denote any prime factors of P, Q, R, S re- spectively ; and it easily follows that is prime to P, Q, & and S, and therefore also to m. A special case of this theorem, which is often useful, is that we can always find a number on capable of primitive representation by (a, 6, c) and such that n is prime to 2D, or, which is the same thing, odd and prime to D. 128. Now suppose that (a, b, c) is a properly primitive form of determinant D, and let n, n’ be any two integers prime to 2D and capable of primitive representation by (a, b, c). Then if we put _ n= aa? + 2bay + cy’, n = a?+ 2bB6 + cd’, we have identically nn’ = a — Dy’, where “= aa8 +b (ad + By) + cys, y = a6 — By. (Cf. Art. 55.) From this identity we draw the following six conclusions :— I. If p is any odd prime factor of D, (n|p) =(n'|p). For since nn’ = «#? (mod p), (nn'|p) =1, and therefore (n|p) = (n'| p). ro4 GENERIC CHARACTERS. Il. Jf D=3 (mod 4), (— 13°) =(—1)2 9, For nn =2— Dy =2?+y? (mod 4), and since nn’ is odd, one of the numbers a, y is odd, and the other even; therefore nn’ =1 (mod 4), whence n=~' (mod 4), and (—1)2@-) =(-1)3 ">, IIL Jf D=2 (mod 8), (— 1)? =(-1)# ">. We have nn’ = 2 —2y?(mod 8), and # must be odd, so that «= 1 (mod 8); also 2y?= 2 or 0 (mod 8): hence nn'= + 1 (mod 8), and therefore n= +n’ (mod 8), n?= 7? (mod 16), and the theorem follows. IV. If D=6 (mod 8), (— 1)2@) +8) = (— 1) 8 a, Here nn = a + 2y? (mod 8), and a is odd; so that nn’=1 or 3 (mod 8) according as y is even or odd; therefore n=’ or 3n’ (mod 8) respectively, and in each case the truth of the pro- position may be verified. This is obvious if n=”' (mod 8); if n = 3n' (mod 8), we have $ (a1) + §( 1) -F! - I) — 4 (02-1) = 4 (nn) 4 4 A= =n' +n” (mod 8) = 0 (mod 2). V. Lf D=4 (mod 8), (—1)2?® =(—1)2@—, For nn’ = «= 1 (mod 4), and therefore n = n' (mod 4). VI. Jf D=0 (mod 8), (— 1)?’ =(—1)2—), and e 1) =— (— Deh. In this case, nn’ = # = 1 (mod 8), and therefore n=’ (mod 8).. It will be observed that these theorems express properties which are common to all odd numbers, prime to D, representable by the given properly primitive form. : Thus theorem I. states that these numbers are all quadratic residues of (an odd prime factor of D), or else all non-residues ; theorem II. asserts that when D=3 (mod 4), the numbers in question are all of the form 44+ 1, or else all of the form 4k + 3; and so on. 129. Let p, p’, p”... be the different odd prime factors of D; and, as above, let n be any odd number prime to D and represent- DIRICHLET’S TABLE OF CHARACTERS. P35 able by the properly primitive form (a, b,c). Then it has been proved that the symbols (n| p), (r|p'), (n| p")... will have values which may (and in fact do) depend on the form (a, b, c), but not upon the particular value of n. They are called the quadratic characters of the form. Besides these, except when D=1 (mod 4), there will be one or more supplementary characters. Putting, for convenience, (—1)2™—-) = x, (— Pe Vy, the supplementary characters are for D = 2, 3, 4, 6, 7, 0 (mod 8), Ws Xo Xo XM Xo x need wp respectively. The totality of all the particular characters of any form, quadratic and supplementary, makes up its complete or generic character. Since any number representable by (a, b, c) is repre- sentable by any other form of the same class, we may speak of the generic character of a class. Classes which have the same complete character are said to belong to the same genus. 130. The following table, taken from Dirichlet (Crelle, xix. (1839), p. 338), shows in a convenient form the assignable characters of properly primitive classes. In every case, S? denotes the largest square which divides D, so that D=PS? or 2PS? where P is a product of different odd -primes p, p’, p”, ete.; finally r, 7’, r”, ete. are the odd prime factors of & which do not divide P. The object of the vertical line which separates the particular characters into two groups will be seen afterwards. I. D=PS?, P=1 (mod 4). S=1 (mod 2) | (n|p), (n|p), ..-- | (lr), |r’), ... S=2 (mod 4) | (7|p), (n|p’), ... | x, (n/n), (n|r’), «..- S=0 (mod 4) | (m|p), (r|p’), ... | x, (n|7), (n7’), ..- Il. D=PS*, P=3 (mod 4). S= 1 (mod 2) | y, (n|p), (vip), ... | (air), (air), ... S=2 (mod 4) | y, (n|p), (n|p’), ... | (air), (air), ... S=0 (mod 4) | y, (n/p), (n|p’), ... | ve (alr), (n|r’), ... 1386 GENERIC CHARACTERS. Ill. D=2P8?, P=1 (mod 4). S=1 (mod 2) | ¥, (|p), (n| p’?), ... | (lr), (rir), ... S=0 (mod 2) | ¥, (2| p), (|p), ... | x, (alr), (2/7), .-. IV. D=2PS8?, P=3 (mod 4). S=1 (mod 2) | yw, (n/p), (r| p’), «.. | (n|r), (nr), .. S=0 (mod 2) | x, Wr (|p), (n|p’), ... | (n|r), (n|r"’), «.. 131. It should be observed that in order to assign the value of a particular character (n|p) for a given’ form (a, 6, c) it is sufficient to find a number n representable by (a, b, c) and prime to p; and in the same way the supplementary character or characters (if any exist) may be inferred from any odd number representable by the form. Now the extreme coefficients of the form are numbers representable by it; and if (a, b, c) is properly primitive, one or other of the numbers a, ¢ must be prime to any given prime factor of D, and one or other of them must be odd. All the particular characters of a properly primitive form may therefore be determined from its extreme coefficients. For example, let the form be (6, 3, 18). Here D=-— 69 =—38.23 = 8 (mod 4), and the particular characters are (n|3), (n|23), and y. From the coefficient 18, which is odd and prime to 3, we obtain | x=(— 18 =+1, (3) =(3)3)=+1, and from the coefficient 6, we find (n]23) = (6|23) = (2|23) (3)23) =(4 1) (41) =+1. Therefore the total character of the form is (n|38)=4+ 1, (n[23)=+1, y=+1; or, as Gauss would express it, 1,4; 43, R23. 132. In the case of improperly primitive forms, the characters will be as in the first line of Dirichlet’s table, except that n will denote an odd number the double of which is represented by the form. Here again, the total character may be assigned by inspec- tion of the extreme coefficients of the form. Thus, if the form be (10, 5, — 4), for which D=65 = 5.13, the EXAMPLES. RS7 particular characters are (n|5) and (n|13). Putting n =— 4/2 = —2, we have (n|5) =(— 2|5) =— 1, (n| 18) =(— 2]13)=—-1. The generic characters of derived forms may be at once inferred from those of the primitive forms from which they are derived. 133. The following tables are given by way of further illustra- tion. For the negative determinants, representatives of the positive primitive classes only have been given. Each line of a table gives the total character of a genus, and representatives of all the classes belonging to that genus. Improperly primitive genera, when they exist, are separated from the rest by a horizontal line. Derived genera are omitted. D=-— 96. (2/3) | x | > i a ee 9G) rte 2.320) cael a as (5, 2, 20), ( 5, — 2, 20) oa < = (3, 0, 32), (11, Seale.) | is Tat Ss | Ge 3, 15), ( if — 3, 15) | D=—99. (n|3) |} (n{11) | te +. Zar 0, 99), (4, it 25), (4, 1, 25) i eHamid oT 20) Comet 20\0 1) O10.) + + Glee) as + CLO EO} D=186. (nj17) | x | ¥ + + | + |( 1, 0,186), (8 0, 1%) i Pee et), | 156) cig Se Ue LO) as SE NICGR SS Lath th 5) ON 3c 45) aes a Ts (— 3, L, 45), (= 3, ale 1, 45) 138 GENERIC CHARACTERS. sb} (n/3) | (n|5) | x Pew OH ges ae 10s 50) Ey a | — |(-1, 0, 150) ee Se 0 a0) — — + |(—38, 0, 50) D=185 (n|5) | (/37) + + 1, 0, — 185) = 3 res eye + coli ey Ib ete) es ses hin GRIN: fe ait} 134. The particular characters of the form (1, 0, — D) are all +1. This form is called the principal form of determinant D, and the class and genus to which it belongs are called the principal class and principal genus. 135. It will be observed that in each of the above examples, — the number of complete characters which actually exist is pre- cisely half that which is a priori assignable. For instance, when © D =— 96, there are three particular characters (n|3), y, Ww; and since each may be either + or —, the number of complete charac- ters possible a@ priorz is 2° or 8, whereas only four of these actually occur. It may be proved by the law of quadratic reciprocity that at least half of the assignable characters of properly primitive classes are impossible. Thus if 7 is a positive odd number representable by a form f of determinant D, then (Art. 59) D is a quadratic residue of m, and therefore if, in the notation of Dirichlet’s table, D= PS? or 2P8?, it follows that P or 2P, as the case may be, is a quadratic residue of 2. IMPOSSIBILITY OF HALF THE ASSIGNABLE CHARACTERS. 139 Applying the generalised law of reciprocity (Art. 42), we have in the first case 7 (Pin) =1, (n|P) = (n|P) (Pln) =(— 109 that is, (n|p) (m| p’)... = XFE, Therefore if D=PS?, P=1 (mod 4), (n|p)(n| p')... =1; while if D= PS?, P=3 (mod 4), x, (n| p) (n| pp). = 1, On the other hand, if (2P|n) = 1, that 1s, if (= 18") (P|n) = 1, we have (n| P) = (— 1) 9) (Pin) (nj P) = ryt? ; hence if Das? b=) (mod 4), p-(a|p) (n|p)eoe = 1; while if D=2PS?, P=3 (mod 4), Vx (m p) (|p). = 1. Comparing these results with the table, it appears that in every case the product of all the particular characters in any line of the table which are to the left of the vertical line of division must be equal to + 1. It is easy to see that this condition excludes precisely one half of the assignable characters. _ The theorem that half the assignable characters are impossible will be subsequently proved independently of the law of reciprocity, thus affording a new proof of that law; and it will further be shown that actual genera always exist for the remaining characters, and that each genus of the same order contains the same number of classes. AUTHORITIES. Gauss: Disg. Arith. Arts. 228-—232. DiRIcHLET : Recherches sur diverses applications de 0 Analyse infinitésimale a la Théorie des Nombres (Crelle, xix. (1839), p. 324, or Werke, i. p. 413). See also Dirichlet-Dedekind, Zahlentheorte, Supplement IV.; and Smith’s Report, Art. 98. CHAIR TE aay 1. Composition of Forms. 136. Ler F = AX*?+2BXY + CY? be any binary quadratic form of determinant D= B?— AC. Suppose that by means of the bilinear substitution Feo ayeen | (1) Y= Gove aii qnzy ab qoyx’ = gsyy Cees ceresene > F' becomes ff", where f=aa? + 2bay+ cy, f =a'e? 4+ Wae'y'+cy?; then we say that fis transformed into f/f’ by the substitution ee Pi; Pe Bs) . Q> N> qa; Ys In all that follows it will be assumed that the coefficients of the substitution are integral; and we shall write Ria ran ieee ge ns (2) S=p.m—pm, T=pgs— pi, U = pids— Psd where observe that PU — Q7'+ RS =0 identically. The substitution, or transformation, is said to be primitive, if P,Q, BR, S, T, U have no common divisor. : Write b—de=d.07-—Uuc—a. and let M,m, m' be the greatest common divisors of A, 2B, C; a, 2b, c; a’, 20, c’ respectively. We may regard the equations which define X, Y, as a linear transformation of the single set of variables 2, y, by means of which the form (A, B, C § , y)? becomes (fa, fb, fc Q x, yp. BILINEAR SUBSTITUTION. 141 The determinant of the substitution is eee ie sa Det oD ne ae (3) got + ny’, qa’ + gay" = Qa? + (B+ 8) aly’ + Ly”, and it follows from the invariant property of the discriminant that (f'by — (fra) (fc) = A’. (B’~- AC), that is, AS RCT 8 NIE eB OM CC OP (4). Similarly, putting eee UA RULES aoe 0 ee oe (5) JoX FY, GE + Osi | = Pa? +(Rh—-S8) «xy + U7’, we have ie OLIN cet, ote ee se (6). Let d6=dv(Q, R+SN, 7), 6 =du(P, R-S, VU); then evidently, by (4) and (6), dm? = DD, d’m? = D8”. Again let fo Oil Caio e Le tu.) then it can be easily proved that k = dv (6, &). For suppose dv (6, &’)=p: then p divides P, Q, T, U, 2R, 28, and therefore either k= or k=3y. In the latter case w must be even, P/k, Q/k, T/k, U/k must all be even, and R/k, S/k both odd. But this is inconsistent with the identity NES XO Coe Tey ay eS ea therefore Keyl Or ono.) Hence Dk? = dv (D&, DS”) TACT RON Cal epee fe Aer (7). It is clear that d/D, d’/D are rational squares; so that putting Od 1 ee ce eee (8), n, n’ will be rational. It has been proved that A2 = n? ee A? = n? Te, p47 COMPOSITION. we can therefore choose the signs of n, n’, so that ANSE 0 A =e having done this, we find by comparison of coefficients a eae tet) ESSN Oe eee (9) ae S20. Leg The form f is said to be taken directly or inversely, according as n 1s positive or negative: and similarly for /”, n’. It may be verified that AA = (92 = Gos) X? + (Pods + PsYo — Pre — Pr) XV +(PiPe— Pos) Y?.......-. (10) identically ; and comparing this with NAS ef we infer that a 9093 _ Pods + aan PrGe — Poi _ P1 Bee pis = nn’...(11). Conversely, if the nine equations Poon, sh SS 2bny Ue= cn, \ OE 0 1 eth 3207 ec (0) NY2 — VoUs = Ann’, Dds + P3Go — Pride — Poi = 2Bnv’, PiP2 — PoPs = Cnn’ are satisfied, then the substitution & “e ‘ a - ae ‘) will transtorm 09 1s 2) 3 F into ff’. Gauss obtains the system (Q) by direct comparison of the coefficients in the identity (A, B, CUX, VP ata, bola. yy xa. 0, cla ye: this leads to nine equations such as Ap +2Bpoqo+ Cg? = ae’, Apypy + B (P+ 2190) + CMH =O, and so on: from these the equivalent set (Q) is derived (see D. A. Art. 235). The simplified method here adopted is due to H. J. S. Smith (Report, Arts. 106, 107). 137. It follows from the identity AX?+ 2BXY + CY? = (aa? + 2bay + cy”) (aa? + 2'a'y’ + cy”) that M divides mm’. It can be shewn that mm’ divides Mk?. GAUSS’ DEDUCTIONS. 143 For by equating the coefficients of «, wy, y? in the above identity, we obtain ; 2 AAWGNE Y\2 ar-a C2) +30(2 2) -e(2) OX 0X Nae Ye OA Oy: Cano ye See Ga Bea eae ae) OX OX OY oY\2 of’ = ae 2B pole | Multiply, in order, by oY\? OymoyY aay? Gye 2bf =2A and.add; thus CAR OVE OXGOY \* oY \? CP axed A OVEN os, (Se ay Oyo =| CA 7 a aaa ye oe a) cae a Every term on the right-hand is divisible by mm’: hence the expression on the left, that is AA?®, 1s divisible by mm’. In the same way we can prove that 2BA?, CA? are each divisible by mm’: and therefore mm’ divides Mé?. Similarly mm’ divides M6”: and hence finally Mk? = dv (M8, M8”) is divisible by mm’. Again let Mil cy (A- BC), i= av (G; 0, c),, mM —du(a, vc); then it can be proved in precisely the same way that mn’ divides EVP’. It will now be supposed that /=1, so that the transformation is primitive. When this is the case, #’ is said to be compounded of f and /”. It follows from what has been already proved that in the case of composition D=dv(dm?, dm?) M=mm’', $= 0 (mod mm) and also that mn’ =Vd’m?/D, and m’n = Vdm?/D will be integers and relatively prime. The second of equations (12) shews that ¢¥#1 divides mm’: and in the same way it divides m’m: hence if m=m and wW=m’, that is, if f, f’ are both properly primitive or derived from properly primitive forms, Ff’ is properly primitive, or derived from a properly primitive form ; whereas in any other case {#1 = 4mm’ = 4M, and 144 COMPOSITION. F is improperly primitive or derived from an improperly primitive — form. 138. We are now confronted by the fundamental problem :— Given two forms (a, b, c), (a’, b’, c’), to find, when possible, a form (A, B, C) compounded of them, each component form being taken in a prescribed way. In order that a solution may be possible, the ratio of the determinants of the given forms must be a rational square. Suppose this to be the case; let d, d’, m, m’ have the same meanings as before, and let D=dv(dm”, d’m?) taken with the same sign as d, d’. Let n=Nd/D, n’ =Nd’'/D, the sign of each being taken positive or negative according as the corresponding form f or f’ is to be taken directly or inversely. Then n,n’ are rational, and mn’, m'n are integers and relatively prime. Hence by the first six of equations (Q), P, Q, R, S, T, U are determined: and it 1s easily seen that they are all integral and relatively prime, so that k =1. The next step is to find eight integers ), D1, D2, Ps» Gor Gr» Yo» Ys so as to give P, Q, R, S, 7, U these known values. Consider the skew-symmetrical system of equations 2,0 — x,T' + «2,8 =0 —a2,U +R — 2,Q = 0 al’ — 2,R +2,P =0 — “8S +24,Q —2.P =='(): these are equivalent to two independent relations; for if we multiply the first three equations in order by R, 7, U respectively and add, we are led to the identity x;(PU — QT + RS) =0; and similarly for any other group of three. Now let 4, @,, 4, 0; be any multipliers whatever, and put oe 6,P te 0.0) ats 0, OR + 0,59 + 6,7, N= — AQ —- 0S + 6,0, = —-OkR—-O6,T — 0,U, GENERAL METHOD OF SOLUTION. 145 then the preceding set of equations may be satisfied (and in the most general manner) by putting Hy - Hs Ly > Xe= Ny 2 M No > Np. We may suppose @,, 6,, 6, 6; to be rational, or indeed integral, and chosen so that m...7; do not all vanish; and then we may suppose 4, #, #, #; to have integral values proportional to No» M1. No» 73+ Call these values q, 1, G2, 93; We may take them so as to have no common divisor, and therefore we can determine four integers My, 1, W:, 73, such that Too + 11Qi + Wee + T3qz = 1. Further, let », 2, ~2, ps be the values of »,, M1, M2, ); When iy UCN CASTa ys cme 72, 7733 then p, Pi, Po, Pss Yor Vi» Yo» Ys Will be a set of eight integers such as is required. We have in fact Poh — AQ =(mP + 7.Q04+ 7k) G+ (mP — 7S — 7,7) Ge = (ToGo + TGs + ToGo + Tags) P + 2 (— GS + HQ — GP) + 73(— Ql’ + mh — q;P) ale. since To + T19r +.ToGo + 137s = 1, and the coefficients of 7,, 7; in the other terms vanish. This is one of the equations which have to be satisfied: and the rest may be verified in a similar way. It remains to be proved that the values of A, B, C derived from the last three of equations (Q) are integral. Returning to the identity (Pa? + (— 8) ay + Uy?) (Qa? + (B+ 8) aly’ + Ty = nn (AX? 4+ 2BXY 4+ CY%), and remembering the meaning of 6, 6’, we see that if (R+8)/8 and (f —8)/d’ are both even, 2Ann’, 2Bnn’, 2Cnn’ are divisible by 260, Now Dé?=dm?, and therefore & = m’n?: similarly 6? = m2n?, and 66’//nn’ = + mm’ = an integer. If, then, as in the case now M. 10 146 COMPOSITION. considered Ann’/d6’=+ A/mm’ is an integer, a fortiors A is an integer, and similarly B, C are integers. On the other hand, if either (R + 8)/é or (R—S)/8’ be odd, then either 2b/m or 2b’/m’ is odd, and hence either m or m’ is even: 2Ann’, 2Bnn’, 2Cnn’ are divisible by 66’, the quotients (neglecting sign) being 2A/mm’, 2B/mm’, 2C/mm’ ; and since mm’/2 is an integer, we conclude, as before, that A, B, C are integers. 139. It will be observed that a form F compounded of f, f’ in a prescribed manner may be found in an infinite number of ways: but it can be shewn that all such forms are properly equivalent. For suppose # = (A, B, C) and #” =(A’, B,, C’) to be any two such forms, and let | & Pry Pa, _ (Ps? Pris Ps i) Gor Ti» Ya. Ys Go> M1» Ya. Ys be two primitive substitutions which transform F’, F” respectively into ff”. | Then Po — Pr qe = Poh — Pde = an"; and similarly for any other corresponding pair of determinants. Now let integers (01), (02), (12), ete. be chosen so that (O1) (Pog — Piqo) + (02) ( pogs — pogo) +... = 1, or say DO, M) (Pada — Puga) = 1: and put ZA, #) (Padu — Pudr) = 4 D(A, #) (PaPul — Pupr) =B, SA, 4) (9x Qu — qu Ir) Bey D(A, #) (page — Pugr’) = 6. Then if v is any one of the numbers 0, 1, 2, 3, apy + By = = (A, ) Pe (Pxdu — Pur) + Ye (PrPu’ — PuPr’)} = ZO #) UPN (Pda = Puede) — Pu! (Pda — Prqe)} = =O, #) Da’ (pda! — pugs’) — Pu (pr'qr’ — pre’) ==, 1) Po’ (pn'qul = Py'qn’) = Pv = (As H) (Prdu — Pua) / =P. EQUIVALENT SOLUTIONS. 147 Similarly Yr + OQ = W' 5 and hence Pre Pul .fOP. +:8q,; op. + Bais | yp, + ofa, YPut Sq, | = (465 — By) (Prdu — Pedr) : therefore ao — By =1. Now if XY, Y are the variables of F, and X’, Y’ those of F’, AT = powe! + py'ay! + pia'y + psyy = (ap + BQ) ve +... =aX + BY, and similarly VY=yX +6Y; | qx’ Ops from which it is clear that Fis transformed into F” by the proper substitution ts AY so that F, #” are equivalent. It is obvious that, conversely, if F is compounded of f, f’ and Ff’ is properly equivalent to F, then F” is also compounded of f, in the same way as F is. More generally, if #” is transformable into ff’, but not necessarily com- pounded of f and f/f’, and if, as before, / is compounded of f, /’, then Ff" contains F. For with the same notation as before, we shall have Pot! — Pid =" (Por — Prd) ete-s and it can be shewn that, as before, P=), TPa, 3. 9, 9P, — 99, » PU — Py QM = (08 — BY) (PrT ue — Pur) * hence ad — By=h, and F” is transformed into /’ by the transformation (‘ a of determinant & : Y that is, Fis contained in /”, Many of the succeeding propositions may be generalised in the same way. 140. Let f be transformed into an equivalent form ¢ by the substitution Mi at then it is clear that F' is transformed Pfr he into $f’ by the substitution X = (py! + pry’) (aw + By) + (p’ + pay’) (ye + by), Y= (gon! + qry’) (aw + By) + (q’ + qy’) (ye + dy), that is, et +Y7pPo, Ait YPs; Bpo a Spo, joyan Be a ao+ 992, An+74G, BOt Sm, B+ 94s 10—2 148 COMPOSITION. Let the determinants of this matrix be called ee ale cla, Ua, and let Dia (atu eas Then we find by direct calculation P’=0P +ay(R—-8)+y7U = n' (aa + 2bay + cy?) =a"'n’; and similarly R’—S' = 20’, T=en. Again, Q’ = (ad — By) Q=(ad— By) a’n, and so R’ +S’ =2 (ad — By) b’n, T’ = (a6 — By) c'n.. Finally, (aq) + 99s) (Bo + 8q2) — (aGo + ¥92) (Bq + 89) = (ad — Bry) (92 — Gos) = (ad — By) Ann’; and similarly the other two corresponding expressions formed as in the last three of equations (©) reduce to 2(ad— By) Bnn’, (ad — By) Cnn’ respectively. : Now if ¢@ is properly equivalent to f, aé— By=1, and we conclude that #’ is compounded of ¢, /’ in the same way as it is compounded of f and /’. If, on the other hand, a6 — Py =-—1, so that @ is improperly equivalent to f, # is still compounded of d and /’, but » has to be taken with a different sign; that is, d@ has to be taken inversely or directly according as f was a directly or inversely. | It follows that in all problems of composition we may sub- stitute for any form, taken inversely, an improperly equivalent form, for example its opposite, taken directly. In future, unless the contrary is expressly stated, it will be supposed that all forms which are compounded are taken directly. : 141. It is now evident that we may speak of the composition of classes; namely, if f, f’ are any two forms belonging to classes Kk, K’, then any form /# compounded of them will belong to a otal, determinate class, which may be said to be compounded of K, K’ and denoted by KK’. ; ARNDTS METHOD. 149 It is clear that the symbols A’K, A’ mean the same thing, because, in compounding two forms, the process has been sym- metrical with respect to the components. A class can always be compounded with itself: this process is called duplication, and the class resulting from the duplication of K is denoted by A®. 142. The composition of forms may be treated in another, and in some respects a simpler way, which is due to F. Arndt’. Let f, 7, d, d’, m, m’ have the same meanings as before; let D=dv(dm?, d'm?), n=Vd/D, vn’ =NVd'/D. Then we see, as before, that mn’ and m‘n are integers and relatively prime. Now suppose that by definition PO wiih ie a 0 nt bn (eaten l= 6 Hie = 0 1 — On: then it follows from the definitions that P, Q, R, S, 7, U are all integers and relatively prime. Let » =dv(P, Q, &); then we can prove the following pro- positions :— (i) The integers ab’, a’b, bb’ + Dn’, are all divisible by pu. (Observe that Dnn’ =V dd’, and is therefore an integer, since, by supposition, d/d’ is a rational square.) We have ab’.mn' = b'mP, ab’. m'n = m’ (ak — bP), ab. m'n = bm’Q, ab.mnv' =m(a' Rh — b’Q), (bb' + Dan’) mn’ =m (UR —c'Q), (bb' + Dnn’) m'n = m' (bk — cP). Now all the expressions on the right-hand of these equations are divisible by w: and since mn’, m’n are prime to each other, the truth of the proposition is evident. (ii) In the next place, aa’ is divisible by pu’. Since aa .mn =a'mP, and aa’ .m'n = am'Q, we see that aa’ is divisible by w. Hence aa’P, aa’Q, aa’R are all divisible by p2 1 Crelle, t. 56, p. 64 (1859). 150 COMPOSITION. Again aa'S = ab’Q — w bP, aa T = ab’ R — (bb' + Dnn’) P, aw U =a’/bR — (bb' + Dn’) Q. Hence, and from (i), we infer that aa’S, aa’T7’, aa’U are all divisible by p?; and since P, Q, &, S, T, U have no common divisor, it follows that aa’ is divisible by p?. Gi) Let aa’/w?= A; then an integer B can be found so as to satisfy simultaneously the three congruences Lope pe pe @ B= he (mod A), p pb 7 BR pbb + Dan pe p- Choosing integers a, 8, y so that aP + BQ+ yh =p, let us put je ‘aah se then it may be verified that this value does in fact satisfy all the congruences. For we have (PB — ab’) = P {aba + wbB + (bb' + Dnn’) y} — pad’ = ab’ (uw — BQ — yh) — pad’ P {ab + (bb’ + Dnn’) y} = B (wbP — ab'Q} + ¥ ((b' + Dan’) P — ab’R}. | Substituting for P, Q, # their values, and writing (b? - a’) for Dn”, this reduces to {8 (bn' — b’n) — ye'n} aa that is, to —(BS + yT) aa’. Hence be (PB — ab’) = 0 (mod aa’) ; and therefore A pee = 0 (mod A), fo fe It may be similarly verified that the other congruences are satisfied: in fact (QB —a'b) = (aS +yU) aa’, {RB —(bb’ + Dnn’)} = (aT + BQ) aa’. T_T ARNDT’S METHOD. Tht Since P/u, Q/u, R/p are relative primes, it follows that all values of B which satisfy the three congruences simultaneously, are congruous (mod A). After some easy reduction, we find that pe? (B?— D) = aa’ fac’? + a’cB? + cc’? + 2b’cBy + 2be'ya + 2 (bb’ — Dunn’) af}, whence B?— D=0 (mod A). Putting a = (, the form (A, B, C) will be of determinant D, and it can be shewn to be compounded of (a, b, c) and ti 0; ¢). In fact, if we put X = pon’ + Bae Se aye BAD s ae : bb’ + Dun’ — B (bn’ + bn) + aa’ YY » Cie etna ke UNTO ; ea te amen all the coefficients of the substitution are integral, and we have identically ; (Gp 4 byt ny D) aie Le + ny (DY AX P BYE VifD, with a similar identity obtained by changing the sign of /D throughout. Multiplying these together, we obtain 1 yp) 2 Dee fa Ale, and therefore, since A= 0 | pe, P=ff; that is, is transformable into ff’. It is easily proved that the six determinants of the trans- formation are P,Q, R, S, 7, U: hence the substitution is primitive, and Fis compounded of f and /’. In every case, then, where composition is possible, we can compound (a, b, c), (a’, b’, c’) into (A, B, C), where Ai = a0, | p27, 152 COMPOSITION. an’ ab enya "B= aL (mod A), le fe b’n + bn’ pe bb’ + Dnn’ le fe C= (B— D)/A. Here D=davidm?, dine). AAA eaeiepank p=du(an’, an, b'n+ bn’). 143. Suppose that kK, Kk’, K” are three classes the deter- minants of which are in the proportion of three square numbers : and let f, f’, f” be any three forms belonging to them. Then by compounding f and f’ we get a form of the class KK’, and by compounding this with /”, we get a form belonging to the class (KK) Kk”. If we first compound f’ with f”, and then the result with f, we get a form of the class K (A’Kk’”). It will now be shewn that the two forms obtained by these different processes are equivalent, or, which is the same thing, that the classes (AK’) K” and K (K'K”) are identical. Let (a,b, c), .(@, U, 0), (a, b”, c’) be the forms, fi fay = d, d’, d” their determinants, and so on; the notation being as before, with the addition of corresponding symbols for /”. Then by Arndt’s process we may compound f, /’ into (4, B, C), or f' say, where A = aa’'/y?, etc. as above. The determinant of Ff is D=dv(dm?, d'm?); and M= dv (A, 2B, C) = mm’. By the same process let us compound F’ and /” into a form D(A al). ; Let A be the determinant of ®, Then A= "dy (Dine dae). Now since D=dv (dm am), Dm’? = dv (dm? m", dm’? m?) : also a Md ems therefore A =dv(dm?m"™, d’m’'?m?, d’m?nv?). Let us write v=Vd/A, v’=Vd'/A, v’=Vd"/A: these quantities are all rational: and moreover m’m’v, m’ mv’, mm'v” are all integers, and relatively prime. ASSOCIATIVE LAW OF COMPOSITION. 158 Further, let N=VD/A= y/n=v' |n’, We have A=Aq jp? =aa'a” |p”, where pw’ = dv (Av, a” N, Nb’ + vB). Now p= dv (an’, a’n, nb’ + n’b), and therefore wp’ divides the following integers: fia Ta CMTS OW GeV ape an.a’N =a'a’y, (nb’ + 0'b).a”N =a" (vb’ + vb). Again, since an’ B = ab’ (mod pA), therefore an’ (Nb” + vB) =a (v'b" 4+ vb’) (mod pAv’”). Hence we see that wp’ divides a(v’b” + vb’); and similarly it divides a’ (vb + vb”). Finally, since (b'n + bn’) B = bb! + Dn’ = bb’ + Avy’ (mod pA), we have (N 6" + vB) (b'n + bn’) = vb'b" + vb") + v"bb! + Avy’v” (mod pw Av”), so that wp’ divides yb'b" + v'b"D + v" bv! + Avy'v”. It is easy to see that the seven integers aa’v”, a’a’v, etc. have no common divisor greater than py’: hence putting py’ =o, we have t o =dvu[aa’p, a’ av’, aa’v”, a(v'b” + vb’), a (vb" + vb), a” (vb' + v’b), vb'b” + v'b’b + vv bb’ + Avy'’v”’). It will be observed that this determination of oc, like that of [ah is symmetrical with reference to f, f’, f”. Take X, Y to be the variables of F’, &, 7 those of ®: then the successive compositions give rise to the identities . g (cue + by + yl) (wat + Diy! + vy VA) =AX+BY+NYVA...... (i), and, |.) = (Ax + BY+ NYA) (a’a"” + b’y” + v"y"/A) ) SRE en ATR Gi) 154 COMPOSITION. Hence by substitution AE+ Bn +V/A = * (aa byt+vy/A)(a'a' +b'y'+y'y'/A)(a a" + b"y" +0" yA)... (111). Now it follows from (i1) that &, 7 are lineo-linear functions of ”y", X, Y with integral coefficients: and from (i) that X, Y are similar functions of a, y, #’, y’ with ig ee SSMU: hence &, » are trilinear functions of a, y, a’, y’, #’, y” with integral coefficients. Multiplying out the right-hand side of (411) and comparing with the left-hand ae we have apg d Ae | foot avy AE Wy ’ / aa v ey eae a a, ce wo LY + HEY a (v'b” + vb’ me y'y!" + a (v'b+vb") ,, a’ (vb’+y'b) ,, , ah a de ala at ae OY YOu: ei vb'b” + v'b"b + v"bb! + Avv'v” __,,, st a | YYYy > and it has in fact been proved that all the seven coefficients are integral and relatively prime. We have oA&E=(ax + by) (aa! + b’y’) (aa! + b”y”) | +A {yp / v'(an+by)y’y” +v'y(a'a' +b’y aa ‘yt vy (ae “4 by" \yy’} —oBn, and hence we conclude that B simultaneously satisfies the fol- lowing seven congruences :— Gi = py’ aa B= 0 aa’b” —v"aa’ B= 0 ab’b” + Aav'’y” — a (v'b" + vb’) B=0 a'b’b + Aa’ — a’ (vb + vb”) B= 0 abl’ + Aa’ — a” (vb' + vb) B= 0 | bb’b” ae A (by'v // - ‘yy -{- bv’) — (vb'b" + v'b"b + vbb' + Avr’y’”) B= 07 aa’b—vaa-B=0 (mod oA). Again since B is determinate (mod A), and since in the above congruences all the coefficients of B are divisible by a, it EXAMPLE. 155 / follows that the seven integers a’a’b, wab’, aa’b”, ab’b” + Aav’v”, ab"b + Aa'yy, a’bb’ + Aa’, bb'b" + A (bv'v” + b'y'y + bv’) are all divisible by o, and that ihe congruences may ae replaced by a ab vow’ —— —-_ 5 = 0i(mod_ A), and so on. Now suppose that by Arndt’s process we first compound /” and f’ into #” and then compound #” and f into (A’, B’, I”): and let a’, A’ be the quantities corresponding to a, A. Then it follows from symmetry that o’ =o and A’=A: hence A’ =A, and there- fore the congruences satisfied by B’ are the same as those satisfied by B: hence the forms (A’, B’, I”), (A, B, [) are equivalent, and we may in fact suppose them identical. This proves that (k’) kK” =K (4K’Kk”), and the symbolical notation for the composition of classes is fully justified, because the commutative and associative laws of multiplication are both satisfied'. Instead of (KK) K” or K (4K’K”) we may write without ambiguity KA’K” and call this the class compounded of 1G TOG ae As an example of the direct composition of three forms, let f=(2,1,2) f’=(41,7) f’=(, 0, 4). Here d=—3, d’=—27, d*=—192, RE TI NN a TE and hence A= 0, 1, y=3, y= 2. We find _ o=av 12,015, 16, 4,8, 12, —16) =o and therefore A=2.3,.4/27=6, The peesccoces to be satisfied by B are 6—6B=0 3—9B=0 8B=0 18+2B=0 (mod. 6), 12+4B=0 12+6B=0 12+8B=0 : whence B=3 (mod. 6). 1 The distributive law is not required, since such a symbol as K+ XK’ will not occur. Gauss writes K+ K’ instead of KK’: this, of course, is equally legitimate, but not quite so convenient or suggestive. 156 COMPOSITION. Taking B=3, we have so that the result of the composition is (6, 3, 2) as (2, 1, 2). This, of course, might have been foreseen, on finding A= — 3; for it is clear that the resultant class will be improperly primitive, and there is only one such class for A= — 3. If we wish to find the trilinear substitution which transforms (A, B, I) or (6, 3, 2) into ff’f", we put b (20-+y + yn) —8) (4a +y/ +3y's/ - 8) (Bx"” +2" —8) =67+3y+y/— 3, and hence by expanding, and comparing both sides, = 6a! a'"y + 9x" va! + 8xa'y" + 2axy'y" + 4a'y"y + 62"yy' — 8yy'y", AX = 2x0'x" —2Qa'x"y — 420" ny! — 40a'y" — Aay'y! — 4alyy — 52x" yy + 2yy'y". 144, It is now evident that if fi, fi...fn are any number of forms whose determinants are proportional to n square numbers, and if Ky, K....4, are the classes to which they belong, then it is possible to find a form compounded of f,, /...f3 and in whatever order the forms are compounded, the resulting form will belong to one and the same class, whose determinant N= dy (dining in Oa, Matt, Oil, Nh rena taken with the same sign as that of d,,d,...d,. This class may therefore be denoted without ambiguity by K,A,...K,. The classes 4 need not be all different: thus we may have such compositions as are denoted by 4,3, K4K,?, and so on. 145. Consider more particularly the composition of two forms If of the same determinant and for which m, m’ are relatively prime. Then if D be the determinant of the form F'= (A, B, C) compounded of them, we have D=d=d,n=n=1, and p=dv(a, a',b+0%. Putting, as before, A = aa’/y?, B is determined by the congruences @ pw Bo Bh Ligne (mod Z~) (KL)=K— K .I~L=1: therefore KL =(KL)+: and so for any number of classes. In particular, the class compounded of any number of am- biguous classes is itself ambiguous. 146. We are now able to compare the numbers of classes belonging to the different orders of a given determinant D. Suppose D is divisible by a square number m?, and let D = Am’. Then there will be an order © derived from the properly primitive order of determinant A. As the simplest representative form of this order we may take (m,0,—mA). If A =1 (mod 4) there will also be an order / derived from an improperly primitive order of determinant A: and we may take for its simplest representative form (2m, m, 4m (1 — A)). Now let f= (a, b, c) be any properly primitive form of deter- minant A: and let us suppose, as we may do (see Art. 127) that a is prime to 2mA: then the form (a, bm, cm?) is properly primitive, and it is easily verified that (ma, mb, mc) 1s compounded of (a, bm, cm?) and (m, 0, — mA). Similarly every class of the order ©’ will contain a form (2ma, mb, 2mc) in which a is prime to 2mA: and this form is compounded of (2m, m, 4m(1—A)) and (a, mb, 4m’c), the latter of which 1s properly primitive. We conclude therefore that every class of a derived order may be obtained by compounding the simplest class of that order with a properly primitive class. Again, every class of the order may be obtained by com- pounding any assigned class of the order with a properly primitive class. or let ® be the assigned class, ®’ any other class, ®, the simplest class, 4, Z properly primitive classes which compounded with ®, give ®, ®’ respectively. Then PB=®O,K, DP’ =0,L=OLK: that 1s,’ is obtained from ® by compounding it with the properly primitive class LA. Now let @,, ®,...@, be all the classes belonging to a given derived order: and let F,, F,...F, be all the properly primitive classes of the same determinant. Then the classes /,®,, F,®,.../,®, DERIVED AND PRIMITIVE ORDERS COMPARED. 159 include all the classes ®; therefore v cannot exceed n. It can easily be shown that » is a multiple of »v. In fact, if n>v, some two or more of the classes F,®,, F,®,...Ff,,82, must be identical. Suppose FD, = FD, =)... = £2). Let FY be any properly primitive class not contained in F,, F,...F,, poumlict nh = iF so that fy = KF,; then KF,, KSI,...K by, will be properly primitive classes different from each other and from the preceding set; and putting KF; = F/, we have eR SD eli pean CD Proceeding in this way, we see (as in a similar case, Art. 18) that £,®,, P,P,...f,P, may be arranged in groups each containing k identical classes, the classes of any two groups being different ; and therefore n= vk, that is, n is a multiple of v. 147. It is evident that & is equal to the number of properly primitive classes which, compounded with the simplest (or any other) class of the derived order, reproduce that class: and the problem of determining & was considered from this point of view by Gauss’, who did not, however, succeed in obtaining a complete solution. The following investigation, which depends on the theory of transformation, is due to Lipschitz’. We consider in the first place two determinants D and D’ = Dp’, where p is a prime. Let f= (a, b, c) be any properly primitive form of determinant D, and let us suppose that ais prime to p. Then applying to f the substitution PC, 3) y, 6 where ad — By =p, or, say, a substitution of order p, we obtain j= Ef —(a;, b;, ¢ ), suppose; a form of determinant Dp*. Suppose that all possible substitutions P are applied to f, and let us examine how the resulting forms may be classified. 1 D, A. Arts, 253—6, 2 Crelle liii, (1857), p. 238. 160 COMPOSITION. Let U = Go ; a where a’é’ — B’y’ = 1, be any unitary substitu- tion: then Pom (CY 3) “Gara oy, gatas) Pe is a substitution of order p. It is clear that the forms Pf and Pf are equivalent, the first being transformed into the second by the substitution U. The substitutions P, P’ may be called equivalent: and we ead only consider the non-equivalent P-sub- stitutions. Since a6 — By =p, and p is prime, it is evident that du(y, 6) =1 or 7p. First, let du (y, 8)=1: then putting a’ =6, y’ =—y, and deter- mining f’, & so that a’&’ — B’y’ =y8’ + 68’ = 1, the substitution PU or P’ becomes & ne Moreover the general values of 8’, 6’ are of the form 8, +6, 6 —ky, where k is any integer: hence h=aB, + 8d, +k (ad — By) =k +hkp: we may therefore suppose that h is replaced by its least positive residue (mod p). Similarly, if dv (y, 8)=p, we may put a’ =4d/p, y’ =— y/p, and then determine £’, 6’ so that a’6’— B’y’=1, and PU= bs ») : Hence every substitution of order p is equivalent to one of the following (p + 1) representative substitutions : 1, °) (0 pl’ ? h a | if ) (h=0, 1, 2...p—1). 148. We will now apply these (p +1) substitutions to the properly primitive form (a, b, c) of determinant D. First let p be an odd prime. Then (2 ta, b, c) = (a, bp, cp), and this is properly primitive, 0, p since @ is prime to p. Again ( 2 a, b,c) =(a, Bc’), where a = 1p b’ = (ah + b) p, ce =ah?+ 2bh4c. DERIVED AND PRIMITIVE ORDERS COMPARED. 161 From these equations we deduce pae=a, pb = pb’ — ha’, pe = pc’ — 2phb' — ha’, and since dv (a, 2b, c) = 1, we infer that dg (G5.20 2c \— lp orp Now ac’ =(ah+b)?—D: and therefore if » divides c’ it will be possible to choose h so that (ah + bY? — D=0 (mod p), and since a is prime to p the converse is true. There are four cases to consider :— (i) D divisible by p?. The congruence in h has one solution, given by ah+b=0 (mod p): this value of h makes b’ and ¢’ divisible by p?: so that in all there are p properly primitive forms (a’, b’, c’) and one form for which du(a’, 2b’, c’) = p*. (1) WD divisible by p but not by p. As before, the congruence has one root, and for the corresponding form, /”, du (a’, 20’, c’) = p. The remaining p forms f/’ are properly primitive. (ui) D not divisible by p, and (D\p)=1. The congruence has two roots given by ah+b+/D=0 (mod p): there are two forms f” for which dv (a, 20’, c’)=p, and (p—1) properly primitive forms. | (iv) D not divisible by p, and (D|p)=—1. The congruence is insoluble, and all the (p +1) forms /’ are properly primitive. In every case the number of properly primitive forms /” may be expressed by p—(D\p), with the convention that (D|p)=0 when D=0 (mod p). Next, let p = 2. Here G oe b, c) = (a, 2b, 4c), which is properly primitive, since @ 18 sagaeess odd. Again iF ita b, c) is properly primitive except when (ah + b? —D=0 (mod 2). M. 11 162 COMPOSITION. If D is odd the congruence has one root given by ah+b=1 (mod 2), and if D is even, there is again one solution given by ah +b=0 (mod 2). In the former case, dv (a’, b’, c’) = 2, and dv (a’, 2b’, c’) = 2 or 4 according as D=83 or 1 (mod 4): in the latter, dv (a’, ’, c') = dv (a’, 20’, c’) = 2 or 4 according as D = 2 or 0 (mod 4). In every case there are two properly primitive forms. 149. Every properly primitive class of determinant Dp? can be derived from a properly primitive class of determinant D by means of one of the (p+ 1) representative substitutions of order p. For let (A, B, C) be any properly primitive form of determinant Dy, in which A is prime to p. Apply to it all possible substitu- tions of order p: then the resulting forms of determinant Dp‘ fall into (p +1) sets, the forms of each set being equivalent, and those of one set, and only one, having a divisor p?. Suppose & Nee B, C) = (pra, p°b, p’c), where ad—Py=p; then (a, b, c) is a properly primitive form of determinant D: and it is easily verified that 5, 2 ae “ (9 Efe 0) =(4,B, 0); that is, (A, B, C) is derivable from (a, b, c) by the substitution. an aoe, Y a perfectly determinate. For if (fe i ie! v', o') = ( » BY a,b, 0) je Moreover the class to which (a, b, c) belongs is = (A, 3B C), then & YA. B, C) =(p'a, p°b, p’e), Ante & tA, B, 0) = (p?0’, p°0', pe’): but all the forms of divisor p? derivable from (A, B, C) belong to the same class: therefore (p*a’, p*b’, p’c’) ~ (pa, pb, p’c) and consequently (a’, b’, c’) ~ (a, b, c). DERIVED AND PRIMITIVE ORDERS COMPARED. 163 150. If now f,, f:...f, are representatives of the n properly primitive classes of determinant D, the first coefficient of each form being prime to p, and if we apply to each of them the (p +1) reduced substitutions of order p, we obtain altogether n {p—(D|p)} properly primitive forms of determinant Dp?, among which will be found representatives of all the properly primitive classes. It remains for us to discover how many of these are equivalent. It follows from the last paragraph, and from the fact that fi, fi...fn all belong to different classes, that any two such equivalent forms must be derived from the same form f/ Suppose then that & AG bc) (4B, GC), Lip aoe ma A @ wha b, Cc) ars (A V B ) C"), where a6 — By = a0’ — B’y' =>, and (A’, BY, C’)~ (A, B, C). If the unitary substitution & 4 ) transforms (A’, Bb’, C’) into (A, B, C), then al’, at H) G OAV, p changes (a, b, c) into (A, B, C) and must therefore be of the form T,—bU;, —cU; a, 4 ( aU;, T+bU Sy, 8)” where (7, U;) is an integral solution of 7?— DU?=1. Operating on both substitutions with ( ° ns ) , we get ee PPD (2 pe) _( %, PNT: —bU;, , —clya, 8) PY, pp —7’, a’ aU;, T,+0bU; Y; 6 ieee A) ee =(p Ala so that the conditions for equivalence are expressed by A=B=Tr=A=0 (mod p). Since & a and (a i) may be taken from the reduced sub- | es _ stitutions of order p, there are only two distinct cases to consider. 11—2 d 164 COMPOSITION. First, let (* A ka ( p ys ie 4 =(¥ ah then 4t. wlan found by actual calculation that A =p? (T; — bU;), B=p {h(T;—bU;) —cU;}, De pal. A= T+ (ah +b)U;j,. The single condition necessary for equivalence is therefore T;+(ah+b)U;=0 (mod p). Since ais prime to p, this gives one determinate value for h, such that 0 — DERIVED AND PRIMITIVE ORDERS COMPARED. 165 In a similar way we find that Besta b,c)oo (Br Gebeac): if (h—h’') T; — pe ee U;=0 (mod p), h(T, —bU;) - = inate cage oe, "(mod p). Let h; be the least positive value of h’ derived from this congruence. It is clear that the value of h;’ is determinate except when there is a solution of the Pellian equation such that haU;,+(7;+bU;)=90 (mod p), mde 0, 1 We also find that that is, if h’= in which case ( (a, b,c) ~ (i RG b,c) by the preceding case. (ah? + 2bh + c) U;_; anda thal; + (1, +6U,) (ha; + (Z; + 0U;)| PY , DON aes h; —h; = If ah? + 2bh +c = 0 (mod p), (R f a, b,c) is not properly primi- tive; rejecting these cases, when they exist, we see that h;’ = h;’ if and only if 7 =7 (mod co). As before, we have a set of o equivalent forms, viz. these are _ derived from (a, b, c) by the substitutions ( ) +3) x Hs) & a i Op Ee? Oe hgh (where observe h,’ = h), or else by the substitutions B “ @ Nhe @ ne i 2 é Hes) a ( ; a) ip dl cen U)) ee Os el Oval according as / does not, or does, satisfy a congruence of the form haU;+(T;+6U;)=0 (mod p). Since the same reasoning applies whichever form (a, b, c) be taken, we conclude that the total number of properly primitive classes of determinant Dp? is ” p—(D|py="2 [1-5 Dip) _o having the meaning above explained, and (D |p) being put equal to 0 if D is divisible by p, or if p = 2. 166 COMPOSITION. 151. By successive applications of this result it 1s easy to conclude that if D’= D§S?, and n, n’ are the numbers of properly primitive classes of determinants D, D’ respectively, , 2s it : w= "8 {1 (Dip). where the product relates to all odd primes p which divide S, but not D, and where o is the index of the first imtegral solution (T,, U,) of J2?—DU?=1 for which U, is divisible by S. The result may also be expressed in the form log (Z'+ Ur/D) | 1 . 7 7 ? Il I Te D ? log (7" + U'/D’) as |P) where (7, U), (T", U’) are the fundamental solutions of 7? — DU? = Nhe and 7" — D'U"=1 respectively. In this form it was originally obtained by Dirichlet, although by means of a very different method, which will be explained later on. v=nS When D’, and therefore D, is negative, the Pellian equation has in general only two solutions, viz. 7=+1, U=0: it is easily seen from the foregoing analysis that all the substitutions of order p give non-equivalent forms, and that n = nSIl {1 — : (D\p)} If, however, D=—1, the equation 7?-DU?=1 has four solutions, 7=+1, U=0, and 7=0, U=+1; the derived forms may be grouped into equivalent pairs, and | — 1)3(2—>) n' =4n8. II \! ~ 1 (a, b, 4c) into (4a, b, c). Changing the sign of U,, put Oy == EU es = Cle y= —auy,, 6, = 4(7, — bU,), then 6, is half an odd integer: a, is divisible by 4, 8,, y. are odd. IMPROPERLY PRIMITIVE CLASSES. 169 Hence (a. +282), $(%2 + 26.) are integers, and it may be verified that (ie b(a,+ ey Yo, ¥(Y2 + 20s) is an integral unitary substitution, which transforms (a, b, 4c) ito (4a, 2a+b, a+b+¢). Expressed in terms of a,, 81, 71, 5, this substitution is (ee 26, — P, | =, (1 — 2y1)/" We conclude, then, that if U, is even, no two of the forms $1, d2, 3 are equivalent, and that if U, is odd, they all belong to the same class. When D is negative, the equation 7? — DU? =4 has in general only two solutions 7 = +2, U=0; the only exception is when D=-—3, in which case there are six solutions 7=+1, U=+1, T=+2,U=0. Thus when D=-—83, ¢,, do, d; are all equivalent: for all other negative determinants they belong to three different classes. The results thus obtained are exhibited in the following table, where n, n’ denote the number of properly and improperly primi- tive classes respectively, and (7;, U,) is the fundamental solution of 7?— DU? =4. I. D positive. D=1 (mod 8), 1 ==, DP=5 (mod 8), Ujodd: n =n, D=5 (mod 8), U, even: n'=3n. II. D negative. As an illustration suppose D=21, ¢=(6, 3, —2). The asso- ciated forms are pi = (3, 3, mi 4), pr oy (12, 3, = 1), 3 (12, 9, 5). 170 COMPOSITION. We have 7,=5, U, =1, and calculating a, 8, y,, 6, it will be found that 21). ij & (3 pena) =D) 3081). Seo ws (u sie 43 3, — 4) =(12, 9, 5), and hence n’ = n, as it should be. It may be inferred from the foregoing, and can be proved independently, that when U, is odd, U, is also odd, and U; even: in fact, the solutions (7, U;) fall into three sets corresponding to the triple grouping of the properly primitive with reference to the improperly primitive classes. Conyposition of Genera. 154. Let K, Kk’ be two properly primitive classes of deter- minant D, and let n,n’ be two numbers prime to each other and to 2D representable by forms of the classes A, K’ respectively ; then it follows from the theory of the composition of classes that nn’ is representable by a form of the class KK’ which is com- pounded of K and Kk’. The generic character of KK" may therefore be inferred from nn‘; and it is easily seen that any particular character relating to A’ may be obtained by multi- plying together the corresponding characters relating to K and A’. For the quadratic characters this is obvious since (nn | p) = (n|p) (n'|p). With regard to the supplementary characters y, Ww, we have (n —1)(n'—1)=0 (mod 4), so that (n—1)+(r’—1)= (mn'—1) (mod 4), therefore d(n—1)+4(7’' —1)=4(mn'—- 1) (mod 2), and hence y (K) y (BK) = (KK): again - (v? —1)(n2—1)=0 (mod 64), whence t(n?n? — 1) =1(r? —1) + 4(m?—1) (mod 8), and therefore ab KB’) = Kk) wv (Kr). The genus to which AK’ belongs is said to be compounded of the genera which contain K, K’; and the genus compounded of NUMBER OF AMBIGUOUS CLASSES. Tal the genera I’, A may be represented by TA. The composition of any genus with itself gives the principal genus, ie. that which contains the principal class. If one of the classes 4’, 1s improperly and the other properly primitive, KK’ is improperly primitive; and it is easily seen that, as before, its particular characters are obtained by multiplying together those of K and K’. If both K and K’ are improperly primitive, AK’ is the double of a properly primitive class, and the characters of this properly primitive class are obtained by multiplying together those of K and Kk’. For if 2n, 2n’ are two numbers representable by A and kK’ respectively, n, n’ being prime to each other and to 2D, the particular characters of K, KA’ are inferred from m and n’: and if KK’ = 20, the characters of Z are inferred from nn’, since 2nn’ is representable by KK’, and therefore nn’ by L. 155. Hach genus of the same order contains the same number of classes. Let IT, IY be any two genera of the same order, and let I contain the classes K,, K,...K,. Suppose KK’ to be any class contained in I’, and let P be a properly primitive class such that PK,=K’'. Then the classes PK,, PK,...PK, will all be different and will all belong to I”, since their total characters are the same. Hence I” contains at least as many classes as I. In the same way [' contains at least as many as IY. Therefore they contain the same number of classes. Number of Ambiguous Classes. 156. The following is Gauss’s investigation of the number of properly primitive ambiguous classes for a given determinant D. In order to avoid a trivial exception, it will be supposed that D is not equal to —1. This case is immediately disposed of by observing that, when D=-—1, there is only one class and this is ambiguous, It will be remembered that a form (a, b, c) is ambiguous if 2b=0 (mod a); also that (a, b’, c’) ~ (a, b, c) if b'=b (mod a), the determinant being the same in both cases. It is therefore only necessary to consider forms of the types (a, 0, c) and (28, b, c). iL g COMPOSITION. We obtain a properly primitive form (a, 0, c) by resolving D into any two factors which are prime to each other, and taking one of them, with either sign, for a. Each resolution of D thus gives rise to four ambiguous forms: but since (c, 0, a) « (a, 0, c), and we wish to find the number of ambiguous classes, we may reject two of the four, retaining those for which |a|<|c|. If n is the number of different primes which divide D, the number of forms which we obtain in this way is 2”. For example if D = — 90 = — 2.3.5, the forms are (+1,.0, +90), (4.2, 0, + 46),0)(4.5;0,418)) (£.9)0)- 2 10)e that is 8 or 2? in all. A properly primitive form (20, 6, ¢) is obtained by taking for b any (positive or negative) divisor of D such that c = (b? — D)/2b is an integer prime to 2b. Now c being odd, c=1 (mod 8), and therefore D=(? — 2bec =(b —cP —?=3 (mod 4) or =0 (mod 8), according as b is odd or even. Hence D must be of the type 8n, 8n+ 3, or 8n + 7 if there are to be any forms of the kind now considered. First suppose D=3 (mod 4). Then if we take for b any divisor of D, b is necessarily odd, b?-— D=2 (mod 4) =0 (mod b), so that ¢ 1s certainly integral and odd: also 2c = b — D/6, so that cis prime to 6 if, and only if, D/b is prime to 6. Since b may be taken positively or negatively, we thus obtain 2.2”= 2+! forms, n having the same meaning as before. Next let D=0 (mod 8), so that 6 must be even. Then since 2c=b—D/b, we see that by dividing D into any two even factors b, D/b which have no common divisor except 2, ¢ will be odd and prime to $b, and therefore also to 2b. In this way we get alto- gether 2”*1 forms, allowing for variation of sign in 0. For instance let D=120=8.3.5: the forms are (+4, +2, $29), (48, £4,413), (412,46, +7), (424, + 12, +1), (+20, +10, 1), (440,420, +7), (460, +30, +13), (+120, + 60, + 29); in all 16 = 24. | In every case the forms (20, 6, c) may be arranged in pairs such as (20,07) men ci) where b’ = 2c —b=— D/b. Se le NUMBER OF AMBIGUOUS CLASSES. eS Observing that 0b -+b’=2c=0 (mod c), we have (20, b, c) ~ (ce, b’, 2b’) ~ (20’, — U’, c) o> (20, B’, ¢), so that we need only retain that form which has the smalle middle coefficient. Thus we have left 2” forms (2b, b,c), for which |b] <|/D\. Now let ~ denote the number of odd primes which divide D, so that n =p or w+1 according as D is odd or even; then the total number, JV, of properly primitive ambiguous forms which we have retained (including both types when they exist) will be according to the following table. D=0 (mod 8), NG oe 2 eee » = 4 (mod 8), eee Pw | ee ere »” = 5 (mod 4), » = ) f Qn — PAs ae ee 2 (mod 4), Site pi = yeep » = 1 (mod 4), »= Qn =— Dr. Comparing this result with the scheme of generic characters (Art. 130) it appears that in every case NV is equal to the number of assignable total characters for the determinant D. 157. In the case when D is negative, the forms retained will be half positive and half negative. We now reduce the system to one-half by rejecting all the negative forms. The 4 positive forms which remain all belong to distinct classes. For every form (a, 0, c) is reduced, because ac, in which case the equivalent form (c, c— b, c) is reduced, because 2b >c gives c>2(c—b), where observe that c—b is positive, because b (which is positive) <— D/b< 2c —b, and therefore 2(c — b) >0. No two of the reduced forms (a, 0, c), (20, b, ¢), (c, e—b, c) can be opposite or identical; so that, finally, the number of positive properly primitive ambiguous classes is 4N. Next, suppose that D is positive. Let (a, b,c) be any one of the NV forms which have been retained. Choose b’ so that b'=b (mod a), and 0/D, the form (a, b, c) must be of the type (20, b, c), where |b|<./D. Putting b’=|b| we have b’=b (mod 2b) while 0<./D—b’p: then H**=1, or H/=1 where f is a positive integer not greater than-y. Let f now denote the least positive integer such that H/=1: then f may be called the exponent to which the class H appertains: and the classes 1, H, H?...Hi™ may be said to form a period. Exactly as in the analogous theory of residues to a prime modulus (see Art. 18), it may be proved that f divides v, so that every class of the principal genus satisfies the symbolical equation bah Again, the period of H™ will contain f/d terms, where d=dv(m, f): in particular, if f is prime to m, it will contain fterms. If fis prime, the period of H™ will contain /f terms, for all values of m. 162. Ifthe principal genus contains a class G appertaining to the exponent v, the determinant is said to be regular. In this case, the period of G comprises all the classes of the principal genus, and the class G enjoys properties similar to those of a primitive root of a prime modulus. For instance, taking G as a ‘base,’ any class of the principal genus may be specified by its index: there will be ¢(v) classes such as G, any one of which may be taken for a base, etc. etc. When the determinant is regular, the principal genus contains one or two ambiguous classes, according as v is odd or even. For if G" is ambiguous, G?”=1 and therefore 2m=0 (mod v): hence IRREGULAR DETERMINANTS. 179 if vy is odd, the principal class is the only ambiguous class in the genus, while if vy is even, G”? is also ambiguous. If, then, the principal genus contains more than two ambiguous classes, the determinant is certainly irregular. This consideration enables us to discover irregular determinants. For instance, the three primes 3, 13, 61 are such that each is a quadratic residue of each of the other two; and the determinant — 2379 =—3.13.61 is irregular, because the principal genus contains, besides the principal class, the three ambiguous classes (8, 0, 793), (39, 0, 61), (18, 0, 183). 163. It may also be shewn that every negative determinant of the form —(8km+3)m?, when k is any positive integer (not zero), and m any odd positive integer greater than 1, is irregular. For let f, = (m?, m, 8km + 4), fo = (4m?, m, 2km +1), and let K,, K, be the classes to which f, and f, belong. Then K, and K, are properly primitive classes of the principal genus, and it can be shewn by Arndt’s formule of composition (Art. 142 above), that woh, and Ke = KK. whence also kK, = Ke = 1. Moreover f, and f, belong to different classes; for either f, or the adjacent form (8km + 4, —m, m*) is reduced; and similarly either 7, or its equivalent (2km + 1, —m, 4m?) is reduced; the two reduced forms are in no case equivalent, and neither of them is the principal form. We thus obtain three distinct periods of three terms, 4, tie iy) re Ky, JeG Ny (dig dee, he): But, in the case of a regular determinant, a period of three terms can only occur when p is divisible by 3; and even when this is the case there is only one such period, namely (1, G’*, G”*) where G is a base of the period of principal classes. The determinant considered is therefore irregular. We see, then, that an infinite number of irregular determinants may be found; it should be observed, however, that we cannot obtain, in either of the ways just explained, all the irregular deter- minants which exist: thus there are prime irregular determinants, such as — 307. It does not appear to be possible to find a general formula which will apply to all cases. 164. Returning to the case of a regular determinant, suppose that G is any class whose period comprises all the classes of the 12—2 180 COMPOSITION. principal genus: then, by the theorem which, for the present, we are anticipating, there is a class H such that G= H?. If v is odd, we may evidently put H=G*"*”, and in fact this proves the theorem about duplication for this special case. The principal genus (1, G,G?,...G’) contains only one ambiguous class: and if A,, A,...Ay,_, are the other ambiguous classes, the remaining genera will be (AieAIG ALG AG): (A. Aneta Ge) cee Ay GAs Gea). Next suppose that v is even. In this case H cannot belong to the principal genus; for if it did, we should have H=G™ and G?"— = 1, which is impossible. There will therefore be two asso- ciated genera, the principal genus _ (ee eel ms), and (CHP SLi ays If C is a class of any other existing genus, all the classes of that genus will be given by (CRCHe CHSC He.) and there will be an associated genus (CH, Cire CH=): It is also evident that every genus contains either two ambiguous classes or none: viz. if any genus contains an ambiguous class A, it will also contain AH” and no others. Half of the existing genera will contain no ambiguous classes. We may, if we please, adopt a similar arrangement when » is odd, by putting H=AG?"*”, A being any ambiguous class, not the principal class. Thus in all cases we have a class H with 2p terms in its period; and the genera arranged in corresponding pairs (A AT Ad one CAL) a) ee deed) where A is ambiguous: each ambiguous genus being associated with one that 1s non-ambiguous, or ambiguous, according as v is even or odd. 165. In illustration, we give the complete tabulation of the primitive classes for D=—365. There are four genera, each containing five classes: so that both modes of arrangement ane possible. EXAMPLE OF TABULATION. 181 Characters Comp. . (5) (75) . (i). (ii). I. ab 0, 365) + ++ 1 1 ( 6, taeeeoL) oe ey (9, —2, 41) Ho Or cae 2, 41) pO Ge (26 — Ly OL} GES Tike II. rae 3, 34) + —-— f ag (15, Dee 0) [Po abe (10, 5, 39) Pe th (15, —5, 26) ee. (11, -—3, 34) ef mane IIT. a TLS) —- + -a Oy (ta; Ty 122) af? dog GUS, inf 20) Of ed, 0" (18, -—7, 28) Cf Saag. (Fone — lst 22) afPL Ong: ig (17, -—3, 22) —-— — +af ag’ (13, -—5, 30) Ey mesg: Ge, oo S88, af > ds Cos) af’ asg oo BU 3) 9 22) af Ps den”: 166. When the determinant is irregular, the classes of the principal genus cannot be arranged in one period; it will be possible, however, to assign a certain number of bases such that all the classes of the genus are expressible by powers and products of powers of these bases: and any particular class may then be specified by means of two or more indices. Suppose that g is the highest exponent to which any class of the principal genus appertains: then the exponent belonging to any class of the genus will divide g. This is proved by shewing (compare Art. 20) that if H, K appertain to the exponents e, f, HK will appertain to the exponent pw, where yw is the least common multiple of e and f. Again g divides v, the number of classes in the genus. For if 182 COMPOSITION. 1, G, G...GI— be any period of g terms, then the classes of the genus may be arranged in sets such as CU, CG, CG*...CGI™, no two of which have a class in common. The quotient v/g is called by Gauss the exponent of irregu- larity. In Gauss’s tables relating to definite forms, which include more than 4,900 negative determinants, the only exponents of irregularity which occur are 2 and 3, with one exception, D= 11907, for which it is 9. According to Pepin, the exponent is also 9 for D=-— 6075. For further details, especially with regard to the choice of bases for irregular determinants, and the arrangement of classes not contained in the principal genus, the reader is referred to Smith’s Report, Arts. 118, 119. AUTHORITIES. Gauss: D. A. Arts. 234—265. Démonstration de quelques théoremes concernant les périodes des classes des formes binaires du second degré (Werke, ii. 266). Tafel der Anzahl der Classen binarer quadratischer Formen (Werke, 11. 449), with notes and corrections by Schering (ibid. p. 521). . SmitH, H. J. 8. Report on the Theory of Numbers, Part 1v. (Report of British Ass. 1862). DiricHLEt-DEDEKIND : Vorlesungen iiber Zahlentheorie (3rd edition 1879) Supplement x. See also Dirichlet: De formarum binariarum secundi gradus — compositione (Crelle, xlvii. (1854) p. 155). ARNDT, F.: Aufldsung einer Aufgabe in der Composition der quadratischen . Formen (Crelle, lvi. (1859) p. 64). Veber die Anzahl der Genera der quadratischen Formen (ibid. p. 72). LipscuHitz, R.: Hinige Sdtze aus der Theorie der quadratischen Formen (Crelle, lii. (1857) p. 238). SCHERING, E.: Die Fundamental-Classen der zusammensetzbaren arithme- tischen Formen (Gottingen 1869). Also in vol. xiv. of the Abhandl. d. konigl. Gesellsch. d. Wiss. zu Gottingen. Poincaré, H.: Sur un mode nouveau de représentation géométrique des formes quadratiques définies ou indéfinies. (Journ. de VEcole Polytechnique, cah. 47, t. xxvill. (1880) p. 177.) See especially pp. 226 and following. The theory of composition, as now understood, is principally due to Gauss. Some special cases of composition were discovered previously, such as (a 49) (al? +.y'2) = (ae! — yy")? + (ay -a'yy?, with the analogous theorem of Euler’s for the product of sums of four squares AUTHORITIES. 1838 The most interesting of these earlier results will be found in Lagrange’s memoir Sur la solution des probléemes indétermines du second degré (Hist. de ?Acad. de Berlin 1767) § vi. Lagrange demonstrates a general theorem, which is equivalent to what would now be called the duplication of the form II (4, +%,0+%,67+...+2,0"), where #,, %y,...2, are the variables, and the product sign extends over all values of 6 for which 6"— A=0, A being any integer. See also the additions to Euler’s Algebra, § 9. In Vol. x. of Crelle’s Journal (1862) p. 357 there is a table, calculated by Cayley, which gives the complete classification of primitive binary quadratic forms for negative determinants from D=-—1 to D=-—100, and for positive determinants, other than squares, from D=2 to D=99. The generic charac- ters are also given, and the composition of the different classes is indicated by symbols as on p. 181 above. The table also contains the periods of reduced forms for the positive determinants. There is a supplementary table relating to the thirteen irregular negative determinants which are numerically less than 1000. CHAPTER Vie Cyclotomy. 167. THE theory of indices and power-residues cannot have failed to remind the reader of the binomial equation w” —1 = 0, upon which depends the division of the period of the circular functions. The analogy is by no means accidental, and there are, in fact, many arithmetical results which are most simply expressed, and most easily proved, with the aid of the complex roots of unity; moreover, as Gauss first pointed out, the problem of cyclotomy, or division of the circle into a number of equal parts, depends in a very remarkable way upon arithmetical considerations. We have here the earliest and simplest example of those relations of the theory of numbers to transcendental analysis, and even to pure | geometry, which so often unexpectedly present themselves, and which, at first sight, are so mysterious. 168. The equation #”—1=0 has m roots, of which only one is real, namely 1. Expressed in terms of circular functions, the roots are given by Wikge 2 ADS ; Ly, = CoS —— +28M — = e2nt/m m m (h=0, 1, 2...m— 1). From this, and de Moivre’s theorem, the propositions which immediately follow may be deduced without any difficulty; in order, however, to preserve, as far as possible, the analogy with binomial congruences, they will be proved independently. If a is any root of «™” —t = 0, and e is any integer, a is also a root, because (a°)™ = (a™)* = 1¢= 1. Let f be the least positive integral exponent (not zero) such that af=1. Since a” =1, it is clear that such an exponent must PRIMITIVE ROOTS OF UNITY. 185 exist, and cannot exceed m. It is called the exponent to which a appertains. ‘The root 1 appertains to the exponent 1, and is the only one of the kind. The quantities 1, a, a’,...a/, where a, fhave the same meanings as before, are all roots of a” —1=0. Moreover they are all different, because if we had a =a? with p< f and q JP ie where a+ #6 < 1, and e, fe’, f are integers prime to p. Now the coefficient of 2”-*-* in the product of f (x) and (x) is equal to | by, a bncr =: Oras Ckh-i + Dra Cp eg as + bra Cha + bn» Cig b+. The first term of this expression is ee’//f’p*t*, and the denomi- nators of the other terms cannot involve p to so high a power as p**8; hence the sum of all the terms must be of the form P + Qp “Rpet 188 | CYCLOTOMY. where P, A are integers prime to p, and y is a positive integer at least equal to 1. This expression is necessarily a fraction, because p is a factor of the denominator, and the numerator is prime to p. This is inconsistent with the assumption that the coefficients of fo or F are all integral; and the lemma is therefore proved. Hisenstein’s proof of the irreducibility of X depends on the following proposition :— If p ws a prime number, the polynomial f (a) vs vrreducible of the coefficient of the highest power of x rs unity, the absolute term +p, and all the other coefficients divisible by p. It is clear that if f(#) can be resolved into the product of two polynomials with rational, and therefore integral coefficients, we must have f(a) =(e" + ba +. +b, et 1) ("4+ Ge" +... ton & +p) The coefficient of # on the right hand is ay Cn—1 EDO gas and since this is divisible by p, we have c,,=0 (mod p). The coefficient of 2? 1s Filo Cn a0 meat 0 ea. and from this we infer that c,_,=0 (mod p). Proceeding in this way, we find that all the coefficients ¢, c.,...¢,-, are divisible by p; hence the coefficient of « on the right-hand side of the assumed identity is +14 60mA + Cobm—.+ ... = +1 (mod p); but this contradicts the hypothesis that it is divisible by p. The | assumed resolution into factors is therefore impossible. To apply this to the polynomial X, we first transform it by changing « into w+ 1; it is obvious that if the new polynomial X’ is irreducible, so also is X. Now the value of X’ is | CY Ee A ey cree sy ANE = ID) me Ga Vai Sle aM Fontes 5} LRT hag ete ion wi Pb the coefficients being those of the binomial expansion (1 + w)?. All the coefficients except the first are divisible by p, so that X’ satisfies the conditions laid down in the preceding proposition ; hence X’, and therefore also X, is irreducible. SIMPLEST FORM OF FUNCTIONS OF THE ROOTS. 189 172. Let r be any root of the equation X =0; then every rational integral function of r with rational coefficients may be reduced, without altering its value, to the form Co Gilet Cult ..e Cpa @ For suppose that #’(r) is any rational integral function of 7, the degree of which exceeds (p— 2). By the process of algebraical division we can establish the identity F(«@)=QX + R, where the degree of Ff is less than (p—1). Putting =r, we have F'(r)= the value of R when #=7'; and this is an expression of the form given above. Since the coefficient of the highest power of # in X is unity, it follows that if the coefficients of #(r) are all integers, the coefficients ¢, ¢,, etc. will also be integers. The reduction thus effected is unique. For if we put Ceti Cl + opal — Cy PO + Or +. Cpl so that (Gq —G)+(Gq—G)) rt... + (Cp. Cp») 1? 2 =D, we must have 6 — Co SIG — CO) Se = Cp — Cp = 05 because otherwise 7 would satisfy an equation, with rational coefficients, of lower degree than p—1; this is impossible, since X is irreducible. The simplest way of making the reduction is to substitute for every term 7’ its equivalent r”, where h’ is the least positive residue of h to modulus p; if, after this, there are any terms in 7?—, we replace r?— by the equivalent expression — (70? + PF + +r 41). More generally let $(r) T)y= fo= ti be any rational function of r; $(7), y(7) being polynomials in 1. Then we have identically fp) = EOD AO) oes VO), Wr) (7?) (7?) WTP) Now the denominator is a symmetric function of the roots of X =0, and may therefore be expressed in a form which is inde- pendent of r; the numerator is a rational integral function of r, 190 CYCLOTOMY. and may therefore be reduced as above. Finally we obtain f (7) in the form Fr) = Co + Gir + Cet? + 00. + Cp ot, where the coefficients, however, will not generally be integers. If the coefficients of (7) are integers, the product AT) Me (0S ear ea(Thia) is a real integer V; and if the coefficients of ¢ (7) are also integers, then ¢), C,-.. Cp» Will be rational fractions, the denominators of which are divisors of NV. In relation to the equation X =0 the quantities V1), W(7"), TP) are said to be conjugate; and the expression N= (7)... is called the norm of W(r) and written Nm.¥(r). These definitions must not be confounded with those of Art. 94. 173. We shall now give an account of Gauss’s theory of the algebraical solution of X =0; but before doing so, it seems desirable to explain the precise object of the investigation. So far as notation goes, the simplest way of expressing the solution is to say that the roots of X =0 are all the values of 1?” except 1; and in a certain sense this is also the theoretically simplest form of the solution. But if we classify algebraical functions according to the character of the irrationalities which they necessarily involve, the matter is quite different. Thus, for instance, when p=5, the equation X =0 is e+ae+oe+at+1=0, and if we put #+ #1=y this becomes y+y—1=0; the roots of which are leon sy oy Yrs seas eae ree Hence the four values of @ are obtained by solving the two quadratics e—yex+1=0, #-ye6+1=0; whence —14/54+V—10 F 2/5 —14+ 5 — V—10 F 20/5 be =e oe ae Se Ol ad 4 Toe : GAUSS'S PERIODS. 191 and although these expressions are more complicated than 11”, of which they are values, still they are to be considered more simple in so far as they do not involve the extraction of any higher root _ than the second; and from this point of view the reduction of the solution of (#°—1)/(@—1)=0 to that of a set of quadratic equations is to be regarded as an essential simplification. In a similar way it can be shewn by Gauss’s method that, in the general case when p is any odd prime, the solution of X,=0 may be made to depend upon a system of auxiliary equations, the character of which is essentially connected with the resolution of (9 —1) into its prime factors. In particular, if p is a prime of the form 2”+ 1, such as 3, 5, 17, 257, etc, the simplest auxiliary system is composed entirely of quadratic equations; and since quadratic equations may be graphically solved by the construc- tions of Euclid’s Hlements, the remarkable conclusion follows that regular polygons of 3, 5, 17, 257,... sides may be constructed by Euclidean methods. 174. The whole investigation is based upon a peculiar method of grouping the roots of X=0. It has already appeared (Art. 170) that if ris any one of the roots, then the whole set of (py —1) roots is given by 7, 77, 7?.... 72. Now let g be a primitive root of the congruence z?1—1=0 (mod p); then the roots of X =0 may be equally well represented by FO a Teh en Ue a r, as before, denoting any one of the roots. Let p—1=ef be any resolution of (p—1) into two factors, and write Mm =P +78 Hr +4... 4790, a — 79 + ryt + perl + aes + ye ee me =e Lote ete be pp pF Det Vos = ge-1 ae yet + pe oe +. ype} These quantities will be called the /nomial periods of the roots. If we change r into r™, n, is transformed into 7,4; where 7 =ind,m; that is to say, we produce a cyclical permutation of the periods. The particular period denoted by m, will depend upon 192 CYCLOTOMY. the choice of g, as well as upon that of r; but it is easily seen that in every case we have the same set of periods, only in a different order. As a matter of convenience, we may extend the notation of the periods by allowing the suffixes to assume all integral values whatever. The equations defining the periods will remain just the same, and we shall have ny =, if k’ =k (mod e). Of course, in the practical calculation of the periods, the exponents of the different powers of r may be reduced to their least residues (mod p). Example. Suppose p=13, e= 4, f=3. Taking g = 2, we have the four periods ma=rt+r a vs, Ny ee eae No = 18+ 7 + 2, mar +1 4 pl, If g =6, the periods are the same, in the-same order ; if we take g=7 or 11, they are 7 f 4 No = o> 1 =Ns> M2 = Noy Ns = Ny. 175. If in the expression for 7, we change r into 7%”, the value of n, 18 not altered; all that happens is that the terms which compose it are cyclically interchanged. With this fact is con- nected the important proposition that Every rational function of r which remains unaltered when r is changed into 1%” may be expressed as a linear function of the f-nomial periods. By Arts. 172, 174 we may suppose the given rational function of r reduced to the form (1) = Ao? + A719 + AM? +... + Apo?” where dp, a, ... Gp» are independent of r. Change 7 successively into 7%, r7”,... 797°: each of these substitutions leaves $ (7) unaltered : therefore b (1) = a7 + ArT + ar? +... = agrl” + ar? + arf? +, = ar Il)? grt ett ~ Rie RATIONAL FUNCTIONS OF THE PERIODS. 193 Hence, by addition, I. b(1) = Goo + ig +... + Apo Mp2 = (Ay + Ue + Aoe + 2.. + Aye) Mo + (Qh + Gera + Mega + ee FO payer) Fees + (Gey + Usey + 0. + Ope) Nea and the proposition is proved. If the coefficients a), a,, etc. are rational, we have p Re =a bono 48 bin, t+ eee ct bee Ne-1 where by, b,, etc. are rational: and if the coefficients a; are integers, the coefficients b; must be integers also, because each power of r occurs in only one of the periods and then with a coefficient 1. In fact, we have in this case ee Di Oe Ogee Dp aje and so on. Exactly as in Art. 172, it may be proved that the reduction is unique, and, in general, that a relation AM + Arm +... + Meta Nea = bom + Oy +... + Oe Mer, with coefficients a;, b; independent of r, implies that a;=);; because if we substitute for the periods their expressions in terms of r, and then divide both sides of the given relation by 7, we obtain an equation satisfied by r the degree of which does not exceed (p—2); this can only be if the equation reduces to an identity. | 176. It follows immediately from the theorem of last article that every rational function of the periods may be expressed as a linear homogeneous function of them; and, in particular, that every rational integral function of the periods with integral coefficients may be expressed as a linear homogeneous function of the periods with integral coefficients. In the practical application of this theorem it is useful to re- member that >y,;=r+r+...70=—-1. As an illustration, let us take the example of Art. 174, in which p = 18, mertr+?, Nose + foes mert+r+r, MH +r 47h. 194 CYCLOTOMY. We have here, for instance, No" — 7 as D4 ae 76 = Dyl0 as 9/2 4. 7 — On + 2n2, NN. = Mm + 3 + 3==> 31 = 2m, “=z 312 — 273, and so on. 177. HKvery rational symmetric function of the periods with rational coefficients 1s a rational number. Let S be the given symmetric function; then it may be re- duced to the form S = ayo t+ mt os + Oe Nea where d), @,,...@e are rational. Since S is not affected by cyclical permutation of the periods, S = Aym + O1N +... + her Mo = AN. +N, +... + Aa = UNe-1 + AN +... + Ac Ne—s- Therefore, by addition, eS = La;2n;= — Za; and S is a rational number. It is, in fact, obvious enough that S=—4(s a= aS 0. In particular, the elementary symmetric functions of the periods, that is to say, 2;, Xminz, etc. are rational integers. Therefore the periods , 7,,... 7, are the roots of an equation F(m) =0 of degree e and with integral coefficients: the coefficient of 7? being unity. | The polynomial /’(7) is irreducible, in the sense that it cannot be resolved into the product of two integral functions of » of lower degree with rational coefficients. For suppose, if possible, that F(n)=$().W(m): then the equation ¢ (7) = 0 is satisfied by a certain number of the periods, say 4, mg,...,, and we may suppose that none of these periods involves 7?-1, because, if 7?— did occur, we could take yy (7) instead of ¢ (m). Let Oa) ae ag ty then, by supposition, @ is rational, and therefore integral (Art. 171), and we have Na + NB + vee +yn+a=0; q but if 7.2, 7g, etc. are expressed in terms of 7 this is an equation in r with integral coefficients and of degree less than (p— 1), and by Art. 171 this cannot possibly be satisfied. The same result may also be very easily deduced from the last paragraph of Art. 175. EQUATION OF THE PERIODS. ED 178. The most direct way of forming the equation #' (n)=0 is by actually calculating the values of the symmetric functions =n, =nine, =ninen, etc. which are the coefficients of F’(m). Prac- tically, however, it is more convenient to proceed as follows. Let » be any one of the periods: then, by Art. 176, we can EXPress 7M, 7M, NN2;--- NNe—, aS linear functions of the periods, so that NN = AyNo + AN, +... + Aca New, 77 bono aP bm ae tea De Ne—1> Nea = Lomo ne Ln +... thr Ne-1° From these e linear equations 4), ,.-. Ne may be eliminated and the result appears in the form (n aS ty), eds btn 2 5 Veet | F()= A bo, (n — b,), ia be, Shieh — Des | — (0, — hy — by — byes Q—len) Thus, for instance, in the example already considered (p = 13), if we take n = » we have. NN — 1 — 22 =0 —m+(y—-1)m —7,=0 Bo + 2m + (n+ 3) m2 + 2m; =9 — No =a lO ae yn —l, -2, 0 fad | —1, 0, —-1l whence ae = 0, 3, 2,7+3, 2 Sil len ol which reduces to ni + 7? + 2n?—4y + 3=0. The reader who wishes for more numerical illustrations should consult Reuschle’s Tafeln Complewer Primzahlen (Berlin, 1875). 13—2 196 | CYCLOTOMY. 179. Every root of F(m)=0 may be expressed as a rational integral function of any one assigned root 7. Suppose, for instance, we put 7»=%,; then we can form the system of equations | —l=y+m+%+...+ eH, No = No» é No? = Apo + Ay, + AeNo +... + Ae Ne, no? = bono + by, + bens + 2. + 0-1 Near, oo = bony +m + lena Bro ae eels and hence, if A is the determinant formed by the coefficients on the right-hand side, we have, for all values of &, A.n, = flea An ce An? 3. Ags aes dene: Yale, wl A,_, are integers. This proves the proposition, provided that A is not zero. But A cannot vanish, because if it did », would satisfy the equation A, + Ain +... + Asam? 1 =0, which is impossible, since #’(m) is an irreducible function. The method of this article affords another way of constructing the equation F’(7)=0: namely by combining with the above system of equations the similar one No = MN + MN, + MyNy +... + Mey Nor, and then eliminating 7,, 72, .-. Ne. As an illustration, the reader may verify that for the trinomial periods mo, 7, 72, N3 associated with p = 13, 3, = — 6 + 4, + 39° + 27°, 387,= 3-2 — N°; 373 = A Ue = 180. Each of the f/ roots of X =0, the aggregate of which makes up any one of the nomial periods, satisfies an equation of the fth degree, the coefficients of which are linear functions of the periods with integral coefficients. It has. already been observed (Art. 175) that the effect of changing r into r”” is to produce a cyclical permutation of the SOLUTION OF X,=(. 197 roots which make up a period: hence all symmetric functions of these roots remain unaltered, and may therefore be expressed as linear functions of the periods. This proves the proposition. Thus, in the case already chosen for illustration, the roots r, 7°, 7°, of which the sum is 7, satisfy the equation — Nv + nov —-1=0; and in the same way the equations w — x + nsx — 1 =0, a — yt? + nc —-1=0, xv — n 2 +n a4 -1=0, are satisfied by (7°, 7°, r°), (r4, 1, r?), and (77, 78, r”) respectively. Remembering that ) may be any root of | F(m) = 1 + 7° + 2m? — 4n +3 =0, and that 7, 72, 7; can be expressed as rational functions of , we see that by the ‘adjunction’ of the single irrationality »,, which is defined by the equation #' (7) =0, the polynomial X,; may be resolved into the product of four polynomials each of the third degree ; so that the solution of X,,=0, which is of the twelfth degree, is reduced to that of one quartic and three cubic equations. | But in the case considered the reduction may be carried one stage further. There are two periods of six terms, C= M+ Me; 7 =m +s, and it is easily verified that G+ &=—1, 66=——3: so that ¢,, € are the roots of | 2+2—3=0. Moreover on, = 7, +7;+3=6,+3: hence 7, 7, are the roots of y?— fy +(+ 3) =0, and in the same way 7, 7; are the roots of y?— fy +(& +3)=0. Hence the solution of ni +n? + 2n?— 4n —3=0 is made to depend upon the system of auxiliary quadratics 2+ 2-3 =0, y — Sy + (f+ 3) = 9, ye — by + (hr + 8) = 0. 198 CYCLOTOMY. The algebraical solution of X,,;=0 may therefore be stated in the following form :— Let € be any root of 2+2—-—3=0, n any root of y? — Fy + (2-6) =0, E any root of a —ne+(€—n)c«c—-1=0; then w= & is a solution of X,, = 0. 181. In the general case, whenever e is a composite number, say ef’, the expressions Gh ar? Spy “ar ee +N (fie) o = +Me41 + Yeti t--. Nf —Ye+) Coa = Nea + Noe—1 + se —a + ++» + Nes are periods, and will satisfy an equation /,(€)=0 of degree ¢ with integral coefficients; and by the adjunction of a root of this equation, the polynomial F’(7) may be resolved into the product of e’ polynomials, the coefficients of which are integral functions of ¢. In the same way, if f is a composite number, say e,f,, any period 7, may be expressed as the sum of e, periods, each of /; terms; and the f,-nomial periods of each such group are the roots of an equation of degree e, whose coefficients are rational functions of the 7s, or, which is the same thing, of any one of them. Proceeding in this way we see that the solution of X, =0 may be obtained from a system of auxiliary equations, the degrees of which are the prime factors of (p — 1); the number of the equations being equal to the number of factors. | The system of equations will depend upon the way in which the roots of X=0 are distributed into successive groups of periods. Practically, it is best to keep conjugate roots r*, r~* together in the same period until the last stage of all, because then the roots of the auxiliary equations will all be real, except in the case of the last, which will be a quadratic with complex roots. ‘Thus, for instance, when p=13, we may begin with the three periods of four terms, and form the auxiliary system 2+2—-424+1=0, YS las) =O, “—nex+1=0, where ¢ is any root of the first equation, and y is any root of the second, CONGRUENTIAL ANALOGY. 199 The solutions of y,=0 and X,=0 are given explicitly in Arts. 358, 354 of the Disquisitiones Arithmetice: the reader may also consult Richelot, De resolutione algebraica wquationis X*” = 1, ete. (Crelle ix. (1832), p. 1), and Cayley, Note sur la solution de Péquation «#” —1 =0 (ibid. xl. (1851), p. 81). It is now well known that there is no general formula for expressing the root of an equation as an algebraic function of the coefficients, when the degree of the equation exceeds 4. This was to some extent anticipated by Gauss (D. A. Art. 359, with the reference there given), but was first proved by Abel. Now the degrees of the auxiliary equations, upon which the solution of X,=0 has been made to depend, are the prime factors of (p—1), some of which may, and in general will, exceed 3. Hence Gauss’s theory of the periods gives us no assurance that the roots of X, =0 may be obtained in the form of purely algebraical irration- alities. That this is in fact the case was discovered by Gauss, whose method was afterwards simplified by Jacobi; but since the interest of the problem is mainly algebraical, and its solution immediately follows from the principles laid down in Abel’s memo on equations which are solvable by radicals, we prefer to pass on to applications which are more distinctly arithmetical in character’. 182. Ifqisa prime of the form Ap +1, where », as before, is an odd prime, the congruence #? —1=0 (mod q) has all its roots real, and the same will therefore be the case with X,=0 (mod q). All the algebraical theory of the equation X,=0 may be applied mutatis mutandis to the congruence X,= 0 (mod q): thus we may arrange the roots of the congruence into periods, and reduce the solution of X,=0 to that of a set of auxiliary congruences, ete. etc. For example, if p=13, g=53, we may take as a system of auxiliary congruences for the solution of X,;= 0 (mod 53) 2+2—-3=0 y?— bCy+(2-O)= A| (mod 53). a —na+(f€—n)«-1 =0) 1 ¥or the explicit solution of X,=0 the reader should consult Gauss, Disq. Arith. Arts. 359—60, and the posthumous paper Disquisitionum circa equationes puras ulterior evolutio (Werke ii. 243); Jacobi, Ueber die Kreistheilung und thre Anwendung auf die Zahlentheorie (Berl. Monatsb., Oct. 1837, p. 127, or Crelle xxx. p. 166); Abel, Mémoire sur une classe particuliére @équations résolubles algébriquement (Crelle iv. (1829), p. 131). See also Bachmann’s Kreistheilung, 8" Vorlesung, where other references will be found. 200 CYCLOTOMY. The first may be written (¢-7)(¢+8)=0; if we take C= 7, the second congruence becomes y— Ty —5=0, or (y + 21) (y — 28) = 0. Putting 7 = — 21, we have e+ 21a?+ 28e%—1=0, whence «=—6, 12, 14. All the roots of X,;=0 are given by x=(—6)*, where k=1, 2,...12. It may be specially noticed that corresponding to the equation F'(n) =0, satisfied by the fnomial periods (Art. 177), we have a congruence F£'(n) = 0 (mod q) all the roots of which are real, and connected by congruential relations precisely similar to those which are satisfied by the algebraical periods. Thus, for stance, when p= 13, we have a period-equation (Art. 178) Te eeel seen it Up the roots of the corresponding congruence, mod 53, are n=14, — 21, —22, —25. If we put 7, = 14, then in order that the relations connecting the roots may be the same as for the corresponding equations, we must write Oi ed Ts se a) ey and so, in general, when any root of the congruence has been chosen to correspond to a particular period, say 7, the relation of | the other roots to the remaining periods is determined (cf. Art. 174). The results of this article were evidently familiar to Gauss, although he did not publish them; see the paper entitled Solutio congruentie «#'” —1=0, which is printed in the second volume of his works (p. 199). It will be seen, later on, what important consequences have been deduced from them by Cauchy, Kummer, and others. 183. For every odd prime p there will be two periods, each containing $(p—1) terms; denoting them by A and B, we may write, In our previous notation, % 4 p-3 A=r trP prt. fre, B=7I9 47h +7P +... +P, CASE OF TWO PERIODS. 201 Since 1, g’, g’,... g?~ are all incongruent with respect to p, we have A =Xr*, where the summation extends to all the positive quadratic residues of p which are less than p; and in the same way B=r*, where 8 denotes any one of the quadratic non- residues of p which are positive and less than p. In order to find the quadratic equation of which A and B are the roots, we may calculate the values of A+B and AB. We have at once A +B=—1, but the determination of AB is less easy. We know (Art. 177), that its value is a real integer; moreover it may be written in the form A Be) ne: where the sum on the right contains $(p- 1)? terms. The term r*+B reduces to 1 if a+ B= p; this gives — 8 = a (mod p), which can only happen if — 1 is a non- residue of p, that is to say, if p=3 (mod 4). Conversely, if this is the case, for every term r* which occurs in A there is a corresponding term 7?~* in B, and when we multiply A and B together, we obtain $(p—1) terms each equal to unity. The remaining terms of the product, +(p—1)? —$(p—1) in number, must reduce to r+7?7+...+7?—1, that is — 1, taken 1 2 oe x (p r= Wey may een times; hence AB=4(p—-1)—4(p—3)=4(p +1), and the equa- tion sesyeltiel by A and B is v+nt+i(pt+1)=0. On the other hand, if p=1 (mod 4), it is impossible that a+ =p, so that in this case 1 ay AB=—1x¥P—¥ -_ 3(p-1) and the quadratic in 7 is m+n —4(p—1)=0. Both cases are included in the formula TP (Pye) 184. If we solve the quadratic we obtain —14£ iP /p U bom PSD Re a, 202 CYCLOTOMY. now when p and r have been chosen the value of A is perfectly determinate, and the question arises how the ambiguity is to be taken when r and p are assigned. We observe that if r is changed into r™, where m is prime to p, A and B remain unaltered, or are interchanged, according as m is or is not a quadratic residue of p. Hence it is sufficient to find the value of A for any one value of r; we shall suppose that r =e», This bemg so, the value of A may be written A — e2tt/p a est t/p A elsni/p aa ak ec: (p—1)2nt/p s=3(p~1) — e2s?rt/p = Gest Instead of this we shall consider the slightly more general expression s=n-1 SSS eer re s=0 where m is any positive real integer, and. the sum consists of n terms. The determination of the value of S has lately been effected by Kronecker in a remarkably simple and elegant manner with the help of Cauchy’s theory of complex integration (Crelle cv. (1889), p. 267); his investigation will therefore be given here before discussing the less direct, although more arithmetical, methods of Gauss and Dirichlet. Consider the expression nian this is a one-valued function of the complex variable z which is finite and continuous for all finite values of z, except when z is a real integer. Putting z=h, a real integer, we have 1 = = 2m th2/n IB ail h) (2) a4 ° ; so that z=h is a simple pole of ¢ (2). Now by a well-known theorem, due to Cauchy, the integral fo (2) dz, taken in the positive direction round any closed contour C which encloses a certain number of poles of ¢ (z), 1s equal to the sum of the values of the same integral taken in the positive direction round closed contours each surrounding one of the poles enclosed by C. ae GAUSSS SUMMATIONS. °038 To apply this to the present case, we choose for the contour C that which is represented in Fig. 6. It consists of a rectangle with two semicircular notches cut out of it: the vertices of the rectangle are + 1y,, $n +ty,, and the terminal points of the semi- circles are at +7%, 4n+%y, respectively. We shall eventually make y, = + and y infinitesimal: and it will be supposed in the first instance that y <4; thus if” is odd, the poles within C are 1, 2,3...4(—1), and if n is even they are 1, 2, 3,... (4n —1). Fig. 6. First suppose that n is odd. The value of {6(z)dz taken round an infinitesimal circle surrounding the pole z= s is a 1 E2ti 0 | Dard = — e2tis?in. Next consider the integration over C. The semicircle at the origin gives ultimately, when y is infinitesimal, Lee fone — Ont : adé = i 5 2 the other semicircle gives nothing, since $n is not a pole. The remaining part of the integration gives | Beat) Gee | aeons | Pd Catia) at 0 ae —Yo | (A), 1 0 Yo , ‘ + [ip ntit art | b (in tt) dt-+ [ig (ty dt Yo an JY where ¢ is a real variable, 204 CYCLOTOMY. Now it is easily seen that the second and fifth of these integrals ultimately vanish when y, =+ # , because 2 e2nt(—ty, +t)2/n $ (— Oy fi t) is 1 — e2ti (in, +t) =e" (1—2t/n) é pp ( t), where w(t) does not become infinite, and y,(1—2t/n) is not. negative ; while p (ty, + t) = enn y (4), where x (¢) does not become infinite. In each case the exponential factor causes the integral to vanish. The first integral in (A) may be written ~i/" 6 (— wd; Yo hence the sum of the first and last =~ i" oD + $id) a, Y1 ; ——4 | e2min | dt ¥ 0 on reduction. With regard to the other two integrals, we observe that ; qn e727 : e 2m it?/n 1 = PUNT) T= yen ) qn e2mt . e72ril?/n ayy ee. a \ es a ke ee p (yn it) i-= (— 12 e27t ? whence the sum of the remaining integrals = if” (b (An -+ it) +6 (4n—1)} df, Y y = (— ihe gon | ' g-amit?/n dt aes jon | : e72mit?/n 5 dt. Yo Yo If we make y infinitesimal and y, infinite, we have ultimately yi ; ) yilNn eri? /n dt = /n en 2miw? du Yo yol Vn oc =/n | Gen di 0 =A) 72 where »/n is taken positively, and A is a constant which is inde- pendent of n. Hence finally, n being odd, s=i(2-1) 4 A (2 + geet) An/n a mis ort S= DIRICHLET’S METHOD. 205 or, multiplying by 2 and transposing, Sa Ann (tx ett), To determine A, put n=3: then S=12)/3, so that 1=2A(i-1); th fi A weal erefore = 5a? % ‘3n and caleneD Wei 1+. This may also be written ae Ss= 18; /n, which is Kronecker’s form of the result. It must be carefully remembered that /n is the positive square root of n. The same formula applies when n is even; the only difference in the work is that $n is now a pole, and the semicircle at $n ulti- mately contributes } instead of zero to the integration round C. The results may be tabulated according to the residue of n to modulus 4. Thus if n= 0 (mod 4), S=(1+2)/n, =l1 =a) it =2 = 0 275) EAN. Suppose, now, that n is an odd prime p; then the expression denoted by A in the beginning of this article is equal to $(S— 1), that is, to a DoF vee ade aD) 2 2 according as p =1 or 3 (mod 4). 185. Dirichlet’s method is somewhat analogous to the pre- ceding, but avoids the use of the complex variable. It depends upon the lemma that if f(«) is a function of # which is finite and continuous so long as 0 + w > 7, 2 [re. cos st dx =7f(0), where the summation applies to all integral values of s. 206 CYCLOTOMY. Observing that, if 7 is a positive integer, (2r+1) 7 ar | f(z). cos sx .da = | f (2r7 + x) cos sx da, 0 QrT (27+ 2) 3 v | F (0)..008 50 dar =| ff (2r+ 2.7 — x) cos sx da, 0 (2r+1) 7 we infer that if h is a positive integer, and f(x) satisfies the same conditions as before so long as 0 + w + 2hz, +o [2hr + 00 7 Qar 2hr 2 F(x) cos sx dz = >, i{ +| ++| | £2) cos se de =00 J () —-2 0 T (2h-1)7r =5 [ (f(a) +f (Qa —«) + f(Qr+a)+...f Qhr —«)} cos sw de =o { f(0) + 2f (2m) + 2f (der) +... + Of (Qh —2. mm) +f (Qn). Consider now the integrals i * cos ¥ To cos at d= 2 | dy, Ihe: 0 ly J +o saps v=| sin de =2 | “Ty 0 : « —0O it is easily seen that these are finite and determinate, in whatever way the upper limit of integration is supposed to become infinite. Writing aa? for «, where a is finite and positive, we find +00 +2 : | cos aa’. da = u/r/a, | sin av’. da =v/y/a, = — 2 ico) /a being taken positively: and hence, if -+- co A= | cos (6 + 2°) dx= wu cos 6— v sin 4, where 6 is any finite quantity, we have +00 5 : d A | cos (6 + ax) ae Ty Now let 8 be a finite positive quantity: then the integral last written may be replaced by +o /(s+1)B > cos (8 + aa?) da —o/ sp where the summation extends to all integral values of s. (s+1)B B But i cos (6 + ax) da =|) cos {6 +a (s8 + x)*} dz, 8 0 DIRICHLETS METHOD. 207 and if we put 8 = 4m, a = 1/S8mz7r, where m is a positive integer, this reduces to 4nutr a [ cos (8-4+a0+ go—) de as Smarr Anat i 401 : re : =| cos (8 ie | cos sx da -| sin (8 bee | sin sa da. 0 Smar 0 Snr Since the second integral on the right hand is an odd function of #, and we may suppose the values of s arranged in the order s=0,+1, + 2,...4h where h is an indefinitely large integer, we have finally | A Anur ted SNAPS cos (8+ r/o ig Writing, for the moment, cos (3 + a) = Mea the application of the previous lemma gives AV Sina = 1 § f(0) + 2f (Qar) + 2f (4ar) +... + 2f (4m—2.17) +f (4mz7)}. Now, if s is any integer, f(4mm + 2s7) =f(2s7r): hence the expression on the right may be replaced by aw {f(0)+f(27)+.. +f (dan) +f (4m +2. T)+.. +f (8m —2 2.77)} =n 5 f 2sm); fe ——_ } cos sa dx. Smr/ therefor eC bs a Sar 4m-1 Sar AV8mr = cos (5457) = 4cos 6 > cos — —sind > Sil 2m 0 2m 2m Since A=ucosd—vsind, and the formula is true for all values of 6, it follows that 4m—-1 2 4m—1 2 Sar oS ao}, sa = cos=— =uV8m/n, = sin =— =vV8m/r. 0 2m 0 2m To determine wu and », which are evidently independent of m, we put m=1, which gives u=v=V47; and hence finally 4m—1 Sar 4m—1 ‘ Scr = = 2r/m. = COs 5 > sin 5 /m Following Dirichlet, we shall write n—-1 tes > erhs’ri/n — d (h, n), Q 208 CYCLOTOMY. where n is a positive integer, and h is also integral but not necessarily positive. Thus the result which we have just obtained may be expressed by the formula dh (1, 4m) = 2 (1 +72) /m, or, which is the same thing, (1, n) =(1 +72) o/n, if n = 0 (mod 4). 186. It immediately follows from the definition that if h’=h (mod n), $(h’, n) =¢ (h, n). Moreover, if a is any integer prime to n, d (ha, n) = ¢ (h, n): n—1 because by definition ¢ (ha, n) = & e(s"rt/n, and when s assumes : 0 the values 0, 1, 2...(n—1), the least positive residues of as to modulus n consist of the same numbers in a different order. Another theorem which we shall require is the following :— If the positive integers m, n are prime to each other, then d (hm, n). d (hn, m) = $ (h, mn). To prove this, we observe that, by definition, zhni(™S 4 RE) /s=0,1,2,...n—-1 fp (hm, n). h (hn, m) ee ey ee \, ea 0, oa ee 4 g? 2 2 Hct ms? nt? (ms + nt) ys nm mn therefore p (him, n). (hn, m) = & ehrtinstnd etna, 8, But the expression ms+nt assumes mn values altogether, and it is easily proved that these are all incongruent (mod myn), because if ms + nt’ = ms + nt (mod mn), we infer that ms'=ms (mod n) and nt’=nt (mod m), whence s'=s (mod n) and t/=t (mod m), whereas in the present case all the values of s are incongruent (mod ) and all the values of ¢ are incongruent (mod m). Hence the integers ms + nt form a complete system of residues to modulus mn, and therefore mn—1 ~ erhni(mst+nt)ymn — SS g2hs’mi/mn — db (h, mn), S, 0 and the theorem is proved. GAUSS’S TRANSFORMATION. 209 187. Suppose, now, that » is an odd number; then, putting m = 4, h=1, we obtain $(4, 2) b(n, 4) =$(1, 4n). As already proved, $(1, 4n) = 2 (1 +7)\/n, and by definition h(n, 4) = 1 4 errs 4 esnmt/s 4 gisnmi/s SVAGl dene moreover, by putting h=1, a= 2 in the formula d (ha?, n) =o (h, n), we find f (4,n) = (1, 2); hence 2(1+ 7") ¢ (1, n) =2(1 +2)/n, or a MG) he ans a/n == Ni OF t4/N; according as n= 1 or 3 (mod 4). Finally let n = 2 (mod 4); then, since 2 is prime to $n, we have d (2,4). bd (én, 2)=4(1, n), but (én, 2)=(1, 2) =0; therefore ¢(1,n)=0. The value of g (1, n) has now been determined for all integral values of n: the results are, of course, in agreement with those of Art. 184, because ¢ (1, n) is the expression there denoted by S. 188. We will now give some account of Gauss’s demonstration, which, as already remarked, has the advantage of not requiring the aid of any transcendental analysis except the elementary theory of the circular functions. Consider the expression —d—a™) d-a" 4) —a4)... 1 — a) Bia (1 —2)(1—#)(1—2@')... (1 — a") where m, pu are positive integers. If m< pu, the numerator involves the factor 1—2*, and therefore (m, w)=0; we proceed to prove that if m <¢ pw, (m, w) is a rational integral function of «. It is obvious, in the first place, that (m, m — pb) =(m, 1), if m<¢ mw; and also that Fp FON. Again, since a 9" = (Sa) ia 1 a), M. 14 210 CYCLOTOMY. it follows that (m, w+1) ale — ret) La qd — ert)! 1 —a)... (1 — crs (—2) (1-2)... (1 —a"t) =(m—1, w+1)+a"" 7 (m-I, pw) =(m—2, w+ 1) + a? (an — 2, po) + om (m —1, p) a skit wt+l)t+a(utl, w)+a°(ut+2, w+... +0" "71 (m—1, p) =l+a(e4+], vw) +a? (442, w+... +a" 1(m—1, p), supposing that m + uw 4 2. Hence if, for any fixed value of mu, the expression (m, m) 18 a rational integral function of w for all positive integral values of m, the same will be true of (m, 4» +1). But (m, 1) =(1—a™)/(1 — 2), which is a rational integral function; therefore the theorem is true for (m, 2), and hence successively for (m, 3), (m, 4), and so on. This proves the proposition. 189. Now let us write l—a2” U-a) 1-2 LAD 1 =1—(m, 1)+(m, 2)—(m, 3)+... This contains (m+ 1) terms, the last being (—1)”, and since, by the proposition just proved, each term is a rational integral. function of #, f(a, m) 1s also a polynomial in a. If m is odd, f(#, m)=0 identically, because the first term cancels out with the last, the second with the last but one, and so on. . Next, suppose m is even. We have identically ia —(m, 1) =—(m—1, 1)-— a" +(m, 2)=+(m-— 1, 2) +a" (m — 1, 1) —(m, 3)=—(m—1, 3)— a”? (m — 1, 2), and so on; whence, by addition J (a,m)=(1 — a”) —( — w") (m — 1, 1) + (1 — a") (m — 1, 2)... =(1—#"™) {1—(m — 2, 1)+ (m — 2, 2) —... to (m — 1) terms} =(1—2") f(@, m—2) =(1 a") (1 2") f(@, m—4) = (1-27) 1 — a) (1 - a)... (1 —2) f(a, 0) =(1 —2#)(1—2*)(1- @)...(1—a™), GAUSS’S TRANSFORMATION. Did 190. Hitherto w# has been any quantity whatever: we will now suppose that #=e"'/", where n is an odd positive integer, and we will write m for (n—1). This being so, we have l—a” l-w#! ee a l—-—«z 1-«z d = —2 1—a™ spel aye 1-2 1—2# : Ll-—gmert1 1 —ae ars ee ee 1 —x* 1 — a oe hence (m, p)=(—1)'ate et , and the identity J (@, m)=(1 — 2) (1 — a) (1 —- 2°)... 1 — a) becomes Lao t+ wet... f+ ee) + + nin) =(1 —2) (1—2@*)(1—-a*)...(1—a"). Since 2, like a, is a primitive root of a”—1=0, we may change z into 2’, and thus obtain L+a?+a°+... + aut) +. gin) a= ( Lies) (le aare’) ot Cla eae) Multiply both sides by Dd oF te HLT ett 8 Carat then on the right hand we have Ee (eee). (ae et) while, if we observe that } (nL +p (wt) =H {nt Int (S41) =1(n—2u—-—1) (mod vn), the expression on the left becomes | 2 MA)* 4 gk (N—3)? 4g R(N—5)7 4 A pk (MT9)? 4 yd (NTI)? Rearranging this, we have finally S=1l+at+at+...+ 0° = (4— a7) (a? —ar)... (a? -— a), Now eo gt a — a 8 — gr = yt 8 — gn-8. and so on: therefore S32 (— 1)8 (a — a) (2? — or) (a — oe)... (2 a) tL (eae) (1 a Lat) =(—1)8"—n, 14—2 212 CYCLOTOMY. because w+) = 1, and «7, a+,... a?"+? are the (n—1) roots of the equation 1+ #+2?+...+4"1=0. Hence S = +/n or + t/n according as n = 1 or 3 (mod 4): and it only remains to determine the signs of the ambiguities. 2har Now a! — a—" = 27 sin are therefore S = 22") 72-9 TI gin a [h=1, 3, 5,...(n—2)]. h First, suppose n =1 (mod 4): then the values of h which make oie : ie negative are 4(n+1), $(r+ 5), ...(n—2), that is, }(n—1) values in all; and the sign of S is the same as that of (— Lye) P (— 1) 1) “2 ike so that S= + /n. On the other hand if n =3 (mod 4) the values of / which make eo PApas saa negative are 4(n+3), $(n+7),... (n— 2), that is, there are }(n —3) values in all. Hence S=72S’, where the sign of S’ is the same as that of (—1ie—) (— 189) = 4-1; consequently S = + 2/n. Gauss proceeds to find the value of S when n is even: this part of his investigation will be omitted here, because it is much simpler to proceed as in the latter part of Dirichlet’s proof given above. 191. + pox + ps0? + yu? + P30 + Des Noes Ue Po = — (So + Piss) = — (Mm — N07) = 2, Bs =a 3 (85 + piSs + p28;) =— $(m0- 0 + 2m) =—1—m, ‘4 ae + (s, + p83 + PoSe + ps8) = —4F (1 — 90? + 2m — Mo — 1°) <* —3(m — no) = 2, Ps=—, Po=1; therefore U = 298 + Qa4 — a? + 227+ 1, V=—a#—-2— 2, and Y = 246 + 0° + 404 — a? + 4074+ w+ 2, | LZ=H+ a+. It may be added that the coefficients of Z are always sym-_ metrical, that is to say, the mth coefficient from the end is always the same as the mth coefficient from the beginning: the same is true for Y if p=1 (mod 4), while if p=3 (mod 4) the corresponding coefficients are equal and opposite. This consideration, of course, greatly shortens the work. 194. It is possible to find the values of p,, p., etc. as explicit functions of p. Thus if we write p=2p’+1, e,=(—1)”, and en = (h|p) for all integral values of h which are greater than 1, we have Sueicn 2 s, = + &n No (h >1); EXPRESSIONS FOR THE COEFFICIENTS. Day's hence, by Newton’s formula, and the equation of the periods, Presa 055 8p, = (1 — 2e, + ep) — 4 (1 + @2) m0, 24n; = 3 — 3e, — 4e; + (1 + 8¢e,) ep —(9 Ge, + 8é; + & P) No; 3. 2’p,=—3 — 36e, — 32¢e, + (26 + 12¢, + 32¢e;) ep + p? — 8 {27 + 9e, + 8e; + (1 + 3e,) ep} m, and so on; and if Y= 22? + ¢,a?' 1+ ca?’ +... we have C1 4c, =3 + ep, 8c; = 5+ (1 + 2e.) ep, 3. 2%, = 105 + (30 + 24e, + 32¢,) ep +p’, 3.2’c,= 189 +(90 + 386e, + 32¢; + 32¢,¢;) ep + (1 + 46.) p?. The quantities e, are periodic functions of p: thus we obtain for the coefficients c,, C2, ¢;, ete. a certain limited number of dis- tinct polynomials in p. For instance, if p=1 (mod 24), eo, =e, =e, =1, and c= yyy (p+ 86p +105); while if p= 5 (mod 24), e, = 1, e, = es = — 1, and LO é 1 < C4 = 395 (PD? — mit + 105) = 3-55 (P — 5)(p — 21). The values of the coefficients above given were calculated successively : 1t is very desirable, of course, to discover a method of writing down the general value of c;, without having to calcu- late the preceding coefficients, but it is not easy to see how this can be done. We may eliminate p,, p,...p:. from the first 2 of Newton’s equations, and thus obtain p; as a function of s,, 8... 8; in the form of a determinant, but the reduction of this determinant to the form a + by is apparently impracticable when 7 is large. 195. It may be observed that, since p is prime, a? —1| =(x—1)” (mod p); hence 4X = 4 (x —1)?* (nod p), and Y=2(¢@—1)?”— (mod p). For sufficiently small values of p, Y may be found by expanding oS CYCLOTOMY. 2(«@—1)2”— by the binomial theorem, and reducing each coeffi- cient to its absolutely least residue (mod p). Thus, when p= 11, Y = 2 (a# — 52+ + 10x? — 10a? + 5” —1) = 20° + at — 2a? + 2a? — x — 2. Legendre, in his Théorie des Nombres, erroneously stated that this rule applies to all cases; he afterwards, however, corrected his mistake. (Legendre: Mémoire sur la détermination des fonc- tions Y et Z qui satisfont a Véquation 4(a” —1) =(a# —1)(Y? 4+ nZ?), n dant un nombre premier (40 F1), Paris, Mém. Acad. Sci. xi. (1833), p. 81. Lebesgue: Recherches sur les nombres, Liouv. 111. (1838), p. 113.) While Legendre’s rule certainly fails for p = 61, and possibly for still smaller values of p, it holds good up to p=31 inclusive. It may, in fact, be verified that when p = 31, Y = 205 + v4 — Ta — 11a? + 2a" + 8a — 34° — 528 + 5a’ + 82° — 8a° — 2a*4+ 11a? + Ta? -— x — 2, Z = a + ao — og — 2a" + a? — 8 — a + 98 — 204 — a? + 8 + The following table gives the values of Y and Z for all primes less than 31. The last case, p= 29,1s due to Legendre; the others were calculated by Gauss. p Y Z 3 | 2a+1 i 5 | 2e?+a4 2 xv 7 | 20° + a —a27—2 e+ ea 11 | 2a° 4+ a — 22° 4+ 2a?7-—aw— 2 +a 13 | 2464+ 24+ 404 -— a? + 40? +042 | @+a+ea 17 | Qa8 + a? + 5at + Ta® + 404 + 72 a + e+ + Qa a + 527+ a+ 2 +e+¢ 19 | 2a9 + a8 — 4a7 + 3405 + 5a — dat O-a+e+at—-v+ua — 32° + 4a? —- a4 —2 23 | Qe + ao! — 5a® — 8a8 — Ta’? — 40% | 4+ a — a7 — 208 — Qa? + 4a5 + Tat + 8a? + 5a?-—a—2 —@+e+a4 29 2x4 + y+ 8a — Bal + gw — 249 8 4. gl — gw 4+ gt a’ + 3a8 + 9a? + 37% — Qa? + a +a—aet+teig —3e+8e2+a¢+4+2 CUBIC EQUATION OF PERIODS. 219 By putting «=1 in the identity 4X = Y?+ pZ, we obtain a representation of 4) in the form 4p = m? + pn’. Thus for p= 17, we have 68 = 847 -17.8? and so on. It may be noticed that when p=3 (mod 4) this process only leads to the trivial result 4n = 0? + p. 2?, except when p= 3. 196. When p=3n+1 there will be a set of three periods m, ™, 2, Which are the roots of a cubic equation ”? os C17? -[- CoN ae C3 = 0, where ¢;, Cs, C; are integers which have to be determined. If g is any primitive root of p, and r = e?""? as usual, we may put N= > pr = Yeah iii the No at Ss pprre All numbers prime to p may be distributed into three classes A, B, C according as their indices to the base g are congruent to 0, 1 or 2 (mod. 3). Numbers of the class A will be denoted by a,a... and in the same way 8 and y (accented if necessary) may be used to indicate numbers belonging to B and C respectively. With this notation we may write - ee where the summations apply to m incongruent values (mod. p) of a, 8, y respectively. The class A includes all the numbers which are cubic residues of p, and is the same whatever primitive root g may be chosen ; if instead of g we take a primitive root g’ such that indy, g=2 (mod. 3) the classes B and C will be interchanged. We observe that 1 and p—1=-—1 both belong to the class A. Also the product of any two numbers of the class A is a number of the same class, or, in symbols, aa =a”, In particular, p —a, or —a@ belongs to A. Similarly a8 =’, boy, / / B = 4. ay= 7; yy =B, J 220 CYCLOTOMY. Returning now to the equation of the periods, we find at once Ss he — (M+ m+) =1. The value of ¢, 18 ¢, = om + M2 + NM = »S, (7276 + rhty + 7yte), Now the congruence a+B=0 (mod. p) is impossible, because it would lead to =-a, and hence 8 would belong to the class A, contrary to definition. In the same way it may be inferred that the congruences 8 + y = 0, y+a=0are impossible. Hence of the 3n? terms of which > (rot8 4+ 78ty + 1%) consists not one reduces to unity. But we know that the ex- pression is a rational integer : hence its value is (r+7?+ ...+7?7), or — 1, taken 3n?/(p —1)=n times. Therefore C,=—n=—4(p-]1). The value of the remaining coefficient is C3 = — Not, = — rat hry, Suppose that ’ denotes the number of distinct solutions of the congruence a+8+y=0 (mod. p); then by the argument used in finding the value of c, it follows that iN aes Se so that c, is determined when X is known. Now a may assume any one of n incongruent values : suppose a’ to be any one of these. Then if a” is determined so that aa” = 1, the congruence a +8 += 0 is equivalent to a’ (a +B+7)=0, or 1+f' +7 =), where §’, y belong to b, C respectively. Conversely, from every solution of 1 + 6’+ y’=0 may be deduced a solution of a +8+y=0, by putting B=4'8’, y=a'7. Hence if the symbol (12) is used to denote the number of solutions of 14+6+y7=0, we have A= 12) 9 and co, = 4 (12) p— ni}, so that everything depends upon finding the value of (12). CUBIC EQUATION OF PERIODS. 221 To do this we consider the system of congruences given in the following table, in which the symbol placed to the left of each congruence denotes the number of distinct solutions of which it is capable. (00) l+tatea’=0, (10) 14+8+a=0, (20) 1l+yt+a=0, (Ol) l+a+8=0, (11) 14+84+8=0, (21) 1+y7+8=0, (02) l+a+y=0, (12) 14+8+y=0, (22) 1L4+y+4=0. It is obvious that (01) = (10), (02) = (20), (12) = (21) ; in fact, the double notation is only used for the sake of symmetry. If we multiply the congruence l+a+B=0 by a number y such that By=1, and put ya=7’, we obtain 1l+y+%=0. Therefore every solution of 1+a+6=0 is associated with one of 1+y+-’=0; and in the same way from every solution of 1+ ++ = 0 we can deduce one of l+a+B=0. Hence COM (22): and similarly (02) = (11). Thus the matrix (00), (01), (02) (LO) Sicily mem) (20) (21) 22) is reduced to the type h 4 k Ae dae > kl 9 where h, j, k, J have still to be determined. The series 1, 2, 3... (p—1) contains n numbers a, and each of these, except the last, namely p —1, is followed by a number a+ 1 which must belong to one of the classes A, B, C, and is therefore of the form p—a’, p— £’, or p—y ; that is, to every value of a, with one exception, corresponds a solution of one of the congruences 1l+a+a’=0, l+a+/f' =0, l+a+y =0. Hence (00) + (01) + (02) =n —1, or Wet ger iL RE A. BR Semen Gis and in the same way Ao ee Re Ai has mov SEO ce Ci (2); because every number 8, without exception, is followed by another number of the series. D2?, CYCLOTOMY. There is yet another relation connecting h, 7, k, | which may be found in the following way. Consider the congruence peep le ori piare boa!) There are (00)=h values of a which make a+1= COS = aie p p] and hence 2k 2 + pZ, + p24. = 6% (k\p) cos = [k=1, 2...4(p—1)] ~ 9 is not altered by changing k into p—k It is proved, in a similar way, that if m is any integer (m|p)? (20 + p& + p*%2) = 63 (i:|p) cos me 8 from 0 to 7, both inclusive Then Suppose, now, that /(@) is a function such that for all values of J (@) =A, cos 8 + A, cos 26 + A, cos 36 + > ip) (=) - A, = (k\p) cos td > (k\p) cos —+ that is, =" ee (|p? Ai+ [py Aa +.-.}, 6S (hip) f(™) = (ot 0a + pte) %(5|p) Au k=1, ana (ea a 3. ) It is known that if 0 +0 $7, therefore Put ies a iat fe beat a ot 8 + 3° 5? 6S (hip) (% — ge) = (+ pes + pte) & (tp) Gaul sie Fi Hert Be Br C= 2y™, 15 226 CYCLOTOMY. where the summations refer to all positive odd integers which belong to the classes (a), (8), (y) respectively, then 6 & (k\p) (= - |= (Z) + pa + p22) (A + p?B + pC). Now let Xa’ stand for the sum of all the numbers a which are positive and less than 4p, and let =f’, Sy’ have analogous meanings. Moreover put iy = 4 (SB + Bef — 23.0’) =} (Za +39! + By’) — Sa =1(2n/ + Da’ — 228) = (p? -1)—- =, 1 Stee AONE dr’. Then the expression 65 (tp) (G—se = = Sip) nit i (Sa! + p&" + p*2r’) 3 2 = i (77) + pm, + pm). Therefore 3 2 (My + pm, + pe) = (2 + PZ + p?Z) (A + p*B + pC). In exactly the same way, 3 2 4 (1) + p?mM, + PMs) = (Zp + p22 + PZ.) (A + pb + pC); and since 3? p (mot Mm +I) =O= (242, +2)(A + B+ C) 37” we have oh Mm = Az,+ Bz,+ Cz, 2 a m =Az,+ Bz+ Cz, 3 2 ae M,=AZ,+ Bz, + CZ,. The solution of these equations gives 2, 2, 2. without ambiguity in terms of m, mm, m, A, B, C. If we put A= A’?+ B+ @?— BC-—CA —AB, ae KUMMER’S CRITERION. 297 an essentially positive quantity, it is found without difficulty that = Am,+Cm,+ Bm,=(C— A) m+ (B - A) m, pA 37? pA B A Cm,=(C— A B 3321 = Bry + Am + Cm, =(C— A)m, + (B— A) mo, BO ea = Omg + Bm, + Am, = (C— A) m,+(B—A)m,. Hence also PP (e— 4) =(C+-A — 2B)m—(A + B-20) m, = BA (i—%)=(C+4-2B)m—(A + B—2C) ma, BO Ge ~)=(C+A—2B)m,—(A + B—2C) my. = Now the quantities A—B, A—C, A+B-—2C, C+ A-2B are all positive. We have, in fact, T 1 1 1 riage the Gnd Tee Gl ae hina 1 =(1+5 +P SPY JEG, pp k where / assumes all odd values prime to p: hence Pe iia Py eh OP . Ea es = i ees hc +B+0); 2 2 therefore A+B40=7 (1-4)

1: therefore B+ C <4, and a fortiors 1 B<4, (<4; consequently A~B>%, A-C>3, A+B-2C0>4, C+A-2B>14. Since m,+m,+m,=0, one of the three numbers m, m,, m, must be greater than either of the rest numerically. Suppose mm, is positive, and the greatest numerically; then m,, m, are both negative, and it follows from the equations previously obtained that 2, 2 — 4%, 2 — 2 are all positive. In this case, then, z is the root which lies between 4/p and 2s/p. Similarly if m, is negative, 15—2 pits: CYCLOTOMY. and numerically greater than either m, or m,, the quantities Zo 24 —%, %— 2 are all negative, and z lies between — »/p and —2/p. In the same way if m, 1s numerically greater than m,) or m,, the limits of z, are determined: while if m, 1s numerically greater than m) or 7, the root z,1s known. In every case, therefore, we have a criterion to distinguish one of the three roots 2, %, 22; and since they are all expressible as rational functions of any one of them, the determination may be completely effected. - | It should, be observed, however, as Kummer himself remarks, that the criterion thus obtained is not really what is required. The calculation of m), mm, m, when p is large, is very tedious, and these numbers are not connected with the value of p in any obvious or essential way. Kummer has found by actual calculation that the limits of z, are —2/p and —/p for p=97, 139, 151, 199, 211, 331, 433; —j/p and +/p for p=13, 19, 37, 61, 109, 157, 193, 241, 283, 367, 373, 379, 397, 487; +/p and + 2./p for p=7, 31, 48, 67, 73, 79, 103, 127, 163, 181, 223, 229,27), 277, 301 3loeoae 349, 409, 421, 439, 457, 463, 499. It is very curious that the proportion of primes in the different classes is nearly that of 1:2:3, and it is much to be desired that some simple method of discriminating the classes might be dis- covered. 198. The present chapter contains only a mere outline of a very extensive theory, which has not yet been by any means completed. Its importance will become even more evident in connexion with the theory of higher congruences and of algebraical integers in general; but before we enter upon this, there are various problems which may be more conveniently discussed at this stage, and among them is that of determining the number of classes of binary quadratic forms for a given determinant. To this the next chapter will be devoted. 229 AUTHORITIES. In addition to the references already given, the following will be found useful :— Gauss: Disg. Arith. Arts. 335--366. Summatio quarumdam. serierum singularium (Comm. Soc. Reg. Gotting. 1811). DrricHLtet: Ueber eine neue Anwendung bestimmter Integrale auf die Summation endlicher oder unendlicher Reithen (Abhand. d. koénigl. Preuss. Akad. d. Wissensch. von 1835, p. 391). EISENSTEIN: Deitrige zur Kreistheilung (Crelle, xxvii. (1844), p. 269). Allgemeine Untersuchungen. tiber die Formen dritten Grades mit drei Varia- beln welche der Kreistheilung thre Entstehung verdanken (Crelle, xxviii. (1844), DaefOoXe1X. 0.719): Ueber die Irreductibilitit und einige andere Higenschaften der Gleichung, von welcher die Theilung der ganzen Lemmniscate abhdngt (Crelle, xxxix. (1850), p. 160). KRONECKER : Sur les facteurs wréductibles de Cexpression x” — 1 (Liouy. xix. (1854), p. 177). KumMER: De residuis cubicis disquisitiones nonnulle analytice (Crelle, Xxxli. (1846), p. 341). Sraupt (v.): Ueber die Functionen Y und Z, welche der Gleichung 4 (a2? -1)/(#-1)=Y*+p7? Genitige leisten, wo p eine Primzahl der Form 4k+1 ist (Crelle, lxvii. (1867), p. 205). _ Stievtses : Contribution a la théorie des résidus cubiques et biquadratiques (Archives Néerlandaises, xvii. (1883), p. 358). This is based upon Gauss’s memoirs on biquadratic residues, which will be considered in a subsequent chapter. Bachmann’s lectures on cyclotomy (Die Lehre von der Kreisthedung und thre Beziehungen zur Zahlentheorte, Leipzig, 1872) give an exceedingly clear and interesting account of the subject, with numerous references to the original sources. CHAPTER VIIL Determination of the Number of Properly Primitive Classes for a given Determinant. 199. THE second volume of Gauss’s collected works contains (p. 269) a remarkable unfinished memoir entitled De nexu inter ~ multitudinem classium, in quas forme binarie secund: gradus distribuuntur, earumque determinantem. From this it appears that Gauss succeeded in determining the number of classes belonging to a given determinant both for’ definite and for im- definite forms; and with regard to definite forms it is possible to make out the method which was actually adopted. Gauss never published his investigation; in fact the first published demonstration is that of Dirichlet in his Recherches sur diverses applications, etc. already referred to. Dirichlet’s analysis will be considered later on; in the present chapter an attempt will be made to present Gauss’s method in a simplified form, and at the same time to avoid the objections on the score of deficiency in’ rigour, to which, as pointed out by Dedekind in his editorial notes on the memoir, Gauss’s deduction appears to be lable. The materials for this simplification are mainly derived from the above-mentioned notes of Dedekind, and from Dirichlet’s great memoir. The principle of the method consists in assuming a number m which is prime to 2D, and constructing two different expressions for the asymptotic value of the average number of representations of m by properly primitive forms of determinant D: that is to say, expressions which approximate to the true value when m is very large. . One of these expressions involves as a factor h, the number of properly primitive classes, while the other does not; and by a NUMBER OF CLASSES. Boot comparison of the two expressions, we obtain h in the form of a semi-convergent infinite series. The semi-convergent character of this series is the principal difficulty of the investigation. 200. Definite and indefinite forms will have to be considered separately ; we shall begin with the former. Let f= ax + 2bxy + cy? be a positive properly primitive form of determinant D=b?—ac=-—A, a negative integer. It is always possible to assign values to # and y, say «=&, y =, prime to each other, and such that wv =f (E, 9) = a& + 2bEn + cn? may be prime to 2A (cf. Art. 127), We may suppose that @ is prime to 2A: because if this were not the case we could find integers &’, 7’, such that &n’ — &=1, and then the substitution (e 4 would transform f/ into an equi- valent form /’ = (qa’, b’, c’) with its first coefficient prime to 2A. It is clear that if f (é, n) 1s prime to 2A, so is f (2Au+€, 2Av+y), where w, v are any integers. We will suppose that &, 7 are both less than 24 and not negative. We have a(a&-+ 2bEn + cn?) = (aE + bn)? + An’, and this is prime to 2A if a&+ by is so. Now let 8 be any one of the ¢(2A) numbers less than 2A and prime to it; then if we put a& + bn = B (mod 2A),, any value of 7 comprised in the set 0, 1, 2,...(2A —1) is associated by means of this congruence with a determinate value of &, which is less than 2A and not negative. Since each number £ thus gives rise to 2A pairs (&, 7), the total number of sets (&, 7) for which 0<&< 2A, 0<<2A, and F(& 7) is prime to 2A, is 2A¢ (2A). The points whose rectangular coordinates are (2Au + &, 2Av+7) form a net, every mesh of which is a square of area 44%, Alto- gether we have 24¢(2A) distinct nets, the nodes of which corre- spond to values of (#, y) which make f prime to 2A. Zon DETERMINATION OF THE 201. Now consider the ellipse represented by the equation ax? + 2bay + cy? =m where m is a large positive integer. The area of this ellipse is am/s/A, and hence the number of nodes within the curve which belong to the aforesaid networks is asymptotically mm 2Ad(2A)_ amd (2d), NTN tat 4A coe Age A Tae that is to say, its value is wmd(2A)/2AV/A+e, where e/m ulti- mately vanishes, when m increases indefinitely. Suppose that m,, 7.,...m, are the positive integers less than m and prime to 2A; and let @(m;) denote the number of distinct representations of m; by the form (a, b, c); then the number of which the asymptotic value has just been found is rigorously 8 (m,) + 8 (m.) +...+ 8 (m,). If we take a representative form from each of the positive properly primitive classes, say fi, fo,...f,, and if we construct the ellipses if =, ye ts Si =™M, (all of which have the same area wm/,/A), then the number of nodes included by all the ellipses, counting each node once for every ellipse within which it lies, is asymptotically ham¢ (2A) 2A/A On the other hand, this number is rigorously qa 1 ie meet 22 Cer) (= 1, 2, os where 6; (m;) denotes the number of distinct representations of m; by the form /;. If we write W(m) for uw, the number of integers less than m and prime to 2A, we have pri iia (25 ya wim) 2AV/A — w(m) ultimately, when m 1s infinite. >A; (mz), Now it is easily seen that v(m) _ $(24) , ane ete WAAGan NUMBER OF CLASSES. Oo hence ultimately, hi HESS Bee Fam) and we have to calculate the value of the expression on the right hand. DS Gy (Wei )ee teste meas (A), -202. In order to find representations of m; by forms of determinant D, we have first to solve the congruence n? = D (mod m;) and then to every solution will correspond a form (m;, 7, 1) which belongs to a class every form of which will represent m;,. Each solution of the congruence gives rise to a group of primitive representations by each form of the corresponding class ; and the number of these groups is equal to the number of the solutions of the congruence. (Cf. Arts. 59, 90.) But by Art. 35, if p, q, 7... are the ¢ different prime factors of m;, which by supposition are all odd, the number of solutions is 0 or 2° according as D is not or isa quadratic residue of each of the primes, In every case, the number of solutions may be written in the form {1+ (D|p)} (1+ |g}... =I (1+ (| py}, where the product extends to all prime factors of m;. Call this number y (m;); then, making use of Jacobi’s extension of Legendre’s symbol, we have x (m:)=1+(D|p)+ (DI q)+-.. - — +(D|\ pq) +... == (D' 6), where the summation extends to all divisors of m; which involve no square factor. If e¢ is the number of solutions of the Pellian equation T?— DU?=1, each form by which m; can be represented at all gives rise to e distinct primitive representations, which form a group. When D is negative e=2, except when D=—1, in which case e= 4. Suppose now that A? is any square divisor of m;. Then the congruence n? = D (mod m;/X*), has y (m;/2) solutions, each of which gives e primitive representa- tions of m;/d2, and therefore e representations of m; for which du (a, y)=2. 234 DETERMINATION OF THE Now if 8” is any divisor of m;/X? which has no square factor, x (mi) = = D8") = = (D|8"’), 7) =1. Every divisor of m; can be expressed in the form 6A”, where 6” involves no square factor, and therefore, on the whole, the number of representations of m;, both primitive and derived, is c= (D/6), where the summation extends to all divisors of m,. since (D But this is also what was previously denoted by 0, (1) + 6, (m;) +e + 0, (m;), so that the double sum which occurs in equation (A) may be written S20; (m;) = eS > (D|6) eee cece veces ve cceces (B). 203. Henceforward we shall write 2 instead of e¢, it being understood that 2 must be replaced by 4 when D=—1. Every number which is less than m and prime to 2A will occur in (B) as a divisor 6. Let 1 be any one of these; then the symbol (D|n) will occur in (B) as often as there are multiples of n which are less than m and prime to 2A. Let [m/n] be the integer next less than m/n; then the number of times (D|n) will occur is wv ([m/n]), or more simply y(m/n), if we agree to take only the integral part of m/n. | Hence (B) is transformed into XO, (m;) = 22h (m/n) (D|n), and (A) becomes har ab (mn) Tm ore) ayepcce CHAO sacri merck i 12" GF VA ho 8 “St ay (1) ( | ) ( ) where the summation extends to all numbers n which are less than m and prime to 2A, It is to be observed that 22a (m/n)(D|n) is rigorously equal to the total number of representations of the numbers m,, m.,...m, by properly primitive forms of determinant D; so that the expression on the right hand in (C) is strictly the average number of repre- sentations for one of these numbers. NUMBER OF CLASSES. 235 So long as m is finite, it does not matter in what order the terms of the sum are arranged; and even when m becomes infinite the series must have a determinate value however the terms are taken, provided that none of them are omitted. We shall effect the summation by supposing that the terms are arranged so that m increases continually as we go from left to right. When m becomes indefinitely large, the sum becomes an infinite series, which may be divided into two parts, for the first of which n is very small compared with m, while for the second it is not. So long as n is small compared with m, we have asymp- totically (m= BOD) yp (mjn) = SE), | w(mjn) _ 1 and hence ‘ie Rehm: Thus the first part of the series reduces to S = (D|n) (i orgy =o lt 0). For the remaining part of the series we may suppose Lt. (m/n) < M, where M is-some finite quantity ; hence for the residue aati i 2 VD Sah Ge aeowe It follows from Art. 46 that the sum of every 46(A) con- secutive terms of }(D|n), in the order written, is zero: hence the limit of %(Dj|n), although indeterminate, is not infinite; and since >(m) increases without limit, this remaining part of the series ultimately vanishes, and we have ultimately har AN or ees = C1) T =e = *(D\n) 230 DETERMINATION OF THE where, for convenience of calculation, the terms are to be written in such an order that m increases continually, and I: (D | n) really stands for Lit. se (D|n), where ” assumes all positive integral values which are prime to 2A, and such that Lt (n/m) =0, when m is increased without limit. If D=—1, we must write Col h=— 2% (—1|n). As a verification, we know that in this case h=1,s0 that we should have 4, i ak al 1=-(1-g4+g-gte.), the truth of which is well known. 204. We will now suppose that D is positive. In this case the equation #?— Du?=1 has an infinite number of solutions, so that every root of the congruence n?=D (mod m) leads to an infinite number of representations of m by one and the same form of a particular class. It is possible, however, to assign certain conditions of inequality by means of which one of these representations may be isolated from the rest. It has already been proved (Art. 89) that if v=& y=7 gives a representation of m by the properly primitive form (a, 0, c), then. the complete set of representations may be obtained from a= t& — wu (bE + cn), y=tn + u(aé + by), where (¢, w) 1s any integral solution of @— Dw? =1. This leads to ax + (b+ V/D)y=(¢ + u/D) faé + (64+ VD) 7} =+ (2+ Uy D)" a& + (0 + VD) 1}, ax+(b—D) y= +4 (T— Uy DY {ak + (b— VD) 7}, where 7’, U have their usual meanings, and n is any real integer. We may suppose that a is positive, because the class to which (a, b,c) belongs will certainly contain some forms with a positive first coefficient, and it is enough to consider representations by any one form of the class. NUMBER OF CLASSES. 237 The different values of av+(b+/D)y, taken without regard to sign, form a geometrical progression, extending to infinity both ways, of which the common ratio is 7+ U)/D. Writing @ for this ratio, it will be possible, in one way only, to choose the sign of the ambiguity, and an integral value of n, so that Vam axv+(b—V/D)y>0“Vam; and by combining these results we infer successively | TN, and 6 fax +(b— VD) y} — 07 {aw + (b+ VD) y} > 0, or (0 — 6) (ax + by) — (0 + 6) y/D> 0, which reduces to U (aw + by) — Ty > 0. Conversely, if the conditions . ap; U (ax + by) — Ty > 0, are satisfied, it will follow that Vam by > 0, az +(b+\D)y>ax+(b—VD)y> am az+(b+V/D)y’ whence ax +(b+/D)y>Vam; while since 6 fax +(b—V/D) y} — > fax + (b+ VD) y} > 0, we have, by multiplying by 6 {az+(b+/D) y}, which is positive, Pam — fav+(b+/D) y} >0, and therefore az +(b+/D)y < OVam. 205. Consider, now, the hyperbolic sector enclosed by ax’? + 2bay + cy? =m, y= 0 U (az + by) — Ty =0. 238 DETERMINATION OF THE By changing to polar coordinates, it is easily found that the area of the sector is ee | dé ~ 2 J9 acos? 6+ 2b cos @ sin 6 +c sin? 8’ where a is the least positive angle for which U (acota+b)—T=0. m " acobO+b+/D |* 4D | °? acot@+b—VD |, m T+U/D =. log wa Ae PLSD Hence A= =27D log (7+ Uy/D), since 12 DU | The argument now proceeds exactly as in the case of definite forms, except that instead of a number of equal ellipses we have a number of equal hyperbolic sectors. Thus equation (A) of Art. 201 is eae by nat log (2+ Uv D)= Lt ne em) eae (m;), and the final result is a oe 1 melon Ties WMD mer eae where h is the number of properly primitive classes of determinant D, and the summation applies to all positive integers prime to 2D. As a verification, suppose D=3; here h=2, the representative . forms being (1, 0, —3), (—1, 0, 3). We have 7=2, U=1, and the series > ; (D|n) is 1 ofc pie eit eli i Magee 5 ie 7 eS ee 7 aes re inh Sa 1 —Vane1 19n4+5 12n+7 Tan411\ It ought, therefore, to be true that 1 Spy Ut ee and this may be verified as follows. NUMBER OF CLASSES. 239 Putting el} = 9, we have x ep ie Li pa\ 1 Uh 1 1 — an pore ee Pero: pt — 3 — — pra a — dL Amd Lec 1 2-11 =2(1—p){o— 5 pia fo —7 Pe" eects Suppose w=7p; then log (7 7) =2ip (1p) -3-7ta-| 1—ip 1+7%9’ = 219 (1 — p)S. Now (1+2p)(1—tp?) 2+(204+1)2 2-3 1 : (l—ip)(1+ip*?) 2—(294+1)t 243 (24+1/3)’ and 2ip (1 — p) = 21 (2p +1) = — 2/8; therefore — 28/3 = log (2 + 1/3)? = — 2 log (2 + 1/3), 1 or S= V3 log (2 + »/3), which is right. 206. It will now be shown that the series ies : (D|n), upon which the determination of h has been made to depend, may be expressed in finite terms. It will be supposed that D is not divisible by any square, since by Art. 151 the class-number for a determinant DS? may be deduced from that for the deter- minant D. We shall therefore have to consider the cases when D=+P or +2P, P being the product of different positive odd primes; it will further be necessary to distinguish each case according as P=1 or 3(mod 4). Altogether, then, there will be eight different cases. I. Suppose D=—P=1 (mod 4). By the generalized law of reciprocity (D|n)=(—P|n)= (nl P), Praha H=%- (n|P) 240 DETERMINATION OF THE Since (n’| P) =(n| P) if n’=n (mod 2P), and (n’| P) = —(n| P) if n’=—n (mod 2P), it follows that 1 ih 1 i 2P—y'9P+y 4P—v' 4P+y where the summation applies to all odd numbers v which are less than P and prime to it. H=3(|P) {i — Now by logarithmic differentiation of lea2 2 we find that Con Aer teas eC 7 ere hence H=5 — ae (v| P) cot 5 This, as it stands, is a finite expression: 16 may, however, be transformed in such a way that the circular functions disappear, and are replaced by purely arithmetical functions. We have v= P—2y, where pw is an integer prime to P and less than 4P; moreover (p| P) = (= 2n)| P) =- (| P) (|B); therefore H=-—- ae pi? | P) > (u| P) tan oS a 2|P)E(u| PS, where r=e?"/P, and the sum is taken for w=1, 2, 3...4(P—1) with the convention that (w|P)=0 when yw is not prime to P. If we write P—w for p, the Sg eessuan remains unaltered, so that =B@lP2OlP) Go. Biber boo cheeg Wena Now if @ 1s any root of the equation w? —1=0, Doh oyaba ra o+l l+o =o —w' + w?—...—@P ea = (ead eal ae : i hence H= a (2|P) & (-1)7 (| P)r. A, @ NUMBER OF CLASSES. 241 But by Art. 191 S(a| Py =(a| P) S| Py A = (a|P)i/P; sos pat therefore H=77p(2|P) > (—1)*(a|P). This may be further simplified: for if a is even, we may put Cade whence (2| P)(—1)* (a| P) = (a’| P); while if a is odd, we may put a eet Le and then (2|P) (—1)* (al P) = (a’| P) as before. The values of a’ in each case are 1, 2, 3...4(P—-1): hat finall fry eke Wag so that finally > phyu2 BL). The number of classes is therefore na PINE 3 (a'| P): or, in words; When D=—P=1 (mod 4), where P involves no square factor, the number of properly primitive classes for the determinant D is equal to the excess of the number of positive integers a’, less than 3P, for which (a’|P)=+1 over the number of those for which (a’|P)=—1,; 2 being understood that (a’|P)=0 when a is not prime to P. When P is a prime of the form 4n + 3, the number of properly primitive classes is simply the excess of the number of quadratic residues of P contained in the series 1, 2,3...4(P—-1) above the number of non-residues. For instance, if P = 11, the residues are 1, 3, 4, 5, while there is only one non-residue, namely 2; hence h=4—1=8, which is right, the positive properly primitive classes being represented by m, 0, 11), (3, + 1, 4). M. 16 242 DETERMINATION OF THE II. Let D=—P=3 (mod 4). In this case na |) =(—1)?”—) (n| P), so that n-3 "lp 1 1 1 1 — — ]}e—-) = eo Ss ee = see es a) onbeats 2P+y wotiwat spu- 1)!*—-) (v| P) cosec 5p ee el ee at ee 1)" («| P) sec pio Writing w= 2’, or w= P— 2p’ en as jis even or odd, this easily reduces to 2Qm’a se > (u'| P) sec “p> where p’, like w, assumes the values 1, 2, 3...4(P — 1). Omitting the accent, and introducing r, we have ae > (| P) soo “A Tp P) seo =e “Bone Now, if w is any complex root of #? —1=0, 0) 1 1 Wo odie =1l+ot+ ot+o°'t+o%+...¢ 0? 14+ 07?"; therefore H=5 — re EE) phere Teel or since PHU ed UN este Wed tL ag Verena shen Te hele A Xv A we have jal = TP ite R), where the values of «” are 1, 2, 3...4(P—1). Finally es ae For example let P=77. Here (@”| P)=0 when a” =7, 11, 14, (a” | Py =1)- when a= 1,74) G6, 9,°10,/138;°16, "16" 17, 19 swe (a”|P) =—1 for the remaining six values of a”; hence h=2(10—6)=8. NUMBER OF CLASSES. 243 The representative forms for the determinant —77 are in fact (1, 0, 7); (7, 0, 11), (2, 1, 39), (9, 2, 9), (8, £1, 26), ire lel), eight in all. III. Let D=—2P=2 (mod 8). Here (D|n) =(—1)#”— (n|P), and frp oe Besen(P). Observing that, if n is odd, (4P + n)?—n?=16P? + 8Pn = +8 (mod 16), we see that the expression H may be written in the form ee eS Aa 1 1 H=3(- I 01P) + gp apy -BP 1 7S (-1)30— Lae >= (— 1) (v| P) cosec £P [v=1, 3, 5...(2P—1)]. By adding the first term to the last, the second to the last but me, and so on, this becomes Fig eo. Pe ieee) Soe ene) 7 mera 2.) = = 4 P 4P P 4P pop Gale whale | Ce (aals oe > pa Ly ap sin Oe mee us Cf 22) (P—7) + 0 cos 2P spo CT Tg : ees? Now if 1s an integer, it is evident that (eee 4 elroy (feed) 705 es +v) 7 lso cos “7p ApS ee ae) gre pe CEA yp) a Site ery meet 16—2 244 DETERMINATION OF THE hence the above expression may be written . (P+1)7 . (P-3)r eens Esp) ATA Ria («= =| P) —_ 2P 4 op Et DT pe (P—3)a Dis 2P . (P+5)a P+5 | P) me ( 4, ae (P+ 5) Zits t. YEE sin > The term = has a coefficient +1 or —1 according as COS 5 P=7 or 3 (mod 8): hence we have, by rearranging the terms, | m2 sin © H = "8" (2|P)¥ (19 (uP) cos [wu =1, 2,3...4(P —-1)]. If we change w into P —p, (— 1)" becomes (— 1), and (| P) becomes —(y|P), while the trigonometrical factor is unaltered: hence EATS, Si = TF @|P)¥ (-1) AP) —— cos —— ts atl Sine Re 1), Writing 2 for the even values of A, and P—2y for the odd values, this becomes, after some easy reductions, similar to those employed in the previous cases, Ajmal __ 72 P gua ET P [w=1, 2, 3...4(P—-1)]. If we write P—y for yw, this expression is unaltered, so that finally sin = — 72 3 @lP) —— wil cos eos XT JE 2X1 T NUMBER OF CLASSES. 245 =H rr — r- where 7 = @2ni/P , and the values of \ are 1, 2, 3...(P—1). = (A|P) a Now if » is any complex root of the equation #? —1=0, @o— ow Faernae ae On OR aa TO Me Ope Oa — cea) where the series ends with + w?-? | w-?+*, Hence = — uN y(A|P) {r—-7 7 -— + rF+...} ry: UGE) a Ry Rye a where the series ends with + (P—2\|P), The value of h is therefore h=2{1—(5|P)+(9|P)—(13|P)+... + (P—2|P)}, where it must be remembered that (m|P) is to be put equal to zero when m is not prime to P. For example let D=—94=—2.47. We find from Gauss’s- table of quadratic characters that 1, 9, 17, 25. R47, 38, 41 N47, 5, 18, 29, 45 N47, 21, 37 R47, hence h=2(8—4)=8. Or again, if D =— 30 =— 2.15, we have h=2 {1 —(5|15) +(9|15) —(13]15)} =2{1-0+0+1}=4. The expression for the class-number may be reduced to the form h=22% (a|P), where the summation applies to all integers a for which 4P (A|P) = 0, we obtain T= [Lea S EP) (OLD) =< ei Pea P yy, and therefore Be 2) (Oey Oy =... This is the same formula as in last case; the only difference is that the number of terms is +(P —1) instead of }(P +1). By reasoning exactly similar to that employed for case IIL, the expression for h may be reduced to the form h=2 (3 (a\P)— 3 (B\P)}, where the summations apply to all values of a and @ such that O (a|P) is used to express that the sum is to p pP qe 8 S be taken for all integral values of a such that (Cf. Dirichlet, Zahlentheorie, p. 275.) 207. The remaining four cases relate to a positive deter- minant. VY. Let D=P=1 (mod 4). Here H=%(D|n) == (n|P) Multiply both sides by VP =S(a\P)r fee 8 9d BA Pe AF 248 DETERMINATION OF THE then, observing that } (A|P)7r"™ = (n|P)& (A|P) 7", we have HP =%(A|P) {n+ BI Th 2 Now if v= pe®, when p is positive and less than 1, 1 He 1l+e : ae, cos 0 + p ere =p ~ 4 © 1 — 2 cos 6 + p? and the limit of the real part of this when p=1 is 1 1 1+cos@ 1 4°81 —cosd 4 Since H,/P is real, it is unnecessary to consider the limiting value of the imaginary part, and we have 0g cots = 5 log cobs NIT H= 575% (a1) log cot 5 | Consequently hlog (T+ Un/D) =2H/P = %(A|P) log | cot 2 I] cot Fy ao oaleae ea I cot where the product applies to all values of a between 0 and P for which (a|P)=1, and to all values of 8 between 0 and P for which (SI em Ns | If we write a= P—da’, and B=P-—{’, when a or B exceeds 4P, the formula is reduced to h log (T+ ON) Oe eat ae with the conditions O (| P) 7s PEERS OXI) jp—zmszom—..}, and Hy/P is equal to the real part of eC Ue tues eae ae 3 5 mw LAr where Sle Ab ee me As before, this is — > (r|P) log cot (F Fale and therefore, with the same notation as before, tan ($+) hlog (7+ U/D) = oeeeaeerre tan (7+ >) 249 This may be written in a somewhat simpler form if we observe that the series of numbers P+ 4a are congruent (mod 4P) to all the integers m between 0 and 4P which satisfy the conditions m=3 (mod 4), (m|P)=1; while the integers -P+4 are con- gruent (mod 4P) to all the integers n between 0 and 4P which satisfy the conditions n=1 (mod 4), (n P)=—1. Hence if b stands for “any one of the odd integers between 0 and 4P for which (P|b)=—1, this will comprise all the numbers m and n, so that wT OT " tan (G+) | baee B=) = Taal 27) i ee TL, tan me II | tan Ge P) 4 bar = IT | tan — AP|" Hence hlog (1+ UD) = log IT | tan “lr Thus when P = 3, the values of b are 5 and 7; whence br etme et bes et ae I tan 7p = tan 12 tan yg =— (2+ v3) = —(7'+ Un/3)’, so that h = 2. 250 DETERMINATION OF THE VIL. D =2P = 2 (mod 8). In this caso = H=¥(—1)" (n|P) V¥P=2%(|P)r%, and therefore HP =3 (| P) [rape St Now it is easily proved that if 1+2 4a — pmi/4 0 2 Cas and |v! <1, ie is fork 1 gg (1+ On) (1+ 0-2) game esiistonrs cat Gober SPs 2,/2, °8 Qe 6x) (1 — Oe), Putting «=r, and attending only to the real part of the logarithm, we find that AT | 7 AT 9 cot ("+ g) oot (BB) II cot (97 +7) oot (4 -7)| Il cot (SF +7) cot (Sr —3)| This may be reduced to the form h log (11+ Un D) = log TI with the conditions ORS le (2P|b)=—1. Thus if P=5, the values of b are 7, 11, 17, 19, 21, 23, 29, 33; and hence > (A|P) log ie and consequently hlog (T' + UD) = log tan ber SP|’ (hn dilgces CLOT ay hlog (7' + Un/D) = log (tan* 40 tan? 0 tan do te 40 a) = log (19 + 610), and therefore h= 2, It may be observed that 4P—b is congruent (mod 8P) to a number a such that O gp?’ we may W I sin om II says [(O 9, hlog B (Dy) =— > (D,\k) log (1 — er!) [k=1, 2, 3....D,—1)], where #(D,) denotes $(7'+ Us/D,), T, U being the least positive integers such that 7? — D,U?= It is to be remembered that (D,|/) is to be put equal to zero when k is not prime to D). As in Arts. 150, 151, it may be proved that if aa Dy’, where D, is a fundamental discriminant, the number of primitive classes for the determinant D’ is log E (Dy) 1 h’=hQ log B (Dy 1 {1— - 7 Dig where / is the number of primitive classes for the determinant D,, and the product applies to all prime factors of Q which do not divide D,. These results are taken from Kronecker’s researches on elliptic functions (Zur Theorie der elliptischen Functionen, Berlin Sitz- ungsberichte for April, 1885, p. 768). It is easy enough to prove that they are m agreement with the ordinary theory; the simplification which is gained is obvious. It may be specially noticed that in the modified theory improperly primitive forms do not occur. The discussion of Kronecker’s very important memoirs must be, for the present, postponed; and in the rest of this chapter only quadratic forms of the ordinary type will be considered. 210. It is unnecessary to enlarge upon the very remarkable character of the foregoing investigation, whether it be regarded as the direct determination of the class-number, or as the ex- 254 DETERMINATION OF THE pression, in terms of the class-number, of the sums of certain infinite series. There are, however, two points which deserve to be emphasized. The first of these relates to the distribution of the residues and non-residues of a given number. For simplicity take the case of a prime negative determinant D=— p, where p is a prime of the form 4n+3. The formula (p. 247) 4 h= > (a | p » combined with the remark that h is necessarily a positive integer, leads to the conclusion that in the series 1, 2, 3...4(p-1), there are more quadratic residues of p than non-residues. It does not appear that any independent proof of this proposition has ever been discovered. If any such proof could be found, it is not impossible that it might lead to a determination of h without the use of infinite series. Similar remarks apply to the other formule for negative determinants. The other point to be noticed is that when D is positive we are able to construct a solution of the Pellian equation by means of trigonometrical formulee; the solution thus obtained being not the fundamental solution, but one of which the place in the series of solutions depends upon the value of h. Dirichlet has verified a posteriori that the trigonometrical expressions which occur in the determination of h do in fact lead to integral solutions of the Pellian equation. For the complete discussion the reader is referred to his memoir (Sur la maniére de résoudre l’équation t? — pu? = 1 au moyen des fonctions circulatres, Crelle, XVII. (1837), p. 286); it will be sufficient to consider here, by way of illustration, the case when D= p, a prime of the form 4n + 1, so that a aid h log (T' + Us/p) = log II tan — ar |" fan — With the notation of Chap. VI, we have cP —] c—1 Y+Z4/p=2I1(@—1r%), Y-Z/p = 211 (a — 1°), where r= er /P, 4 = Y?— 7, NUMBER OF CLASSES. 255 Let f, g be the numerical values of Y, Z when «=1: then tial ot ae Hence f is a multiple of p, and if we put t Seal st BSE Reo Be then /’, g’ will be integers, and Af em HU Similarly, iff’, g” are the values of Y and Z when «=—1, f? we pg’? =A Hence ie OND AG het pas UE Gi 19) TE ie") f+ INp ff’ -o Np Wart) (+7)? II tan bm l that is, een Marmmg yen oof) noi B) - (fp = D00. Aa ait vp) . Now if p=1 (mod 8) it is easily seen that /”, 9’, f’, g” are all even; for if we suppose g’ to be odd, g”=1 (nod 8), and f?=pg? —4=5 (mod 8) which is impossible ; and similarly for /’, g”. Hence f/f” — pg’q” and f’q” 9 ‘are both multiples of 4, and II tan 2 as = (t+ w/py, TI tan — p where (¢’, wv’) is an integral solution of ¢? — pu? =—1. Consequently (¢ +u/p)? =t+ unr/p, where (¢, %) is an integral solution of ? — pw? = 1. Next suppose p= 5 (mod 8). ‘Then from the equations Vp (f' + op) = 211 (1 — 1%), (f" + op) = 2U 1 + 7%), it follows that Vp (f +9/Vp) (f" + op) = 401 (1 — 17) = Aal (1 = rey, 256 DETERMINATION OF THE since (2|)=—1; therefore Vp (f+ 9'Vp) Sf! + Vp) = 2p f — hy) and hence —2(f" + 9p) =(f — 9 py: (f' — g'lpy Ff +9py _ (f + —e 16 2 It may easily be shown that whether f”, g” are both even or both odd, 1 ” 3 where (¢, w) is an integral solution of #—pu?=1. This is obvious when /”, g” are both even: if they are both odd, it follows from the fact that, to modulus 8, : St’ (ff? + 8p9") =f’ 14+ 38.5.1) =0, and similarly gf (po? + 8f”) = 9’ (5.14+8.1)=0. It is evident that f’, g” cannot be one even and the other odd; hence in every case when p= 1 (mod 4) where (¢, wu) is an integral solution of ?— pw=1. It may be observed that when p=1 (mod 8), f”=2 and g’ =0; consequently Y-— 2 and Z involve the factor «+1. For instance, when p= 17, Y—-—2=2(@4+1) (20% — a + 62 + a + 3a? + 441) =a (#+1) (a+ 2a 41) (20 — 2? +4 2” +1), 4=u(ex+1)(#+1)(# 41). It may also be noticed that it follows from the foregoing analysis that when p=5 (mod 8), and the equation ?— pw? = 4 does not admit of odd solutions, the class-number is divisible by 3. This agrees with the results of Art. 153. CHAPTER IX. Applications of the Theory of Quadratic Forms. 211. IN order to acquire complete familiarity with the theory of quadratic forms, it is indispensable to work out a considerable nuinber of special cases. The student should have no difficulty in doing this, with the help of the examples which have been already given; and he cannot do better than draw up a complete classifi- cation for a series of positive and negative determinants, afterwards comparing his results with the tables of Gauss or Cayley. It is not easy to construct a large variety of exercises distinct from these direct applications: in fact, in the present state of the theory, every problem of a distinctly new type is apt to present unex- pected difficulties, and its solution often requires the invention of new methods, and even of new principles. There are, however, a few problems of great interest to which the theory of quadratic forms has been successfully applied, and some of these will now be considered. 212. The first is the discovery of all the integral solutions, if any exist, of the general indeterminate equation h = an? + 2hary + by? + 29x + 2fy +¢=0. Following the notation usually employed in analytical geometry, we shall write caelonc A=be—-f?, B=ca-g, C=ab—h’, F=gh—af, G=hf—bg, H=fg—ch. M. Ws 258 DIOPHANTINE EQUATION Suppose, first, that neither b nor C is zero, and that C is not a negative square. Then the proposed equation may be multiplied by OC, and the result written in the form (Ca — G+ Chat by+fP=-— If we put Ca—G=X fon eta, oe (1), he+by+f=Y then whenever (2, y) is an integral solution of ¢=0, (X, Y) is an integral solution of X¥24.0Y2=—dA...2 Sel, Nee ore ae (2), and conversely, if (X, Y) is an integral solution of this equation, a, = aria pe ras une vet eet (3), CY -hX+bF toe ERC gives a solution of ¢=0, provided these values of # and y are integral. If Cand DA are both positive, there are no real solutions of (2), and the proposed equation is insoluble. If C is positive and bA negative, there will be only a limited number of integral solutions of (2), if, indeed, there are any: so that it may be discovered by a finite number of trials whether there are any integral values of « and y. If, on the other hand, C is negative, and (X,, Y,) is any integral — solution of (2), there will be associated with this an infinite numb of solutions, expressed by the formulae + sae cere HY Sty ie where (¢, w) is any integral solution of #+ Cu? =1, with ¢ positive, and the ambiguities are independent. If the upper sign is taken in each ambiguity, the corresponding values of x, y are _tX,—-ucCY,+G memairarsae to _ t(CY,-—hX,) + uC (X,+hY,) +oF ii Cee a i, Coe OF THE SECOND DEGREE. 259 With respect to the modulus bC, the residues of the values (T’,,, Un) which satisfy the equation 72+ CU,?=1 form a recurring series (see Art. 87); the same will therefore be true of the residues of the numerators of the expressions for # and y. Consequently, a limited number of trials will determine whether any of the values of « and y are integral. The process of trial has to be applied separately to each case arising from (4) by variation of sign in the ambiguities. By way of illustration, take Gauss’s example e+ 8ey + y+ 2a—4y+1=0. Here A=—36, C=—15; and it is easily seen that all the integral solutions of A? — 15 VY? = 36 are given by +X=6t, +Y=6u, where (¢, w) is any positive solution of # — 157? = 1. We have 7, = 4, U, = 1, and the series of residues of (7, U,,) to modulus 15 is (4, 1), (1, 0). The general values of , y are by (3) (Gj) w=1(2t+3), y=—2(4¢415u41); (i) v=—-1(2t-3) y=2(4t4+ 15u-1); and these are integral if we suppose that ¢ is chosen in (1) so that t=1 (mod 15), and in (11) so that t=4 (mod 15). For instance i=1;u=0 gives 2=1, y=—2; and ¢t=4, u=1 leads to e=—1, y =0 or 12. If b happens to be zero, the equation @¢=0 may be multiplied by aC, and the work proceeds as in the former case. If a, b are both zero, and h is not zero, the equation may be written (hat+f)(hyt+ 9 =So- 5 hence ch must be even, and if this condition is satisfied, then putting fg—tch=m, we break up m into the product of two factors a, 8 in all possible ways, and find by trial all the integral solutions (if any) of ha+f=a, hy+g=B and ha+f=B, hy+g=a. where a=. 17—2 260 DIOPHANTINE EQUATION Returning to the general case, suppose C=—m?, a negative square: then supposing b is not zero, the equation in X, Y is (X+mY)(X —mY)=—DbA; and this is solved by putting X+mY=a, X—-—mY=B, where a8 =—bA, and examining all the different cases to see whether integral values of X and Y can be found. If this is the case, the corresponding values of « and y have to be examined. A particular case of C being a negative square is when a or 0 is zero, or again when a and b both vanish: these cases have been already considered. Suppose, now, that C=0; then | ax + 2hey + by? =m (ax + By) where m, 4, 8 are integers; and if we put ax+ By =z, we have me + 2gxu+2fy+c=0; and hence mBz + 2gBa + 2f (2 — ax) +cB=0 or Se ac 2 (fa — 9B) The solutions, if any exist, of the congruence mBz + 2fz+cB=0 [mod 2 (fa—g8)] will make x integral; and then these solutions must be examined separately to see whether the corresponding values of y are in- tegral. Here, again, there are various special cases which may occur: it is not worth while to discuss them in detail, but the following — example may serve to illustrate the general method. Let the equation be ba + L2vy + 12y? + 4a — 2y — 85 = 0. Putting «+ 2y =z, we find that 327+ 5a —-2—85=0; hence we must have 32?—z=0 (mod 5), leading to z= 5t or 5¢+2. First, suppose z= 5t¢; then w=—15P4+t4+17 2y = 15+ 4¢-17=—1 (mod 2), OF THE SECOND DEGREE. 261 therefore t must be odd: let ¢=2u—1, then “= — 60u?+ 62u+1 y= 30uv’?—26u—3, where w may be any integer. If z= 5t+2, then e=—15?-—11t+15 sy= 1d?+16¢-13 and here again ¢ must be odd; so that putting t=2u—1, we obtain the second set of solutions x= — 60u? + 38u+11 y= 30uv-—14u—7. If, in the general case, A= 0, the equation ¢ = 0 assumes the form (ax + By +) (aa + By +7) =0, and its solutions are found by solving separately the linear in- determinate equations ax+ By+ty=0, av+P’y+y7=0. It is hardly necessary to add that if a, h, b all vanish the equation d = 0 is not of the second degree. 213. Another problem to which the theory of quadratic forms may be applied is that of finding out whether a given number is prime or composite. Theoretically the question may be answered by trying whether the number is divisible by any integer less than its square root; but when the given number is very large this method becomes impracticable. The principle of the method which we are about to explain consists in discovering by trial a quadratic form by means of which the given number m, or any multiple of m, may be represented ; the determinant of this form is a quadratic residue of m (Art. 59), and therefore of every prime factor of m. All possible factors of m must therefore belong to a certain set of linear forms (Art. 46), and it is therefore unnecessary to try any divisors not contained in the set. If another quadratic form, with a different determinant, can be found, by which m, or a multiple thereof, can be represented, the number of trial divisors may be still further reduced; and by proceeding in this way we may at last reduce the trial divisors to a sufficiently small set. 262 DISCOVERY OF Before going into the general theory, it is advisable to give an example. Let m=173279; then it will be found that 2m = 589? — 3.112. Hence 3 is a quadratic residue of m, and therefore every prime divisor of m must be of the form 12n +1 or 12n +11. Moreover /m < 417, and the only numbers below this limit, comprised in the above linear forms, and exclusive of those which are obviously composite, are 134-235,.5 AT, B95 (61S ie eso 7, 107, 109, 131, 133) 157,216 (el 7 OS isi alas 227, 229, 239, 241, 251,. 263, 277, 311, 3138, 323 337, 349, 359, 373, 383, 397. If these divisors are tried successively it will be found that 241 is a factor of m; in fact 173279 = 241. 719. Since there are 78 odd primes less than 417, the number of trial divisors is, by this method, reduced by more than one-half. It happens, in this case, that the resolution may be effected much more easily by observing that 8m =7212 — 4= 723.719 =3.. 241.719, whence m= 241.719 as before. It will often be found that the methodical application of the process leads to the required resolu- tion into factors in some such simple way as this: the advantage of the general theory is that each trial gives some information as to the character of the factors, and thus reduces the number of trial divisors. In general, if we can express any multiple of m in the form km = aa + by’, where 1t may be supposed that a and 0 are free from square factors, and wz, y are integers, one or both of which may be unity, it follows that — aba’ = b’y? (mod m), and hence that — ab is a quadratic residue of every prime factor of m. By means of decompositions of this kind, taken separately or in combination, it is usually not difficult to find a number of small quadratic residues of m; each of these imposes certain conditions upon the linear forms of the divisors of m, and the number of trial divisors is correspondingly reduced. PRIME FACTORS. 263 Suppose that, by any method, we have discovered a number D which is a quadratic residue of the odd number m; then the number of the solutions of the congruence a= D (mod m), is equal to 2 where » is the number of different prime factors of w (see Art. 35). The congruence may be solved by the method of exclusion explained in Art. 47, and then from the number of solutions we may at once infer the number of different prime factors of m. Moreover if « = x, and x = a, are two solutions, we have Ly — L,? = (@, + Lp) (&, — 2) = 0 (mod m), so that if #,+ 2, 1s not a multiple of m, it must involve a factor of m, and this can be discovered by finding the greatest common divisor of m and (#,+4,). Thus the method not only enables us to decide whether m is prime or composite, but helps us to find the factors of m when it is not prime. It should be observed, however, that when m is a power of a single prime, we are unable in this way to detect the composite character of m. Instead of solving the congruence z= D(mod m), it is usually more convenient, to find by trial all the representations, or at least all the groups of representations of m by reduced forms of determinant D. Supposing that D is negative, say D=— A, there will, in general, only be two representations (a, y), (— 7, — y) in each group; and it is obviously advantageous to choose, if possible, a value of A such that there are only a few classes in each genus. The particular genus which has to be considered can be found at once by determining the generic character of m. If possible, a value of A should be chosen for which there is only one class in each genus. Of such determinants sixty-five are known, and are given in the following table; the Roman numeral prefixed to each group denotes the number of genera. Th 2.34521. Pimeeuso 102313 Lo elGels) 22.25, 28, 37,°58. IV. 21, 24, 30, 33, 40, 42, 45, 48, 57, 60, 70, 72, 78, 85, 88, 98, 102, 112, 1380, 133, 177, 190, 232, 253. VIII. 105, 120, 165, 168, 210, 240, 273, 280, 312, 330, 345, 357, 385, 408, 462, 520, 760. XVI. 840, 1320, 1365, 1848. 264 DISCOVERY OF It is highly probable, but has not been proved, that there are no other values of A which satisfy the condition in question. 214. Gauss has explained (D. d. Arts. 323—6) a tentative method for finding all the integral solutions, if any exist, of the equation au + by? =m, in which a, b, m denote given positive integers, without any common divisor except unity. Let p be any prime which does not divide a, and let p” be the highest power of p which divides &. Take any power of p, say p*, and let 2, m2, m3, ete. be the quadratic non-residues of p*. Let the solutions of the linear congruences aztbn=am, az+bn,=m, az+bn;= m, etc. (mod p**”), be Z=%,, 2=%, 2=2%;, etc. (mod pt”). Then it is clear that if z; is a quadratic residue of p"*”, and if x? = 2; (mod p*t”), the value of Mm — ax? Baye is an integer congruent to n; (mod p“), and therefore cannot be a square. Conversely, if we suppose that Mm — ax? ala ae (mod p*), we have au = m — bn; (mod p*t’) == OSs and therefore = 2, If, then, & 6’, 6”, etc. are those of the numbers z; which are. quadratic residues of p“t’, and if + & +&, + &”, etc. are the solu- tions of the congruences e=0 #@=0, a=", etc. (mod prt”), it follows that no values of # which are congruent (mod p!*”) to any of the numbers + & + &, + &”, etc. can possibly lead to integral solutions of the proposed equation. In most cases v will be zero; and in applying the method we may put successively uw = 1, 2, 3, etc., and then for any particular value of mw it is sufficient to retain those non-residues (n;) of pp” which are residues of lower powers of p, since those which are non-residues of lower powers lead to values of « which have been already excluded. PRIME FACTORS. 265 The value of « must in any case be less than Vm/a; and by the process of exclusion just explained the number of values which have to be tried may be reduced to any extent that may be desired. As an application, let it be required to factorise 781727. It will be found that this number, which is of the form 4n +4 3, is a non-residue of 3 and a residue of 59: this agrees with the generic character of the form (38, 0, 59), for which A=177, one of the sixty-five special determinants given above. We now proceed to find all the positive integral solutions of 59a? + 3y? = 781727. It will be observed in the first place that «< 116, and that a = 1 (mod 3), or e=+1 (mod 3). Within the prescribed limits there are 77 numbers of these forms, namely eet, Omer tbo LLo; Now take the ‘excluding number’ 5; then since 59=—1 (mod 5) and 781727 = 2 (mod 5), while the non-residues of 5 are 2 and 3, we have to solve the auxiliary congruences —2+6=2, —2+9=2 (mod 5), whence 2,=4, 2, = 2 (mod 5). The first of these leads to the exclusion of all values of « which are of the forms 5n + 2. In a similar way the excluding numbers 7, 11, 13 lead to the rejection of all values of « which ave of the forms in, Tn+2; lln +1, 1ln+2; 13n, 18n+ 5, 18n4+6; after which only the following twelve remain :— 4, 11, 25, 29, 41, 71, 74, 80, 94, 95, 106, 115. It is easily seen that # cannot be a multiple of 4, so that 4 and 80 may be rejected; then by actual trial of the remaining values we find that 781727 = 59.292 +3. 4942 = 59.747+3. 391° = 59 .115?+ 3.227. Since these representations are all primitive, we conclude that 781727 is the product of three different primes. 266 DISCOVERY OF To find the actual values of the factors, we observe that since 3.4942 4+ 59. 29%= 3.2274 59.1157= it follows that 3 (494? . 115? — 22? 29°) = 0. Now 494.115 —22.29=56172, and it will be found that dv (56172, 781727) =4681. Hence also 781727 + 4681 = 167, which is a prime. The other prime factors may be found in a similar way, and the final result is that (Slizi—olroL Lore a (mod 781727), 215. In general, suppose that ax? + by? =m, as? + by? =m, where, as before, a, b are positive and prime to each other and to m. For simplicity, take a, y, x’, y’ all positive. Then a (ay! + xy) (ay’ — ay) =(y? — y?) m = 0 (mod m) Therefore one of the numbers (#y'+2’y), (ay’-a'y) is a multiple of m, or else each of them has a factor in common with m. Now ne = (aa + by) (ax? + by’) = (aun — byy’)? + ab (ay’ + a’yy; therefore, except when ab=1, (wy’+a"y) is certainly less than m, and hence dv (m, xy’ + #’y) 1s a factor of m. lf ab = 1, then a =b=1, and me = (aa! — yy’)? + (ay’ + ayy, and here again wy’ + ay 0; so that to make & positive we must take the upper sign in the ambiguity. The condition that & may be less than & gives | DUn eta Alye, and therefore DU? (&—m)>(T-1) ®, leading to 2(T-1)2&>mDU? >m(T?—1); and hence ge A, ee PRIME FACTORS. 269 Conversely, if this condition is satisfied, £’ = 7 — _DUn will be positive and less than &; so that if the equation a?— Dy?=m is capable of solution there will be a suitable value of # which is positive and less than vi meu The corresponding value of y will be less than that given by pape pale 2 2 CU —l)m 2D In the same way, if the equation that is, it will be less than ny x — Dy? =—m admits of solution, then by writing it in the form (Dy) — Da? = mD, we see as before that there will be a value of « which is positive and less than wv eke and that the corresponding positive (T+1)m value of y is less than one Suppose, now, that there are two representations e?—Dy2=m, 2«2— Dy?Z=m, such that ; (T+1)m (lL) re Oe coy eet) 0 A my | ie | Tare Pi PiPi a contribution LS VOD ESM S OI ST It follows from this that the sum above written is equal to the number of integers less than m which are divisible by one at least of the selected primes; subtracting this from m, the remainder expresses the number of integers in the series 1, 2,...m which are not divisible by any of the primes. (Compare Art. 7). For example, the number of integers not greater than 50 which are prime to 2, 3, 5, 7 is 50 — 25 — 16-10 -7+483+5+3+43+241 —l1-—1-0-0+0=12; they are, in fact, 1, 11, 18, 17, 19, 28, 29,31) 37, 41, 43; 47. It will be sometimes convenient to write P(M; Pr, Pas-+-Pr), for the number of integers, including unity, which do not exceed m, and are not divisible by any of the primes p,, po,...p,. With this notation, we have h(M; Pi, Po---Pr) =[m] i (1 = ‘een (1). It should be observed that when p,, p....p, are the different prime factors of m, this expression coincides with ¢ (m) as defined in Art. 7; and also that the value of the expression is not altered by adding to the series of selected primes any number of primes — which exceed m, because these additional primes only contribute zero terms to the sum which has to be calculated. It will now be supposed that p,, p., ps, etc. denote the successive primes in their natural order, so that pi = 2, Pp. = 3, ps = 5, and so on. Instead of f(m; p,, ps,...p,) we shall write ®(m, r), and we shall use F(x) to denote the number of primes which do not exceed a. (Observe that w need not be an integer.) THE DISTRIBUTION OF PRIMES. 275 It follows at once from the definitions that SDT (NL) oa VE vase Me eva (2), because every number, not greater than m, except unity, is divisible by one at least of the primes p,, Dats DE Gry More generally, if n is any positive integer, SCs ONG i) ad hae (3). Let pqs. be the first prime which exceeds 4/m, or which is the same thing, let a= F'(s/m); then since the only numbers in the series 1, 2,...m which are not divisible by any of the primes Pi, P2-+-Pa are unity and those primes which exceed /m but not m, and the number of these primes is F (m)— PF (/m), that is, F'(m) —a, we have It is easily seen that this relation continues to hold good, provided that a is an integer satisfying the conditions IE) ee (WT) eee tee. (CD the argument is, in fact, the same as before. For example, if m = 50, F'(/m) = F(7) =4, and ® (m, F' (W/m)) = ® (50, 4) = 12: hence F (50) =12+3=15. Thus, in general, the tabulation of primes up to p, enables us to calculate the value of F(m). The calculation of ® (m, F'(/m)) is, however, impracticable when m is very large; it is therefore necessary to-make use of a few transformations to obtain a manageable formula. | 7 We observe, in the first place, that if w, y are positive and y> 1, Biale. y Yel from which, and the identity n ¥h =a i m |*-!1 1 i(1-—)= Ul (1-— -|2 | Tl (1-—), Lm] 1 Pi brn] 1 oo Pat 1 Pi it follows that @D (m,n) = Bm, n-1)- (| ],n—1) bee (6). n Put n=a=F(/m): then by (4) and (6) F(m)=a-14 &(m, a—-1)-0(|™ | a—1). a 276 THE DISTRIBUTION OF PRIMES.’ Let a =F (4/ e ; then, in general, F (~) Hoilewue Pa * and consequently another application of (4) gives DeSean Ey Ga wiih) r(™)+a-2 = (Gl ae )+0(m, yest) Provided that #(pe,) ¥4-24 F(\/ 52) this may be again transformed in the same way into P (m)=(a- 1) + (w= 2) + (@-3) — F(™) - F( * Pa Pai + P(m, a— 2), and so on, until at last we obtain F(m)=(a—1)+ (a—2)+...+(a—v—1) } -F(™)_P( a )-.-F/ w ) Lee (7), Pa Pam Pa-v+i | + @D(m, a—v) ) | with the conditions 3 Eas) oS eee (4/ aye while the inequalities r(B)eonr147(y/), are not satisfied. Now since py_y < Pa—v41 we have Area RS) (ne Pa-v + Pa-—v+i so that if Mm #) (oe) + ak it follows a fortiort that F (eee ; ar Sal : ta-—p THE DISTRIBUTION OF PRIMES. 277 Consequently the only way in which the process of transformation can be stopped is when a— r(4/ a : yas 225) while a—v-1a-—-v—l. Thus the conditions of inequality cease to hold good when a—v=F(¥/m), or y=F (Vm) — F(y/m); and it is easily seen that they do not break down before, because ata Wits it follows a fortiort that serif) If, then, we put /'(°/n)=b, and suppose that in (7) v has its critical value (a ~ 6), we obtain finally F(m)=4 (a—b +1) (a+b —2)+ ®(m, b) . s F(S) RATT eee ene ah (8), b6+1 [o=F (Vm), b=F(/m)]. whenever come 278 THE DISTRIBUTION OF PRIMES. As an illustration of this result, suppose m=50; then =a we Dias and the formula gives F'(50)=4.3.4+4 ® (50, 2) — (#10) + #7} =6+17-—(444)=15, which is right. In applying the formula to a large number m, ® (m, b) is calculated by the repeated application of equation (6). For the details of the actual computation, the reader is referred to Meissel’s papers; the following table gives his results: n F(n) 20000 2262 100000 9592 200000 17984 300000 25997 400000 33860 500000 41538 600000 49098 700000 56543 800000 63951 900000 71274 1000000 78498 10000000 664579 100000000 5761460 219. In a memoir presented in 1850 to the Academy of St Petersburgh, Tchébicheff determined, in an explicit analytical form, a superior and an inferior limit to the number of primes between the limits a and 8, both of which are assumed to be greater than 1. Before we give an account of Tchébicheff’s investigation, it may be as well to trace the connexion of ideas by which it was probably suggested. Let 8 >a>1, and let ~ be the number of primes which exceed a but do not exceed 8; also let @(x) denote the sum of the logarithms of all the primes which do not exceed x Then 8 (8) — 8 (a) THE DISTRIBUTION OF PRIMES. 279 is the sum of the logarithms of all the primes p which satisfy the conditions a Il, ok faa pet and therefore the two sums last written have the same value. Since p was any prime whatever, the proposition follows, and we have 1 ian) i Tc) = log I [a]= = 0 (=) , m, t Me It is convenient to write wr (a) = 0 («) +O (x) +O (a) +... 1 = 20 («)'; and with this notation, we have P(e) = (=), 221. We now come to the application of Stirling’s theorem. By an analysis, which need not be reproduced here, Serret has proved (Alg. Supér. 2nd ed. p. 212, or Todhunter Int. Cale. Chap. 16) that log Ila > 4 log 27 — «+ (« + 4) log x 1 and log Iw < 4 log 27 —a+ (w@+9) loga+t so. THE DISTRIBUTION OF PRIMES. 281 In the first of these inequalities change w into (w+ 1), and subtract log (w+ 1) from both sides ; thus log IIx > $ log 2a — (a +1) +(a-+ §) log (w +1) > flog 2 + (a +1) {log (w+ 1)—1} —4 log (x +1). Now it will be found that the expression y (logy — 1) — 3 logy, increases with y, if y>2; and hence, observing that [#]+1> 4, we find that, supposing « > 1, T (x) = log I [#] > $ log 27+ a (log «—1)— $ log a. Moreover from the second inequality, since 1<[«#]<.a, and (y + $) log y—y increases with y, if y > 1, we infer that T (x) <4 log 2n —x +(x+4) loga+ 7 4 log 27+ aloga—a—tloga, T (a) < 4 log 27 + wlogx—x«#+4 logz+ 74; THE DISTRIBUTION OF PRIMES. 283 hence “P@+T (5, 50) > og 2 + 35 @ log w — ao 39 108 30 - 81 a — log «+4 log 30, ax T(w) +1 (55) < log 2m + $4 @ loge — 3 log 80— ha + log # —4 log 30 + §, r(5)+7(5)+7(5)>3 log 2a + 34 « log x — # {fh log 2+ flog 3+ § log 5}— 354 —gloga + § log 30, T(5)+2(5 J+ L(G ) <$log 2m + ghalog a —a {Llog 2+ tlog3 +4 log 5} — a+ § log #—$ log 30 + ¢. From the first and last of these four inequalities T(o) +1 (5) -12(5)-2(§)-2(5) >a {hlog 2+1log 34 flog 5 — 54 log 30} — flog a + blog > — 4, and from the other two, T (a) + 1 (5) - r(5)-T(s) -1 (5) if T (w) + 1 (55) - £5) r (5) r (=) > Ax — $ loge —1. Strictly speaking, these results are only proved for values of « which exceed 30; but it is easily verified that the two last in- equalities hold when # lies between 1 and 30; therefore they are true for all values of # which exceed 1. 284 THE DISTRIBUTION OF PRIMES. 222. By combining these formulae with those previously obtained (p. 282) we infer that Wr (a) > Aw —$ loga—1 (a) — (§) < Ae § log a. The first of these inequalities gives an inferior limit for w (a) ; to deduce from the other a pee limit, we put f(a)=§ dat 7 Hog 182) + 4 log «; then we have ve) 4 (5) 0 (x). As already proved, (x) > Ax —$ log #—1, wh (a*) <% Aa® +. ane g (log + 4loga+1; hence W(x) — 2p (a) > Av —12 Au? — ack (log x)? — 13 log # — 3, and comparing this with | O(a) < (@) — 2a (*), we have O (x) > Aw —12 Ax — 3 stage F (log x)? — 15 log # — 3. In a similar way it will be found that O(a) <% Ax — Ax? +3 Hogg (108 2) + Hog 2 +2 For convenience let these last inequalities be written O(a) > d:(z), 8 (@) < $.(#); then if a, 8 are two positive integers such that 1 $i (B) — $2 (a), 0 (8) — 8 (a) < $.(B) — fr (). It has been remarked at the beginning of the investigation that if « is the number of primes between a and 8 ,, 9 (B) — 8 (a) log 8 ; 0 (8) — O(a). b< ) log a 286 THE DISTRIBUTION OF PRIMES. consequently (8)— $40) +f dh. a log B Be (B)— hr (a) log a By putting a=2, we obtain superior and inferior limits for F (8); but in order to secure a practical approximation, it is best to suppose that a and @ are both large. The expressions for the limits of ~ are very complicated ; the essential point is that ¢, (8) and ¢,(8) are of the form P8—Q8!+ R (log 8), where P, @ are constants, and R (log z) is a quadratic function of loge with numerical coefficients. There will be more than & primes between a and £ if I log B < ¢s (8) — ¢ (a). Now dr (8B) — 2 (4) —A(8 — fa) — A (1a S30?) B Tea ((log ibe —} (8 log 8 + 2 log a) — 5 > A (B— fa) — ip AB? — ——, (log 8)’ — 4f log B — 5, since, by hypothesis, 8 > a. A fortiori, therefore, u will exceed k, if klog @< A (8 —$a)—42 Ap! 15 2 5 ~ 8 log 6 (log 8) = af log B = Oy, or Tees YIN oie ® (log 8)? 5; — 5 CP +B) log B= In particular, putting k =0, there will be at least one prime between a and 8 if 25 a< $0 = 28) = nb Tee 6A log 6 ‘08 8) — sag lg B 6A’ 224. Tchébicheff employs this result to prove a theorem the truth of which was conjectured by Bertrand (Journ. de U’ Ecole Polyt. cah. 30); namely that there is always at least one prime between a and 2a— 2 if a> Z. THE DISTRIBUTION OF PRIMES. 287 In the inequality at the end of last article, put B=214—3; then it becomes a<§(2a—3)-—2V2a—3 REET {log (2a — 3)}? 125 25 — 5qq log (2a — 3) — 63° Since, when 2 becomes indefinitely large, (log x)*/x ultimately vanishes, it is easily seen that this inequality holds good for all values of a which exceed a certain finite value. This limiting value is found by changing the inequality into an equation and finding its greatest positive root. According to Tchébicheff, this lies between 159 and 160; Bertrand’s postulate is therefore proved for all values of a which exceed 159, and it is easily verified for all smaller integral values except 1, 2 and 3 by actual experiment. 225. The crucial point of the investigation is where the inferior and superior limits of w(x) are deduced from the ex- pression e+ B(5)- 2 (5)- 2 (3)- 7 (5): for convenience this may be denoted, after Sylvester, by [1, 30; 2, 3, 5], and, in general, [a,, Ag ,«+»Am 5 1> Dat On, zn (7)-29 (5). The advantage derived from the use of the combination i, RIDE ap ah Ba] is twofold; in the first place, when the expression is written in the form may be written for SAmp (7), the coefficients which do not vanish are alternately +1 and —1, and this leads to the determination of a superior and an inferior limit of y(«); and in the second place, on account of the relation 1+3=4+4 +4, the term «log a, which occurs in the limiting expressions for 288 THE DISTRIBUTION OF PRIMES. T (x), ete., is eliminated. The expressions for the limiting values of [1, 30; 2, 3, 5] or PAaw (*) consequently assume the forms Ax + R, (log x), and Ax+ R, (log x), where, as above, A =4log2+41log 3 +1 log 5— +, log 30, and £, (log ~), R, (log x) are rational linear functions of log a. It is evident that similar results may be obtained from the expression [@,, Qs,-.-dim; 01, b.,...b,] provided that m | ny] aA in fact, it will be found that m+n OAR AAO Le i id died esa Bea ee 5 te loga+B m+n 2 < Axv+ log 2 + B’, 1 i where (Vey (-, log ai) —> ( 5, log b : and B, B’ are certain numerical constants, which may, if we think fit, be replaced by positive or negative integers, appropriately chosen. . The function [a, d:,...dm; 6), bo,...b,], or [a; b] say, may be expanded in the form 7 SO (=) pane 64) where the coefficient C;, will depend upon the relation of 7 to the a’s and the b’s; in fact, if r is divisible by p of the a’s and by q of the b’s, C.=p—gq. Hence if w is the least common multiple of A, Ag,++-Am, 0,, by,...b, 16 follows that C,=C, when s=r (mod yp). Consequently the coefficients form a recurring series with mw terms (or fewer) in its period. The sum of the first 7 coefficients is easily seen to be eal ee ae el +Lal+~*Lal- Lal Lil THE DISTRIBUTION OF PRIMES. 289 Call this S,.: then Therefore in the series fet cas ypsiene hie there will be one term which is negative, and numerically greater than any of the negative terms which precede it, while it is not numerically less than any of the terms which follow it. Let this be S;,; then it is easy to see that none of the expressions Cris (Cri + Chto), (Choi + Cre + Crs) sn bees (Chea + Opec +..0+ Chu)s can be negative. Remembering that y (z) is never negative, and cannot increase when z diminishes, we infer that h+p x Foy) h+1 G cannot be negative. In the same way the sum of the next group of w terms, namely h+2u x Ss” Ob (*) h+ptl is is not negative, and so on. Therefore & x [a3 B=2 Cay (2) +P, 1 where P is certainly not negative. Comparmeg this with the inequality } m+n [a; bl< Aat 5 log «+ B’, we conclude that. h > Op (=) < Ace" log a+ B ewes (i) 1 In the same way, if S; is the first of the sums S,, S,, S;...to attain the maximum positive value, we find that [a; == Cr (2)-Q where Q is not negative; and hence that _m+rn k SHO (=) 5 ee ads? a ay (ii). 1 r 2 1 Tt may exceptionally happen that S,=0: this is the case, for instance, with fi, 30;.2, 3, 5). M. nS 290 THE DISTRIBUTION OF PRIMES. 226. We will now suppose, for simplicity, that a;=1. Each of the inequalities (i) and (ii) involves > (#), and a certain finite number of terms Chr (=) besides. For these other terms may be substituted their inferior or superior limits as found by previous approximations; the inferior or superior limit being put in for each term in such a way as to leave each inequality valid. For instance, if in (1) a particular term C,ap (*) is positive we must substitute for y (*) its inferior limit, while if C, is negative the superior limit of (=) must be inserted. Exactly the oppo- site rule must be applied in (11). When this has been done, the resulting inequalities give two new superior and inferior limits for (a), and these may afford closer asymptotic values for ia than any previously obtained. When / and & are inconveniently large, simplicity is gained by first suppressing on the left-hand side of (1) any group of terms (not including yy (#)) which is known by previous approximations to be ultimately positive when « is very large; and in the same way im (11) we may suppress any group of terms the value of which is known to be ultimately negative. If, as we suppose, the first asymptotic limits adopted are those of Tchébicheff, derived from [1, 30; 2, 3, 5], it is evident that the - result of any finite’ number of applications of the process just. explained will be of the form (a) > Ax + Q, (log z) < A’x + Q, (log a), where A, A’ are constants, and Q, (log «), Q, (log) are rational integral functions of log « not exceeding the second degree. Since log # and (log «)* are both negligible in comparison with x, when # is very large, we may consider that A and A’ are asymptotic limits of ue , and the nearness of the approximation may be estimated by the approach of A’/A to unity. This ratio A’'/A is called by Sylvester, to whom this extension of Tchébi- 1 We say finite, in order to avoid the risk of the coefficients of Q, (log x) or Q, (log x) becoming infinite, THE DISTRIBUTION OF PRIMES. 201 cheff’s theory is due, the regulator of the approximation. Thus Tchébicheff’s original process gives a regulator § = 1:2. 227. The choice of groups of selected terms which may be omitted from Sylvester’s inequalities is much facilitated by the following considerations. Suppose that at any stage of the approximation we have obtained asymptotic values for (a) in the form Yr (x) > Ax + Q; (log x) < qAx + Q, (log 2), so that g is the regulator. Let m, « be positive integers, and m< yp. Then we have x x ne ee vv (| —f (=) > (4) Az+Q (log x), where Q (log z) is a new quadratic function of log . Now if (7-2) is positive, or which is the same thing, if w>qm, we thus obtain an inferior limit for W(v/m) —W(a/p) which is ultimately positive when «@ is large enough; while if fuAxv—Slogr—1; therefore (2) > Aw—$loga—1 + (5) - + (55). x ase Also (54) oT — § (log # — log 24) —1, x 6Ax 5 A a (55) <5 99+ Pagal (log # — log 29) + 4 (log « — log 29) +1, therefore a fortiort (2) >pAxv+q (log xv)? +7, log «+s, where 7, G1, 71, 5 are certain numerical constants, and in par- ticular a Le Ce ae {Mowery me ay tear In a similar way, since V ¢ (1) —(6) + 7) — (10), we have (2) < Aw t §log e+ (7) — wp (a )+¥ Ge THE DISTRIBUTION OF PRIMES. 293 and hence (a) < tAx+ u, (log #)?+ v, log a+ w,, where ¢,, %, V;, W, are numerical constants, and in particular h=l+$ (b+) $= 38. We may now repeat the process, employing the new asymptotic x are values of (5) a (5) etc.; and it is clear that after 7 succes- sive applications, we shall obtain results of the form Wr (x) > pA + q; (log 2? + 7; log w+ s;, br (a) < t:Av+u; (log cP + v; log «+ w;. To determine the coefficients we have a set of linear difference equations with constant coefficients ; these have been completely worked out by Hammond. It will be sufficient here to consider those which are satisfied by the coefficients p;, t;. It is easily seen that Pin = ae Pi— go k +L, tin=Gti)i-tpitl BE ies tai: The initial values, obtained from Tchébicheff’s inequalities, are Po a iF ty g ; and the complete solution is pi = 84953 + Po’ + Qpi', ‘i ti = 33599 + Rp’ Spi’, where P, Q, R, S are numerical constants, and p, p, are the roots of the equation It may be verified that p and p, are both proper fractions, so that the asymptotic limits of oe obtained in this way are ultimately 5107 By foray a tA = 39595 A =-9226107.. By applying this process to the schemes imG070-0205,002, 70210] and 4, Gi) a 2108 231 11 5o: eeu at, 11, 105d, 294 THE DISTRIBUTION OF PRIMES. Sylvester has succeeded in reducing the inferior and superior asymptotic limits of ual to ‘9461974... and 1:0551851... each of which is more nearly equal to unity than the corresponding value obtained by Tchébicheff. The scheme [1,1 68 10) 14,9105 372," 3, 5,7, «115, 18) 385; T0019 leads to still closer limits, namely "95695... and 1:04423... It appears to be very probable that the true asymptotic value of ve@) o is unity; of course, we cannot expect to prove that it is so by any approximative process such as that which has been described. 229. It must be carefully borne in mind that it has not been ee proved that approximates to a definite limit when # becomes infinite ; it oe indeed been proved that for all values of « A+” Sie Skea where A = 1:04423..., A’ = "95695... and 7, 7’ are quantities which | are very small when @ is large; but this is quite consistent with the hypothesis that ee continually oscillates between two finite | limits, without tending to a single definite value when # =o. It may, however, be shewn that aa. Poincaré’s proof of this proposition is extremely simple and elegant. To avoid confusion in the use of brackets, we shall write THE DISTRIBUTION OF PRIMES. 295 E(«) instead of [xv] to denote the greatest integer which does not exceed «. Then if the function a (#) is defined to mean 1 or 0 according as « ¢ 1 or « <1, we shall have E(w) =a(w) +4/( 5) +a(a)\ +... +a (2) 4 be in fact, since a (=) =1 for n= 1; 2, 3.2.4 («), while a (=) = Orior all integral values of n which exceed #'(«), the series is simply 1+141+... to H (a) terms: that is, it is H(«). ~ 1 Now let Sn=l+agtgte. +7, Vienv=E@) + #(5) +E (5) us +B("); then since, if p > 1, jv P Ve it follows that S,.H(“2)-n+1< Re BR Te ed OA OD reich ee (1). : op ae Uae Again Mloptese= ue < lo ; 8 G 8 pp 8 p-1 and therefore log (n+1)— log 2<8S,—1< log n, whence a fortiore log (1+ 1)<8, <1+logn. From this and the inequalities (1) we conclude that E (a) log (n+1)—n+1< V (a, n)< E (a) {1 + log n}....(2). Pee (2) — B (0) +8 (5 ) + +2 (=) + =28 (7); 1 then since # (=) =0ifn>F (za), it is clear that V (a2) = V {x, Eb («)} 5 consequently by (2) E (a) log {EF (#)+ 1}-# (w)4+1< V (@)< H(«) {14+ log £(a)}. 296 THE DISTRIBUTION OF PRIMES. Dividing by w log w, and making @ infinite, we see that V(@) _ fg LOR if, Now let @ be any assigned quantity which is greater than 1, and suppose, if possible, that there is only a finite number of positive integers w, for which | ar (a) < an. Let a be the greatest of these; then for all integral values of a which exceed 2, we shall have ap (w) + a It will be possible to choose a finite positive integer b such that for all positive integral values of x pa) >a(e+1)—b; we might, for instance, take b = # {a (a+ 1)} +1. We may infer from this, that, for all positive values of # which exceed unity, ap (xv) > ak (x) — ba (x): in fact, if « > 1, we have #(#) ak (x) — ba (xz) change « successively into and add; then since the pa ee ge 20 ona inequality only fails when (a), /(@), a(«) all vanish, we have a ~\— bala SY Ga) re eg ee that is, T (x) >aV (a) —bE (2), (2) V@ _, F@ and therefore : slogz” wloge , sloge But from Tchébicheff’s inequalities it appears that Ta) _, pio locWian ? also the limit of #' («)/« log w is zero, and that of V («)/x log & is 1; we are therefore led to the absurd result that 1 >a. Consequently THE DISTRIBUTION OF PRIMES. 297 it is impossible to assign a finite value of a, such that whenever L> x, v(x) < ax; the inequality h(x“) < ax is therefore satisfied by an infinite number of positive integers. It may be proved in exactly the same way that if @’ is any assigned quantity less than 1, the inequality | Wr (“) > aa is satisfied by an infinite number of positive integers. Consequently if Wy (x)/a has a definite limiting value, it must be unity. Perhaps the argument becomes clearer with the help of a geometrical figure. Suppose the curve y ao to be constructed; then if A, A’ are Sylvester’s limits, and k, k’ assigned proper fractions, however small, it has been proved that as we proceed to infinity along the axis of #, the curve ultimately lies wholly between the lines y= A+h, y= A’—k’; and also that however far we may proceed along the strip enclosed between the lines y=1+m, y=1-—m’, where again m, m’ are assigned positive quantities, we shall always find points of the strip which belong to the curve, no matter how small m and m’ may be. Therefore if the asymptotic form of the curve is a straight line parallel to the axis of #, this line must be y=1. 230. It is now easy to prove that if z o has a definite limit when « is infinite, that limit must be unity. For suppose a is any assigued quantity greater than 1; then, by last article, the in- equality (wv) < az is satisfied by an infinity of integral values of x; but 0(x)< W(x) always (Art. 220); therefore the inequality 6(x) (x) — 2 (Va), and abe (/@) < Sa ; consequently 0 (a) > (a) — 12/2, O(v) (a) 12 “2 iol mie bake? Now if a is any assigned quantity less than 1, there will be an infinity of integral values of « for which OES and among Mfr 298 THE DISTRIBUTION OF PRIMES. these there will be an indefinite number for which =. is less than assigned quantity 7, however small; therefore there will be an indefinitely great number of positive integral values of « for which O («) Tes: > Lea). Now if a’<1, we may always put a =a—~y, where a is also less than 1 and 7 is positive; consequently if a’ is any assigned quantity less than 1, the inequality O(x2)>aa« will be satisfied by an infinite number of positive integers. ig converges to a definite limit, It follows, therefore, that if this must be unity. 231. Let F(x) denote the number of primes that do not exceed #; then from the definition of @ (a) it is obvious that F(a) log « > 0 (a), consequently whenever @ (a) >a, we shall have AL NC ee is Combining this with the result of last article, we infer that | whenever a< 1, there is an infinity of integers for which AL We may also prove that if a >1, there is an infinity of integers _ for which Ax log x” EF (a)< To shew this, write, for the moment, F(a) =n; then because the nth prime number in order, say p,, 1s greater than n, we have 0 (x) => log p, > S log r 1 1 eta): that is T[F (a)| < O(a). Now for all values of « which exceed a certain limit T (x) > ba log «a, THE DISTRIBUTION OF PRIMES. 299 where b is any assigned quantity less than 1; consequently for all such values of « 0 (w) > bF (a) log F (a). But, as already proved, F(x) log # > @(«), and therefore log F(x) > log @ (a) — log log «; hence 6 (x) > bF (x) {log 8 (x) — log log x}, 1 0 (x) 2 he. "EAL log 6 (x) — log log a" Let pearaner = f(y); then /’ (y) has the same sign as log y — log log # — 1, and is therefore positive if log y > 1 + log log a, that is, if y > e log x. Provided, then, that this inequality is satisfied, f(y) mcreases with y (w remaining constant). Now since when «w is large, log # is negligible in comparison with , it is easily seen that both @(«) and awx (where a is positive) exceed ¢é log « for all values of « beyond a certain limit. Hence whenever « is sufficiently large and @ (x) < ax we shall have F148 (@)] a, it will always be true that F(a)< ax AX log (ax) — log log x & log a a & b log x’ provided that # exceeds a certain definite value. Now if ¢ is any assigned quantity greater than 1, it will always be possible to assign values to a, a’, b so that , a ; ana 1 Mee Sy Nia Ue nds F 300 THE DISTRIBUTION OF PRIMES. and since the inequality @(#) 1, it follows that F (1 the principal value of the integral (in Cauchy’s sense) is to be taken. It is not very clear how this principal value is defined ; however, there will be no objection to writing Fe ; | Jie li (a) —la (a), where x and a are both less, or both greater than 1. There will, of course, be no difficulty in understanding what is meant by are 9 log # taken along any path of integration which does not go through the point #=1, provided that the particular value of log 0 with which we start is definitely assigned. 302 THE DISTRIBUTION OF PRIMES. Gauss, in a letter to Encke (Werke, 11. p. 444), states that the number of primes which do not exceed # is approximately i — 5 the integral being given without any indication of the limits. In his tables of the frequency of primes, a comparison is made B between the values of F(8)—/'(a) and | = for successive intervals of 100,000 beginning with « = 1000,000, and ending with — a= 2900,000; the comparison is also made for the intervals 10°... 2.106 and 2.10°... 3.10% It should be observed that Gauss’s enumerations of the actual number of primes in the different intervals are not very accurate. 233. The mere fact that, so far as the enumeration of primes has been hitherto effected, the formula spe [ie log x is approximately correct, does not in any way prove that this is a proper asymptotic formula. It seems clear that Gauss was led to it simply by observation, and it does not appear that he ever ac- counted for it in any theoretical way. The only satisfactory attempt to determine a general analytical formula for F(z) appears to be that contained in Riemann’s celebrated memoir | This is confessedly incomplete, and the analysis which it contains is very peculiar and ditficult: but because of its great import-_ ance, some account of it ought to be given. On the properties of the function I‘ (z) for a complex variable, which will have to be assumed — in the course of the investigation, the reader may consult Prym, Zur Theorie der Gammafunction (Crelle, Ixxxu. 165) and various — papers by Bourguet and Mellin in vols. 1. 11. i. and vii. of the Acta Mathematica. . If « and s are complex quantities, we may define «* to mean pea | + slog «+5 (log a)" +... and since log isa many-valued function, «* is many- EAMG also. If we write x Log #= i ue 1 &@ where the integral is to be taken along a path from 1 to « which does not surround the origin, the general value of log w is THE DISTRIBUTION OF PRIMES. 303 Log «+ 2kai, where & is any integer, and the general value of x® is therefore = ezksni es Log we In what immediately follows we shall suppose that «* stands for ess, so that for instance when «@ is real and positive, and s=a+ fi, as = e(@tBi) Loge | = Lose feos (8 Log x) + 7sin (8 Log #)} where Log # is the ordinary real logarithm of z. This being so, then, whenever the real part of s is positive and greater than 1, we have where the product on the left applies to all positive primes p, and the sum on the right to all positive integers n. This is easily seen by observing that, under the conditions stated, we may write Le a. 1272 say pasar See ee ge, se LES Ea ee La 8 cay ee — 1 + ae he Fe ee Bee where the series are absolutely convergent; and since n~* can be expressed in one way only in the form ns = Dian jee mye. a where p,, Ps, p; etc. are different primes, and this term occurs once, and once only, in the expansion of (1 + 2-*+...)(14+3-%+4...)... (1+p*+...)... the proposition follows. Now if ” isa positive integer, and # a real variable, we may write | Ce Otani | e¥ ys dy =n-* I(s) 0 0 and hence é ig * a dx U (s) 2 Phan =| (ez) peas da il id Ps 304 THE DISTRIBUTION OF PRIMES. We will now suppose the integral as dar er — 1 to be taken first along the axis of real quantities from «=+ 0 to x” =e, where € is a very small positive quantity, then in the positive direction along the circumference of a small circle of radius ¢, with its centre at the origin, and finally along the axis of real quantities from «=e tov=+o. After going round the small circle, log # mcreases by 272, and if we take w* to mean (as above) exp | s aan sl it is evident that, after describing the circle, foi becomes taal itr the integral round the circle vanishes ultimately when e is infinitesimal: and therefore the whole integral is if bins 2 ee Les her Sco) || a =f Lei Ai: [” ¢- da = 21e™* sin 7s i} a =, Ls ae 1 ears It is easily seen that if the real part of s exceeds 1, Comparing this with the previous result, we have Pane Gis of —l os if CAS ae da 2 sin ws I (s) Sn-*§ = — ies | -/ Fs re the integral on the right being taken along the path above defined. Observe that (— x)** is to be taken as e—™ 6) a5, where a is to have the same determination as in the equation D(s)= | On” ae 0 For convenience we may suppose 2° = e&—) Les”; but any other determination might be adopted, provided that the proper corre- sponding value of I‘ (s) be taken. THE DISTRIBUTION OF PRIMES. 305 The path of the integration denoted by I may be modified in any way consistent with not including any of the zeroes of the function (e” — 1). If, now, we define the function €(s) by means of the equation : oe OE: 2 sin 7s I’(s) (saa ee €(s) is a one-valued function of s which is finite for all finite values of s except s=1, and vanishes when s is a negative odd integer. Moreover when the real part of s exceeds 1, €(s) = Xn~. When the real part of s is negative, the point =+ (a real quantity) is not a-pole of the integral © (— 2) dx oo ee Mt and consequently the value of this integral, taken as above explained, is equal to the sum of the values of the same integral taken in the negative direction round infinitesimal circles each surrounding one of the poles +27, +4771, etc. The value of the integral taken round the pole 2n77 is (a — 2n7) (— x) ge ape a — 2ar7v. Lt = — 2art (—2n71)*1: 2=2nmi Gea and in the same way that round the pole — 2n7ri is — 272 (2n71)s1. Sheeene these values in the equation which defines ¢(s), we have 2 sin ms I'(s) £(s) = (Qa) {8 + (— i} Sn = (ny {e+ (— a} (1-8) or 2 cos > I (s) €(s) = (277)8 € (1 —). If in Legendre’s formula 1 n—1 cdi ho P(@)P(2+;)..0 (e+ )=@n) > nim (nw) we put n=2, ¢ =5 we obtain re CF')-e 2-76 2 iS aol (s), 306 THE DISTRIBUTION OF PRIMES. 1l+s se 7 eRe - Begs. TE (ecg 7) Uh (35 =e ey eee sin iat Dak cos — and therefore, eliminating [ (- 5 *) : | 1— naeey 6 ar (yt (—F : *) cos 5 = qr T (5). Hence the relation connecting €(s) and €(1—s) may be written in the form r(G)s@-aar(=* or, which is the same oa a rar(s\e@am a T(>*\¢a-9) This naturally suggests the introduction of ‘the function T 2 oT (5 5) 6) instead of €(s); this function is unaltered by the change of s into 1—s. Since nt 7 2 (5) =| ent 92" dx 0 ) $1 =s) it follows that ai (5 =) E(s)= i ar (x) 3 dx, where wh (a) = Sern, 1 The function y(z) is one of those which occur in the theory — of elliptic functions; in fact, writing K’/K for #, and putting o-tK'|K — 1+ 2 (@) =1 + 2¢ + 2q¢+ 2¢°4+ ... 2K ar e Interchanging the moduli, so that # becomes : , we have 1 PHI Ss 1+2y(2)=)/== =o (1+ 240) ei 8 1 8 = $ and hence [ ar (a) «2 * da =| ab (x) 22" dx +{ ar (x) 2° * dar “0 0 1 ; 1 s sd 's =| ty (5) + 4a%- +} a2" da +/ h(x) a2" da 0 re 1 =[ +) {or 4083) det = THE DISTRIBUTION OF PRIMES. 307 Now write s=$+ti, 2G) i (8) 00) =£0; then, on substitution, we find ea agian Df + ) {orats is oval a ou is =$-@+)[ 244 @) 00s (48 log x) du where, of course, the determination of cos(4¢log #) must be in accordance with that of m2T ( 5) in order that ¢(s) may be one-valued. 7 In order that the real part of s may be greater than 1, so that xn-* may converge, we must have t=a— pr with B > 4. If, in the expression for &(¢), we suppose the integration to be taken along the axis of real quantities, and log # to denote the real logarithm of #, we obtain an expression of the form E(t)= A,+ A+ Aott+... where A, A, A,... are real coefficients. This particular value of £(t) is always real when ¢ is a pure imaginary, or, which is the same thing, whenever s is real. If sis a real quantity greater than 1, this value of &(¢) coincides with the real value of s(s—1) -* /s bs The coefficients A; depend upon integrals of the type ee Ned ey Bo) 21). (log 2)” de and it is easy to see that these diminish very rapidly as 7 increases, and that the expansion of &(¢) converges absolutely for all finite values of t. __ Since, when the real part of s exceeds 1, the function of s with which &(¢) then coincides, never vanishes for finite values of s, it follows that the finite roots of the transcendental equation & (¢) = 0 must all be of the form T=at+Pi with B } 3. 20—2 308 THE DISTRIBUTION OF PRIMES. Now the number of roots of &(¢)=0 contained within any simple contour is equal to = [a log & (t) taken in the positive direction round the contour. Riemann states without proof that the number of roots of & (¢) = 0 contained within the rectangle bounded by the lines x=0, «=T, y= +4, is found by this method to be approximately when 7’ is large; the error being comparable with . It follows uh from this that the frequency (or ‘density’) of the roots in the neighbourhood of the line «= 7 is asymptotically dsl Te rened ey eT. TP lig (8 Se ~ Oar 8 Dar’ and hence we may write log £ (t) = 3 log (1 - “) + log £ (0) where the summation applies to all the roots 7 of the equation &(t)=0. Riemann adds that, although he has not succeeded in proving it, it seems very probable that all the roots of &(¢)=0 are real. Let us now denote by F(x) the number of primes which are less than x, when « itself is not a prime; when «@ is a prime let_ F'(«) stand for the number of primes less than «a, increased by 3 2 so that at every point where /’(#) changes abruptly F(«x)=1{F(#@—0)+F (a+ 0)}. Let s be a quantity of which the real part exceeds 1; then log €(s) = —& log (1 — p-) =p +42p-" 4 42p-* +... leo) ioe) Now p= | CER Daria s ii cea meas feo p p* and so on; and if these values are substituted in the expansion of log €(s) we obtain an expression of the form s | f CQO PGES THE DISTRIBUTION OF PRIMES. 309 where f(#) is obtained by taking 1 for every prime which is less than x, 4 for every square of a prime which is less than a, } for every cube of a prime which 1s less than w, and so on; that is f(a) = F (a) + 4F (2°) +4F (a) +..., because if p? < a, it follows that p< #?, and the number of primes for which this condition is satisfied is #’(#), and similarly for the other terms, There will of course be abrupt changes in f(x) when w 1S a prime or a power of a prime, but since these critical values are separated by finite intervals, the expression | MG) 0 eagle J perfectly definite. When 2 is less than 2, f(x) =0, so that we may write oR San i Dee ia? a wip f (a) oe dx provided that s=a+ bi, with a>1. Observing that f(x) is real throughout the integration, we have log (8 oy =| f (2) a {cos (b log x) — 7 sin (6 log x)} d log a, and from this, by Fourier’s theorem, it follows that Sra f(a) = ie * log & iOg and therefore, multiplying both sides by 2x", Serif (a) =f” BES ards, a—-tn cos (b log w) + ¢sin (6 log «)} db the integration being taken along a straight line, so that the real part of s remains constant. From the way in which F(x) was defined it will be seen that this formula holds good for all real values of , including those for which f(x) changes abruptly. Since HESAGE: vanishes when s =a + 1, we find by partial s integration that re. ae tp ee danlog € (3) rif) =— egal jae 5 *) weds 310 THE DISTRIBUTION OF PRIMES. From the definition of £@) 2, - C(s) =~ = E() (s—1) ,/s r(5) eee £ (2), ) and therefore log £ (8) = 5 log —log (s — 1) — log r(G +1) + log £(t) =5 log m — log (s 1) log (5 +1) + > log it “4 aay + log & (0). Hence i (“82 ©) d (1 8 ne ah mae 1 2\2 +5,[j Slog {1+ SSP] - BLO. Now S n=imn 8 Ss — log P(5 + 1) = Lt ( > log (1 + 5.) —5logm) M=0 \n=1 and therefore Also aes (s—4)) | _ d{1 s 1 s Ts Sqi+ | =a 28 ( - ye) +38 (1 al 1 r+t = 08 7 Consequently 1 THE DISTRIBUTION OF PRIMES. Pay ae where C= log & (0) + & log (1 + iz) - LE The integral att a+nt [ : ads = | ae = ds I q-0i 8 U- 0% Se tye | +” cos (6 log 2 : ae (6 log x) db oft log | &008 (0 log x) db = 2x log of eae = 21a" log x. me—*8% = Dart log & (as may also be seen by replacing the straight line of the original integration by a small circle round the pole s =0). All the other integrals in the expression for 27r7f(x) log « may be reduced to the type at+ot d ie A ar log (1 -5)| ads. Call this ¢ (8); then #B=[" a a@= \ ae Sloe ie rds B a-ai B—S’ on integrating by parts. Now if the real part of 8 is less than the real part of s, so that the point @ is on the left of the line along which the integral is taken, we may change the path of integration into a small circle described in the positive direction round the poles=@. Hence the value of the integral is 27108 log x pC es x = 271 log « | “w®—-da or Qari log | a daz, 0 (0) according as the real part of 8 is negative or positive. \, Hence in the first case e m»B—1 (8) = 2mi log « | oe - OL THE DISTRIBUTION OF PRIMES. and in the second p(B) = Brilog a | [ tes des Bh, where A and B are constants ares upon the way in which log (1 - A) is defined and on the paths along which the integrals are taken. In the second integral the path from 0 to « must not go through the critical point «=1. We may, if we like, suppose that the integration from 0 to a is taken along the axis of real quantities from 0 to 1 —e, then along a semicircle of radius € from (1 —e) to(l +e) and then along the axis of real quantities from 1+etow. Similarly we may suppose the integration from » to x to be taken along the axis of real quantities. The values of 8B are —2, — 4, —6, etc., unity, and the different values of $ + 77; and since a, the real part of s, is supposed greater than 1, we have finally ~ v a ate + ett Uae tues sf log a ie da: ; +f (ae Bee ines - C © dx “gt cos(tlog a) da f- da: aebt eS ee ’ » log # 2s | log # Lf (eo) log ae | C’ being a constant, the value of which will depend on the deter- mination of log # in the integrals. Observing that [ : (a) dx only differs from I ” b (#) dx by a constant, and that f(2) =4, we may write, for # > 2, 9 a-* cos (7 log #) da AO in ce o 23: | 2 log « ir da: — | ee 22 (a@—1)loga *’ and take log « to mean throughout the real logarithm of #; the integrations, also, being taken along the axis of real quantities. Neglecting terms which are comparatively very small when « is large, PHONE) See 93 o 2 cos (7 log uw) log a“ THE DISTRIBUTION OF PRIMES. SS It will be remembered that f(a) = F(e) +4F (a) +4 F(a) +..., and from this it may be inferred that F ()=(0) 4) -@) - oy +i f(0)-4f(e)-af(e")-.. =e @yeoey f (ar), UL where m assumes all positive values not divisible by any square, and wu is the number of different prime factors of m. To prove this, let n be any positive integer greater than 1, and let m, m’ be any two conjugate factors of so that mm’=n: then 1 1 Ne ley a. the coefficient of F’ (a) in >(— 1) a ef (a) 1S Ss geek ae ee 1) ¥(-Iyee 5-2 3 (-1y where the sum is obtained by calculating pw for every divisor of 1, 1 and vn inclusive. But if » contains & prime factors, it is obvious that S(-1¥=1-k+ RD SCS DA maa bees IAT because there is one factor of n for which 4 =0, there are & for k(k—1 which p= 1, rye Y for which = 2 and so on. 1 Thus all the functions («”) disappear except /’(x), which occurs once with a coefficient 1: and this proves the theorem. Substituting the values of /(«), f(2*), etc. in the formula, we obtain an expression for F' («). ‘i da: a me oe I, x (a? —1) log a’ and let A denote the constant 314 THE DISTRIBUTION OF PRIMES. = gt cos (t log x) dx then if we neglect the integrals | i we obtain 2 Og x the approximate value P(a)=A +3 (-1¥ = 1 (a), where Tey {<2 bee Since A is a finite quantity, it may be omitted witheut sensibly affecting the formula when «# is very large. We conclude from this result that Gauss’s approximate value is ultimately too large; and this is confirmed by the comparisons given by Gauss for the second and third millions. AUTHORITIES. LEGENDRE: Dune lov trés-remarquable observéee dans Pénumeration des nombres premiers (issai sur la théorie des nombres, 2nd ed. (1808) Part Iv. § Vili.). Gauss: Tafel der Frequenz der Primzahlen (Werke ii. p. 435); followed by a letter to Encke (ibid. p. 444). TCHEBICHEFF: Sur la fonction qui détermine la totalité des nombres premiers inférieurs a une limite donnée (Mém. de Acad. de St Pétersbourg | (Savans Etrangers) vol. vi. (1851) p. 141; reprinted in Liouv. xvii. (1852) p. 341). Mémoire sur les nombres premiers (Mém. de VAcad. de St Pétersbourg (wé supra) vol, vil. (1854) p. 15; or Liouv. xvii. (1852) p. 366). RIEMANN: Ueber die Anzahl der Primzahlen unter einer gegebenen Grosse (Monatsber. der Akad. d. Wiss. zu Berlin for 1859; or Werke, p. 136). ‘ MEISSEL: Ueber die Bestimmung der Pr anaes innerhalb gegebener Grenzen (Math. Ann. ii. (1870) p. 636). Berechnung der Menge von Primzahlen, welche innerhalb der ersten hundert Millionen natiirlichen Zahlen vorkommen (Math. Ann. iii. (1871) p. 523). Roce: Zur Bestimmung der Anzahl Primzahlen unter DEL LEG Grenzen (Math. Ann. xxxvi. (1890) p. 304). SYLVESTER: On Arithmetical Series (Messenger of Math. vol. xxi. (1891); see also a previous paper by the same author, Amer. Journ. iv. p. 241). PorncaRé: Hxtension aux nombres premiers complexes des théoremes de M. Tchébicheff (Liouv. (4) viii. (1892) p. 25). | EXAMPLES. $15 EXAMPLES. 1. Prove that if, to the base 10, | Ee ee ae 1004 ae See 1028G102bLr ot 8082 2 lo __ 125? | =d lo i we 8 794.1296 °898.100 ” then 196 log 2 = 59 + 5a+ 8b — 3c — 8d + 4e ; and find log 3 and log 41 in terms of the same quantities. (See Gauss, Werke i. p. 501.) 2. Verify that 2%*?+4 ] = (211 + Qet14 1) (241 — Qet1 4 7), and hence factorise 2+ 1. (This application of a familiar algebraical identity appears to be due to Aurifeuille: see Lucas, Théorie des Nombres, i. p. 326.) 3. From the facts that 10%=1(mod 9) and 10™=(—1)™ (mod 11) deduce the ordinary criteria for the divisibility of any number by 9 or by 11. Prove that a number expressed in any scale of notation is divisible by a given number m if a certain linear combination of its digits is divisible by m ; and shew how the simplest combination in question may be discovered. For example, prove that a number andj... dd, expressed in the scale of 7 is divisible by 19 if (d+ Ag+Q, +...) +7 (de +05 + Og +...) —8 (G3 +a¢+ a+...) = 0(mod 19). 4. If is an odd prime (except 5), and a < p, the fraction a/p may be expressed as a pure circulating decimal, and the number of figures in the period is equal to the exponent to which 10 apper- tains, mod p. Calling this exponent /, the expansions of ig OL pea An will give rise to (p—1)/f groups of periods, the periods of each group being derived from each other by cyclical permutations of the digits: hence if one period of each group is known, and also 316 EXAMPLES. the index of 10 to any primitive root of p, the index of any number may be determined. (D. A. Arts. 312—318.) 5. If 4n+3 and 8n+7 are both primes, then 2#*?—-1 is divisible by 8n + 7. (Euler.) The numbers 2?—1, where p is prime, are known as Mersenne’s numbers. According to Mersenne (Cogitata Physico-Mathematica, (Paris, 1644) preefatio generalis, Art. 19) the only values of p, not exceeding 257, for which 2?—1 is prime, are 1, 2, 3, 5, 7, 138, 17, 19, 31, 67, 127, 257. Seelhoff has proved that 261_] is prime, but this is the only exception to Mersenne’s statement yet discovered. See W. W. Rouse Ball On Mersenne’s Numbers (Mess. of Math. xxi. (1891) p. 34) and Lucas, Théorie des Nombres, i. p. 374. 6. Prove that if 2?—1 is a prime, then 2? (2? —1) is equal to the sum of its aliquot parts. (Euclid ix. 36.) A number which is equal to the sum of its aliquot parts is called a perfect number. No method of finding perfect numbers, except Euclid’s, has been discovered: it is not even known whether any odd perfect numbers exist. Euclid’s formula includes all even perfect numbers: of these the first six are 6, 28, 496, 8128, 33550336, 8589869056, and three others have been calculated (Rouse Ball and Lucas, as above). 7. A pack of 52 cards is shuffled in the following way. The top card is removed, and the card originally second is placed above it; then the card originally third is placed below the two cards already removed, and so on; the card which is at the top of the unshuffled part of the pack at any stage of the process being — placed at the bottom or top of the other packet according as its place in the whole pack was odd or even originally. Prove that when this process has been repeated 12 times the cards come back to their original places. In general, for a pack of 2n cards, the original order is first | restored after m shuffles, where m is the least number for which (On the history of this problem, known as ‘ Monge’s shuffle, see Bourget, Liouv. (8) vii. (1882) p. 413.) 8. Deduce from the theorem 2d (d) = n (Art. 8), that 1s Op ROL germ 1 RIND pe 2 (mya 10100 vw. 3) PST ay iP Ta a. 4 14.0 On ipeeeer 5 EXAMPLES. ST where there are m rows and columns, the last column consisting of the natural numbers 1, 2, 3...m, and the elements of the rth column (r0, are both odd primes, then 3 is a primitive root of p, provided that 2”*+? gq > 9°”. (Tchébicheff.) 14. If p=4n+3 is a prime number, 2 2n+1)i1+G1)*=0 (mod p), where » is the number of quadratic non-residues of p which are less than 3p. (Jacobi, Crelle ix. p. 189: the problem is Dirichlet’s.) 318 EXAMPLES. 15. Verify the following congruences: 2!+1=1.3 (8), 12!41=0 (187), 28!41=18.29.(29?), 4!+1=0 (57), 16!41=5.17(17), 30!4+1=19.31 (81), 6!41=5.7 (7), 18!412=2.19(19), 40!4+1=16. 41 (41). 10!+1=1.11QP), 22!412=8. 28 (23%), Can any rule be discovered for finding primes p such that (p —1)!+1=0 (mod p?)? 16. If 2n+ 1 is an odd prime p, | (2n) != (— 1)” 2 (v!)? (mod p?). 17. Can any rule be assigned for deciding a priort whether the diophantine equation 2? — Dy? = + 4 admits of integral solutions in which w, y are both odd ? (Cf. Art. 153 and Cayley, Crelle liii. p. 369.) 18. If m and n assume all positive imtegral values, the expression (m+n—1)(m+n—2) 2 assumes all positive integral values without exception and each value only once. (Cantor.) m+ 19. The rth series of polygonal numbers being defined by the formula 4 {rn?—(r—2)n}, prove that the series will contain an infinite number of squares, unless r is the double of a square number. (Euler, Comm. Arith. 1. p. 9.) 20. If #(«#) denote the integral part of #, then (1) m and n being any two positive integers such that n is not a factor of m, mi mee. ~ lop oA ee hs ee cienisre ict E (3 =——=+ — in n (ee) a n nN (11) m and n being any two odd positive integers prime to each other, rae, (me) Ww—-Llm n-1l 1 *2@-) km _,, 2kr > —|=- —— —- = an co k=l ( n Sa, 7 4 QM 2) gE4 n n (iii) if m and n are prime to each other, and both = 1 (mod yp), h=(n-Wie sf k= (n-1)/ “as ES g 1 “e es) E, ns 1) "e (=) sf (m a 1) h=1 n m M (Hisenstein, Crelle xxvii. p. 281.) k=1 EXAMPLES. 319 21. If » is an odd prime, and @ an odd number prime to p, prove that (a|p) =(— 1)”, where ae! ¢ 2) op — Oe eet kat aa kar} 7 T= 2p 4 k=1 (Eisenstein J. c.) 22. Let @(k) denote the excess of the number of divisors of k which are of the form 4n + 1 above the number of those which are of the form 4n + 3; then m m m 30 (k) = E(m) -£(%) +2(5) -E (7) oe the summation on the left extending from k=1 to k= E'(m). Deduce from this that the number of points whose coordinates are integers (exclusive of the origin) contained within the circle x +y?=m is four times the number contained within the area bounded by the line #=0 and the hyperbolas y(4e+1)=m, y(4e+3)=m. (Eisenstein, Crelle xxvii. p. 248 and Gauss, Werke 1. p. 292.) 23. Prove that when y assumes all positive integral values from 1 to # ( ue , the number of the quantities a of which the fractional part is less than ? but not less than + is m m m m - B(5)-2(7)+2(5)-2 (a) + Generalise this proposition, and shew its connexion with last example. 24. Kronecker has stated the following proposition :— Let p=4n +3 be a prime, and let (a, b,, ¢,), (de, be, C2), etc. be the properly primitive reduced forms of all those negative determinants — A for which p is representable by the principal form «? + Ay?, with y uneven; then the roots of the congruences a,x? + 2b;4 + ¢; = 0 (mod p) will all be real, and will form a complete system of residues to modulus p; double roots being reckoned once only. Verify this forp = 7,11, 19, etc. (Berlin Monatsber. 1862, p. 304.) 320 EXAMPLES. 25. Prove that the sign of the symbol (m|n), where m, n are both odd, is the same as that of nee So eek) h=1,2,3... é(™m-—1) nm (, TU 2 b= 208 et and deduce the law of reciprocity. (Kronecker. ) 26. If Gis a large positive integer, we may write asymptoti- cally 2 3 | Xd (m) =— G?+A, 1 7 where A< (slog G+404+4)G +1, C being Euler’s constant. (Mertens, Crelle lxxvii. p. 289.) 27. Representing by h(A) the number of positive properly primitive classes of determinant — A, then, when @ is large, we have asymptotically An 218; ’ | ene Eb where B=l+yBtatpte- (Mertens /. c.; and Gauss, D. A. Art. 302, Werke 11. p. 284.) G pai ay == 1 INDEX. [The numbers refer to the pages.] AUTHORS QUOTED: Hurwitz, 131 Abel, 186, 199 Jacobi, 31, 42, 199, 317 Arndt, 149, 182 Klein, 131 Aurifeuille, 315 Kronecker, 55, 56, 186, 202, 229, 253, Bachmann, 199, 229 319, 320 Ball, 316 Kummer, 224, 229 Baumgart, 55 Lagrange, 30, 31, 58, 127, 130, 183, Bellavitis, 31 271 Bertrand, 286 Lebesgue, 218 Bourget, 316 Legendre, 38, 52, 55, 58, 131, 218, 300, Bourguet, 302 314 Burckhardt, 31 Lipschitz, 159, 182 Cantor, 318 Lucas, 315, 316, 317 Cauchy, 202 Meissel, 273, 314 Cayley, 183, 318 Mellin, 302 . Crelle, 31 Mersenne, 316 Dase, 31 Mertens, 320 Dedekind, 58, 131, 182, 230 Poincaré, 131, 182, 294, 314 Degen, 131 Prym, 302 Dirichlet, 31, 38, 45, 50, 55, 58, 62, Reuschle, 195 130, 131, 1385, 139, 166, 182, 205, Richelot, 199 212, 229, 230, 252, 254, 317 Riemann, 112, 302, 314 Hisenstein, 41, 56, 187, 188, 229, 318, Rogel, 273, 314 319 Schemmel, 317 Euclid, 191, 316 Schering, 182 Kuler, 30, 31, 55, 181, 270, 271, 316, Seeber, 131 317, 318 Seelhoff, 267, 316 Fermat, 15, 30, 90, 98 Serret, 186, 280 Fricke, 131 Smith, 30, 130, 131, 142, 182 Gauss, 6, 14, 19, 21, 30, 31, 38, 45, 53, Staudt, 229 55, 58, 74, 180, 189, 142, 155, 159, Stieltjes, 229 171, 176, 182, 184, 187, 190, 199, Stirling, 279, 280 200, 209, 212, 214, 229, 230, 264, Sylvester, 30, 282, 287, 290, 292, 314 267, 302, 314, 315, 320 Tchébicheff, 268, 270, 278, 286, 300, Glaisher, J. (see Corrigenda), 272 314, 317 See Oed Wwe Waring, 31 Hammond, 293, 317 Wilson, 16, 31 Hermite, 127, 131 Zeller, 56 M. Bik 322 INDEX. [The numbers refer to the pages. ] Abelian equations, 186 Absolute value of a complex quantity, 103 Adjacent forms, 68 Algebraical solution of cyclotomic equa- tions, 190, etc. Ambiguous forms, 68, 79, 100 classes, 100; number of 171, ete. Argument of a complex quantity, 103 Associated forms, 79 Automorphic substitutions, 88, etc., 100, 123 Binomial congruences, 22; analogy to binomial equations, 184, 199 Binomial equations, 184, etc. Canon Arithmeticus, 31 Characters, generic, 134, etc.; table of, 135; how determined, 136; half the assignable characters impossible, 138, 176 Circle, division of the, 184, etc. Classes of quadratic forms, 66, etc.; number of ambiguous, 171; number of properly primitive classes, 230, etc., 247, 251, 253 Complex numbers, 103, etc. Composite numbers, 2; their resolution into prime factors, 3 Composition of substitutions, 63, 117 of forms, 140, etc.; direct or in- verse, 142 of classes, 148 of genera, 170 Congruences defined, 7 elementary theorems concerning, 1,9 limit of the number of solutions of, 9 linear, 10; simultaneous linear, 13 quadratic, 32, etc.; number of solutions of a quadratic con- gruence, 36 associated with the cyclotomic equations, 199 Conjugate complex quantities, 104 Continued fractions, 81 Critical points, 110 forms, 120 Cubic char acter, 224 equation of periods, 219 Cyclotomy, 184, etc. Definite and indefinite forms, 59 Derived forms, 60 Determinant of a quadratic form, 59 Diophantine equation of the second degree, 257 | Discriminants, fundamental, 253 Divisors of a number, 2; methods of discovering, 261, etc. Divisors of a quadratic form, 52 Duplication of forms and classes, 145; every class of the principal genus may be obtained by duplication, 178 Elliptic substitutions, 106 Equation of the periods (in cyclotomy) 194, 200, 219 Equivalence 61, etc.; proper and im- proper, 62; geometrical theory of, 103, ete. Euler’s function ¢ (n) 4, 30; the formula Xo (d)=n, 6 Exclusion, method of, 53 Exponent to which a number appertains, 17 Factors of large numbers, methods of discovering, 261, etc. Fermat’s Theorem, 15, 18; Euler’s ex- tension of, 16, 18; history of, 30 Forms, defined, 57; properly and im- properly primitive, 60; definite and indefinite, 59; equivalent, 61; sim- plest representative, 87 Genera of quadratic forms, 132, etc. ; composition of, 170 Geometrical representation of complex ~ quantities, 103 Geometrical applications of the theory of numbers, 184, 191 Hyperbolic substitutions, 106 Identical congruence, 9 transformation of 4 (a? —-1)/(«—-1), 215 Improper equivalence, 62, 100 INDEX. 323 [The numbers refer to the pages.] Improperly primitive classes, their num- ber, 167 Indices, 21, etc. Infinite, infinitesimal, 112 Irrationalities essentially connected with an equation, 190 Trreducibility of (a? —1)/(~-1), when p is prime, 186, 188; of the equations of the periods, 194 Irregular continued fractions, 81; de- terminants, 178, etc. Irregularity, exponent of, 180 Jacobi’s extension of Legendre’s symbol, 42 Laws of operation, 1, 104 Least residues, 7 Legendre’s symbol of reciprocity, 38 Linear divisors of a quadratic form, 52 Logarithm of a complex quantity, 104 Loxodromic substitutions, 106 Mobius’s circular relation, 106 Modular functions, 58, 131 Modulus of a congruence defined, 7 of a complex quantity, 103 Nets, 124, etc. Nodes, 124 Norm of a complex quantity, 103 of a net, 125 ; Normal form of linear substitution, 105 Opposite forms, 68, 157 Orders of quadratic forms, 67; number of classes in different orders com- pared, 158 Pellian equation, 90, etc., 130; trigono- “metrical solution of, 254 _ Periods of reduced forms, 77, 121 of properly primitive classes, 178 of the roots of the cyclotomic equation, 191, etc. Polygons, regular, which may be con- structed by Euclidean methods, 191 Powers, residues of, 17, 30 Prime numbers, 2; enumeration of, 273; distribution of, 278, etc. Primitive roots of a prime, 19, etc.; of a composite modulus, 26; of unity, 185 representations of a number, 59 quadratic forms, 60 Principal form, class, genus, 138 Proper and improper equivalence, 61 Quadratic residues and non-residues, 17, etc. equation of periods, 201 Reciprocity, law of quadratic, 38, 44, 176, 212; history of, 55 Reduced forms, 66, etc.; geometrical theory of reduction, 103, etc., 124 Relative primes, 2 Representation of numbers, 58, 65, 97 Representative forms, 72, 87 Residues with respect to a modulus, 7; quadratic, 32, etc. Resolution of a number into prime fac- tors, 3 Roots of a congruence, 8 of unity, 184, etc. Sheafs of equivalent triangles, 123 Stationary points of a linear transfor- mation, 105 Substitutions, linear, 61, etc., 88, 105, etc.; bilinear, 140, etc. Summation of Gauss’s series, 202, etc. Supplementary formule in the law of reciprocity, 37 generic characters, 135 Symbol of reciprocity, 33, 42 Transformation (see Substitution) Triangles, equivalent, 116 Unitary substitutions, 61 Unity, roots of, 184, etc. Wilson’s Theorem, 16, 30 Cambridge: ¥. : ‘PRINTED BY C. 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