■ • ■ ft m I n LIBRARY OF THE UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN 510.84 no. 131 -140 cop. 3 The person charging this material is re- sponsible for its return to the library from which it was withdrawn on or before the Latest Date stamped below. Theft, mutilation, and underlining of books are reasons for disciplinary action and may result in dismissal from the University. To renew call Telephone Center, 333-8400 UNIVERSITY OF ILLINOIS LIBRARY AT URBANA-CHAMPAIGN NOV 0113(2 SFP 1 9 i( % L161— O-1096 Digitized by the Internet Archive in 2013 http://archive.org/details/threenewmersenne138gill . 5/0.84- XIL lor DIGITAL COMPUTER LABORATORY UNIVERSITY OF ILLINOIS URBANA, ILLINOIS REPORT NO. 138 THREE NEW MERSENNE PRIMES, AND A CONJECTURE by Donald B. Gillies May 27, I963 Revised June 2k, 1963 Revised January 22, 1964 If p is prime, M = 2 - 1 is called a Mersenne number. 2 If u, = 4 and u. , = u. - 2, then M is prime if and only if u , = (mod M ) 1 I + 1 i ' P p - 1 p This is called the Lucas test (see Lehmer [3])- The primes M ^n , M , , and M, 1p _ which are now the largest known primes, were discovered by Illiac II at the Digital Computer Laboratory of the University of Illinois. The computing times were 1 hour, 23 minutes, 1 hour, 30 minutes, and 2 hours, 15 minutes respectively, and the calculations were checked by repetition. This brings to 23 the number of known Mersenne primes, namely for p = 2,3,5,7, 13, 17, 19, 31, 6l, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 42 53, 4423, 9689, 99^1, 11213. The numerical values of M £„ , M , , and M are shown in Table IV. The Lucas test was applied to all M , p < 12,143 for which no factor is known and the residue u , wa.s printed in modified octal (one base 4 digit and p - 1 ° l4 octal digits per word). The last five octal digits of these residues are shown in Table V for all p < 12,l43. As an indication of the speed of Illiac II, the residue for Mo took 100 hours on Illiac I (D. J. Wheeler), 5.2 hours on an IBM 7090 (Hurwitz [l]) and 49 minutes on Illiac II. The three values agree. An Illiac II word has a U5-bit mantissa and a 7-bit exponent. The kk program checks the multiple precision arithmetic modulo 2 - 1. Four errors were found in the residues given in Hurwitz [l], namely 3637 53313 not 67413 3847 14400 not 57652 ^397 44327 not 40174 4421 03013 not 25131 -1- and ten errors were found in the results of M. Berg and S. Kravits, privately communicated, for the range 6000 < p < 7 000. In no case has the .Illiac II program obtained different results on different runs* The author Is indebted to J. Brillhart for a list of prime factors < 2 3k of Mersenne numbers. For the range 5 000 < p < 17000 a wider range factor table was computed by Illiac IT and Table VI gives these factors. These were used to exclude some values M from the Lucas Test, The very large gap in p, kk2 3 to 9689, between successive Mersenne primes raises the question of what is the distribution of Mersenne primes and of factors of Mersenne composites. I. J. Good [6] suggested that the number cf prime M p < x is asymptotic to 2, 3 log log x, and D. Shanks [7] in effect suggests approximately ~~2—— i os i n£r v r+ ... 1n K n . 2 " y log 10 s L ^ g x ' ' wlil be suggested in what follows that loiT log log x ls the co ^ect asymptotic function. Since the number of Mersenne primes Is a slowly varying functi, D of Xj a stronger assumption will be made, which implies the above and can be tested by studying the distribution of prime factors of Mersenne composites. It has beer, proved by Fermat and Euler that all factors of M must be of the form 2kp + 1 and simultaneously of the form Q& + L . This means that, depending on p, k Is constrained to one of the sequences 1, 4, 5, 9, 12, . . . and 3, k, J, 8, 11, 12, . . . Thus the smallest potential factor is either 2 P + 1 or 6 P + 1 and thereafter potential factors are spaced, an average of ^p apart. i„ the conjecture that 'ollows, we suggest that statistically, this theorem sets a lower bound on potential divisors, but does not change the expected probability density of divisors above that lower bcund-the low densi.1 , potential divisors is just compensated for by a high conditional probability that, any one of these actually divides M . P The prime number theorem implies that a randomly chosen integer in the range s to s + A s is prime with probability ~ -L_ ass a s ,^ h , As J logs ' > s ro wt - - 0, and that the probability that such a number is prime and also divides some large random number N is ~^~ . The probability that there is -2- no factor of N in the range A < s < B < vN is given asymptotically by ■;. B ds s logs A log A log B CONJECTURE. If A < B < n/m then the probability that there is no divisor s of M in the interval [A,B] is given asymptotically by log A log B if A > 2p or by log 2p log B if A < 2p , and prime divisors are statistically independent. Thus Mersenne numbers are asserted to have the same likelihood of being prime, and to have the same statistical distribution of factors as integers of about the same size which are selected to have no factors less than 2p but are otherwise randomly chosen. If true, this conjecture implies (l) The probability that M is prime is — — *j. (2) The expected number of primes in any interval [x, 2x] in p log 2x is 2+2 lof log x which is asymptotically equal to 2. (3) The number of prime factors less than B < \T M is Poisson - P distributed with mean = log log B log 2p The observed frequency of Mersenne primes in the interval [x, 2x] in p is shown in Table I: X 5 J 10 20 40 80 1 1 160 320 6Uo 1280 2560 5120 Number of primes in [x, 2x] 2 3 1 1 3 2 1 2 3 2 The total number of prime factors less than 2 of Mersenne numbers in ,36 the interval 5000 < p < 7000 was estimated as it (7000) - it (5000) log log 2- log 12000 / and was also counted using the factor table prepared. Similar calculations were done for other intervals of length 2000 in p and the results are shown in Table II: Range of p 5000 to 7000 7000 to 9000 9000 to 11000 11000 to 13000 13000 to 15000 15000 to 17000 Factors < 2 observed 223 209 206 192 175 169 Predicted number 226 205 206 192 181 -k- This agreement appears very good, with perhaps a tendency toward fewer than the expected number of factors for large values of p. Finally a check was made on the independence of factors found, namely whether the number found per value of p follows a Poisson distribution. The exact distribution of the aggregate sample would be a sum of Poisson distributions corresponding to each value of p but since the mean varies only 15$ over the range, a comparison is made with the Poisson distribution obtained from the observed aggregate mean, in Table III: Number of prime factors < 2 1 2 3 k 5 Observed p with that frequency 513 1*95 199 76 12 1 Predicted number 525 k-jk 215 65 15 2.6 -5- 1. A. Hurwitz, "New Mersenne Primes," Math . Comp . , v. l6, 1962, pp. 2^9-251. 2. S. Kravitz, "Divisors of Mersenne Numbers 10,000 < p < 15,000," Math . Comp . , v. 15, 1961, pp. 292-293. 3. D. H. Lehmer, "An Extended Theory of Lucas' Functions," Ann , of Math., v. 31, 1930, pp. 419-448. 4. H. Riesel, "Mersenne Numbers," MTAC , v. 12, 1958, pp. 207-213. o 5. H. Riesel, "All Factors q < 10 in all Mersenne Numbers 2 P - 1, k p prime < 10 ," Math . Comp., v. 16, 1962, pp. 478-482. 6. I. J. Good, "Conjectures Concerning Mersenne Numbers," MTAC, v. 9, 1955, pp.120, 121. 7. D. Shanks, Solved and Unsolved Problems in Number Theory , Spartan Books, Washington, 1962, p. 198 4782202 7880546120 2952839298 6600059097 4149717240 2236500851 3345109918 3789509426 6297027892 7686112707 8945868247 2098152425 6319306585 0526768340 8748083442 9433264797 4258932476 2368833102 1633208954 8473548057 9994334130 9825989013 7438061871 0958104314 8680813778 3215304967 1560156328 2624414040 3981432076 2203627219 0408590790 5372034752 5610556407 1579263867 8752409855 7335652265 6108542128 5773210578 7905232886 5035355873 6156793636 5588992571 1574420153 8320917524 2184304691 8811427400 6621355593 0351685370 3976812686 3857503762 2778794958 0582081831 2617257010 0349820651 2529872677 2334895109 5346937568 3037038373 9996967715 8578890563 9115522613 4054957071 8452415821 9208223766 4420590145 9333065700 9722153962 5768534237 7048613857 8089775621 3011678112 9916640736 1746606697 8081867579 6691467124 6073712904 2005884089 2318638773 7867675292 8869537970 6698096740 6053530122 8535390369 6549022478 4924649007 9548986785 0331465554 6475504501 6861873548 6696437455 2614120640 7829496224 5202778896 2138602665 9331476876 9632208950 4278791624 6515193H3 2783175655 3779377194 5246753958 1928148666 8576584019 5907201794 1334958297 0319393884 3888104945 4604034208 7536563628 3321520731 6161430072 1769371426 2385175405 2084521466 5313301183 5519625918 4955693849 9025348780 3767164770 7593063443 6840084466 2559374434 5169031599 9349137664 6589689726 1419901530 4906547819 0562271712 2494707073 9716300953 7757434413 0792050.186 5532234466 5456456957 7433188504 4978250148 6634673721 3039209989 4852145190 9982328787 7248665051 3010816769 9028925187 1925006694 7215706536 2162486962 4056925686 5554296221 5522115604 2777866254 5936998801 0701861626 0147647429 5459830183 6512753634 6273267588 3060701410 5592548291 4977433929 7173680765 6109595999 1130918978 8238350131 6356726614 3596921823 9977196933 8743954039 9662367558 0528211207 1363963708 5805605116 0781770985 4525769880 3233381293 9272752101 9446295274 9031383555 1985197095 9288852564 1530178921 8675141014 5412030961 9127093436 9039522098 2803176689 4206132557 2349643638 4030564873 4929088422 5786292887 4722312190 3238528103 409-1824306 6189477407 2726552428 4693304474 8614549420 7679904175 9447165838 2816714104 3583120679 0501914527 5262875703 3997470720 6016882562 8274042701 7032260672 7980343479 5264257500 9185981307 7719322455 3947639606 0658821432 6603156141 4907405576 9805516626 5044447583 7567115164 9018119344 2236859424 1518437953 8933576545 2129944054 8553451558 5927342456 1825146813 7147206062 8778102124 0923708021 4922983496 3517952727 0302962970 1569276865 1165505008 0407282674 2523626446 9571076976 88.66137302 7893136096 7438271901 7385508484 6633734761 2084356798 3065059558 0729351106 5754424080 7550667082 9872337797 6867493698 3584523095 6389961206 1631863439 1967112086 4643846494 7096323007 2729200912 5861472679 9976249670 9852769503 5357539244 1620265772 0741248683 5922028289 8331114083 3923302433 9177979769 9031142584 3619350936 7544838111 944088L276 3588084204 4518049124 5438588418 0800945275 6266680576 2895476338 4641305107 7537732470 8249580453 3355717481 9650250708 1975046642 2826105697 5105642897 9895118219 2885976352 2290558989 4875761464 2139910911 5558645058 189926968:2 6225754111 Table 8V: Mgggcj in decimal 346 0882824908 5121524296 0395767413 3167226286 6890023854 7790489283 4450062208 0983411446 4364375544 1537075336 6448674763 5050186414 7070933237 3970608376 6904042292 6578964799 3709760358 4695523190 4548491005 0504149809 8185402835 0715968356 2252941968 0597622813 3454473972 0849260904 8551927706 260549H79 3590389060 7959811638 3872143299 4278763633 0955774381 9484486647 1124967685 7988881722 1203300082 1469684464 9561469971 9412692128 4336206463 3138595375 7720046244 2029064681 3260875582 5748847048 9384243989 2702368849 7864306309 3004422939 6033700105 4659538630 2009073043 9444822025 5909740670 0597330570 7995078329 6313093873 9885080198 4162586351 9452291304 2562936679 8595874957 2103117374 7796418895 0607019417 1750600195 7152430032 3636319342 6579851623 6047451209 0898647074 3078036229 8307038193 4454864937 5664799180 4258775574 9758539033 1573508289 1029392359 3527586171 8501994255 4854671861 0745487724 3988072960 6244911940 0666801128 2582409581 6458261761 8617466040 3480205646 6825143718 2554927847 7938099174 9580255263 3255265564 5774589415 0848953969 9028185500 5787087622 9329803338 2857354192 2825902216 9602665532 2108347896 0205168654 6011466757 9815060562 4748005507 1718250333 7375022675 0754417851 2950738594 3306845408 0269822896 3986562732 5971753720 8729564907 2830289749 7713585508 6795150871 0859216743 2185229188 1167063744 8496498549 0944505412 7744407940 7989539857 4694527721 5216658088 5754560477 4088429155 2729294869 6897496141 6149197398 4545283589 4324473601 3876096437 5051469921 5032683744 5270717186 8409183217 0948369596 2800611845 9574614558 9068811190 2531018735 9531915610 7319196071 1505984880 7002708870 5842749605 2030631941 9116692210 6176157609 5672419481 6062598903 2127984748 0810753243 8263209391 3796444665 7006013912 7836052300 2267434295 1943256072 8066126011 9378719405 1514975551 8754925215 4264394645 9638539649 1330969777 6533329401 8221580051 8288927807 2368602128 9827103066 1811511896 4131893657 8454002968 6001242039 1376964670 1839835949 5411248456 5597312460 7377987770 9207170671 0824505707 4572201550 1589959176 6244957768 0068024829 7667592059 2995410164 2247764456 7122214980 3657927708 4129255555 4281704557 2430846389 9881299605 1922731398 7291200902 0608820607 3576207589 2299473666 4058974270 3581178687 9875694315 0786544200 5560346962 5309399653 9559323104 6643003914 6465805452 9650140400 1942589755 2675534768 2486246319 5145149318 8170905972 5887801118 5028119055 9073677771 1874328140 8867867428 6302108275 1492584771 0129645183 3651979717 3751709005 0567364596 4696355331 3698192960 0026738958 3289299126 7383457269 8052599895 5997501176 6642010428 8854608569 9446442834 1952529487 8748841059 5750197438 7863551192 0421085580 4692460582 5338329677 7 19 469 1145 9901921324 9849688100 2118996828 4941331573 1640563047 2548086892 1823441538 1995905838 5241278684 0833479611 4199701017 9297835565 3650755329 1582986542 4622534682 7207503606 7407459569 5812738374 8717825918 5274731649 7058209518 1312905519 2427102805 7302314555 4793628499 0105092960 5584971257 7978984921 8399970574 1589767415 4830708629 1454847245 3672457262 2450131479 9926816843 1046444943 9022250504 8592508547 6189478888 9552527898 4009881962 0001486857 5640253136 5091456281 2719155485 8275085907 8914699790 1942622488 3789465551 Table IV: Mg^^x in decimal 281411 2013697373 1333931529 7584258419 1818662382 0136007878 9241934934 5515176682 2763138107 1509474563 3257074198 7893085350 7153734244 5016418881 8017893905 4870941439 1857257571 5657587064 7841835674 7070674633 4971880530 5087541682 1624325680 5558260711 1069194660 7460873056 9653608305 7159024277 4934226866 1839663091 8543346251 4537484258 6559823862 3504602922 7507801410 9071633484 3954778109 3397260096 9096770918 4394455575 4221115477 3437602069 7965006708 7884993478 0129772778 7853280743 2236554020 9315718023 1042992316 7588432457 0361041108 5096043976 9038450365 5140223496 2538566575 1207169661 6973527322 3611192684 6454751701 7345270113 7914817510 7820821297 6289467956 3109896076 7492250494 8342540733 3441412162 7833939461 5392125289 3201072613 6689293688 8156654916 7139517471 0452663709 1757556037 7415685576 6515513827 6137272816 9669265352 9666363787 2865397699 4160910777 7183593536 0026801245 1763345149 0439598324 8238364572 5121940639 1432635639 2256045560 4239600430 7799361927 3799005864 0042076309 2320813392 2624929420 7631295326 8033818471 5552558206 5930888994 8665570202 4038158563 1357894977 9767046261 8453279567 2576728920 5262311752 0147862478 1333183401 5084475386 7605266122 1734057972 1237414485 8037255554 6302200953 6301008145 8675247046 0461886203 9093555206 1953282409 5189510704 0793284825 0954625301 5187282399 7171764140 6633158043 0900861194 2578380931 0647489915 9440747632 8437785848 8254239211 7061493829 4029485257 1629792993 8894069587 7375448948 0811083452 9339432780 8452729789 8341351401 9391241966 1799488795 2103282581 1274221870 0634541149 7436572872 3284342636 9348804878 9934719624 0539596785 7676150371 6001966502 5216825011 7793178488 0120005054 2282136255 0520509209 7244598958 5236682747 7851619190 5032548531 1502940313 2178989005 1957511943 0134027728 2750390683 6511205878 9506019875 3121882187 7886570240 0729178418 6518589977 7885103067 4394589610 8645258766 4156928256 6417447061 6155305144 8522738845 4963505925 5410606458 4273238641 0950668763 6514447514 2690949529 5321992421 2594695157 6550091585 2117342092 3275882065 3276254086 1796505296 2053572563 5556040560 9783211154 7555908988 4338169197 4761581716 1606620557 3070003771 9473001345 1815560750 1590278421 6490142254 4571224546 9567932349 7089495466 8425436412 3477855761 9451003013 9080568383 4207726286 1872264610 9707506566 9281028000 5396170434 3991962002 0597945655 2777491388 3237756792 7200655457 6864079217 7441559278 2723508250 9284368355 4596679150 2296761018 5424578782 0420087274 0286172126 8457658875 3605769491 2241098665 9257736066 6241467280 1589886055 2548634588 0882227855 5057065092 7654941505 4547677180 6182965528 6626300550 9222254318 4597681941 2672760304 7460344175 5810292985 2017122655 5234439676 8165099191 2757420655 4807719021 8754158915 8087152904 9187829508 4121534009 1041975651 3021540478 4366041784 4675775899 8632083586 2079922540 8516265457 5406771169 7073232159 8828494377 9122171985 9556058979 0229178176 8286548287 8781804150 6065546004 7164104095 4857772017 5746887332 4068550450 6958262105 0431635658 5511384093 4900213323 7246346337 3977427405 8966758275 4420512857 4874581960 3352520056 5722951959 2569288171 5752767022 6045091173 5069504025 0166677552 1493207364 3654199488 4770105659 0937200575 7899989580 7757751266 2111505790 571744941? 2220160705 5024591611 6705990451 5042562065 1828929775 8303095152 4505497722 3951496482 1601858628 8614465019 3601771054 6777505109 2630309947 4739761857 6207575447 7254414271 5556242856 0865669527 1576359850 4544797181 6718801659 8695475251 4650565557 1845717916 8756691403 2072497856 8586718527 5866024596 0233528551 3944980064 3270302781 0422414497 1883680541 6897847962 6739147608 7696392191 Table 8V: ^11213 ' n decimal -9- Table V: Lucas Residues for p < 11,400 p residue 2 - 3 - 5 - 7 - 13 17 19 31 ■ 61 89 ■ 101 72066 103 55476 107 - 127 - 137 77550 139 53004 149 04577 193 06362 199 67307 227 63716 241 62747 257 35250 269 41506 271 67351 293 74572 307 40635 P residue 313 02246 331 77175 347 76076 349 67177 373 67521 379 32311 389 14135 401 20721 409 45336 421 74216 433 45210 439 06636 457 05600 467 22017 479 26225 503 77275 509 10315 521 - 523 67657 541 25253 563 75077 569 57547 599 55245 607 - 613 25474 631 57303 -10- P residue 647 74350 653 61575 661 50426 673 24350 677 04110 691 46704 709 26643 727 60377 733 77214 739 55065 751 43611 769 26011 787 17045 797 55244 809 56736 821 56017 823 11022 853 46414 887 36277 919 62325 947 71416 971 71262 983 07153 991 54514 997 26343 1061 50556 P residue P residue P residue 1063 33064 1069 26061 1087 26637 1109 67204 1123 22707 1151 47342 1163 47576 1171 02010 1217 76613 1237 55742 1259 56251 1277 74272 1279 - 1283 75032 1289 65347 1291 61137 1297 22725 1301 40241 1307 03053 1319 31014 1373 56733 1409 17120 1427 65142 1429 15715 1471 43741 1483 52161 1487 70562 1493 03173 1549 52273 1553 07421 1567 02570 1571 64574 1579 36064 1597 44266 1601 42154 1613 12030 1619 62340 1621 20507 1657 57656 1669 37217 1699 42750 1733 62344 1747 57322 1753 66055 1759 22017 1783 15265 1787 02767 1831 54121 1867 51104 1873 77122 1889 42676 1907 73435 -11- 1933 67541 1949 76457 1951 47123 2027 74437 2029 31211 2053 35065 2087 34206 2089 44263 2129 13006 2137 76527 2143 22504 2153 26730 2161 00314 2203 - 2221 74014 2237 76774 2239 67122 2243 40670 2267 73073 2269 45440 2273 21661 2281 - 2287 15072 2293 13215 2311 15173 2341 07415 UNIVERSITY of ILLINOIS UBRAM P residue P residue P residue 2547 32152 2557 73755 2571 65041 2577 01201 2381 21005 2383 42577 2423 40520 2437 44263 2441 63033 2467 50433 2473 25401 2477 46576 2503 23011 2521 54751 2551 06004 2557 32321 2633 17651 2647 17616 2659 52257 2671 07552 2683 73666 ?OZ 77061 2713 61363 2719 16103 2729 56435 2777 74644 2789 76456 2791 54413 2797 36762 2801 26322 2803 77752 2833 00365 2851 63335 2861 26370 2879 04010 2887 55047 2927 50235 2957 72234 2969 26336 2971 53516 5019 25616 3049 73343 3061 32511 3079 76331 3089 66237 3109 33761 3169 32333 3187 42211 3203 40242 3209 37347 3217 - 3229 64405 .12- 3251 16002 3259 44224 3271 52222 3301 72013 3307 62061 3313 10050 3531 51270 3343 76415 3371 57040 3373 36120 3389 64705 9407 47335 3413 50261 3449 26453 3461 03241 3463 57665 3467 23046 3469 21765 3547 75574 3559 45350 3583 42507 3607 45062 3613 16120 3617 35431 3631 14530 3637 53313 P residue P res 5 due P residue 3643 04606 3671 04031 3673 01626 3677 32350 3691 54715 3697 53745 3709 06427 3727 75371 3739 22413 3769 00747 3821 52075 3833 45453 3847 14400 3877 46507 3881 34503 3889 30737 3919 16520 3929 63331 3943 33442 3947 46201 4007 17770 4027 60265 4049 31260 4051 63236 4079 17563 4091 55650 4093 26670 4111 20437 4133 66046 4157 43640 4159 62544 4201 532UL 4219 51756 4231 51457 4241 11012 4253 - 4259 46007 4261 55632 4283 74774 4327 75542 4339 41356 4349 74465 4363 61114 4397 44327 4409 51070 4421 03013 4423 - 4451 06734 4463 17523 =13= 4481 70216 4493 36053 4519 01571 4523 22235 4547 62126 4583 46556 4591 47243 4637 65312 4643 51444 4651 00707 4663 52442 4673 40333 4679 14305 4703 54013 4721 04420 4729 40137 4751 77401 4783 77350 4789 02364 4799 04305 4817 70020 4831 33213 4877 75412 P res 8 due P residue P res 8 due 4889 2 4410 4909 61113 4937 26525 4951 22271 4969 20752 4973 03354 4987 71175 5023 35472 5077 27063 5081 74607 5099 67662 5113 20010 5153 52273 5309 40357 5333 44244 5351 05171 5407 51133 5419 70701 5443 51737 5479 17227 5503 26142 5527 41614 5573 34740 5581 31446 5647 40775 5653 50244 5669 57031 5689 32731 5693 47014 5749 47641 5839 24031 5851 37460 5857 11252 5869 00764 5879 52670 5923 16616 5953 32461 6007 07707 6037 2U20 6043 21605 6047 37000 6053 53471 6073 41646 6079 15712 6091 02043 =14= 6133 42630 6151 63451 6211 71252 6217 07377 6221 24166 6229 06517 6247 00472 6269 57356 6299 71037 6337 21676 6359 51351 6451 23476 6469 51252 6547 06546 6571 67142 6577 45051 6581 74210 6599 77554 6659 75241 6679 13275 6701 07636 6709 05700 6733 35544 p residue p residue p residue 6763 01753 6791 41143 6823 14573 6833 26431 6857 63102 69H 63345 6971 65345 6991 50365 7069 07313 7109 76754 7121 11254 7127 70746 7177 64624 7213 35572 7229 21270 7237 44431 7243 75126 7247 21344 7309 16030 7321 03466 7331 55736 7333 40632 7351 43525 7369 60027 7433 13456 7477 34613 7481 56621 7489 31253 7507 62414 7523 54030 7559 01556 7577 74000 7603 42610 7607 04024 7621 43263 7649 17277 7699 15637 7703 11617 7723 56523 7727 65716 7753 00553 7757 35327 7793 01165 7817 00650 7867 12212 7927 35532 7937 36343 7949 55402 8009 55062 8069 61334 8089 24237 8117 07547 -15- 8147 32143 8191 03624 8233 75260 8263 73755 8291 31062 8297 63521 8311 72342 8329 64615 8363 55516 8369 76051 8389 21672 8443 14744 8447 05235 8521 47055 8527 06705 8543 63267 8581 17603 8609 52627 8647 12277 8681 35061 8707 37065 8753 32153 8619 24665 8831 51436 8837 25460 p residue p residue p residue 8861 22326 8863 03214 8887 23230 8893 47223 8923 41014 8941 01034 8999 36025 9011 74242 9013 56660 9041 56532 9091 53605 9133 17234 9151 36255 9187 75427 9203 23113 9209 75536 9227 07577 9241 25121 9257 62553 9277 02036 9281 62622 9319 62503 9377 66277 9413 17541 9433 25035 9437 43446 9463 63616 9473 02247 9533 43134 9551 33274 9587 05303 9623 26262 9631 02445 9649 10710 9661 27003 9679 75063 9689 - 9697 24211 9719 32015 9721 13527 9749 37416 9767 03263 9769 51126 9781 34012 9787 21740 9839 12364 9857 23533 9871 32316 9867 33416 9901 15526 9907 01401 9929 44612 -16- 9931 25643 9941 - 9967 56346 10037 34070 10061 30610 10069 23577 10079 70743 10099 H311 10159 47661 10169 37134 10177 75465 10259 72321 10273 50763 10303 21613 10313 45100 10343 40266 10369 21740 10399 12436 10453 62061 10463 52513 10477 67452 10487 05675 10501 77772 10531 35724 10597 42547 10639 70051 p residue p res i due p residue 10667 55573 10709 21742 10711 33125 10723 26405 10729 55703 10771 50555 10781 00164 10789 27651 10831 42672 10909 63265 10937 47503 10939 63332 10957 64103 11003 57656 11027 10024 11057 60335 11069 64651 11093 03231 11113 24506 11117 57045 11131 50752 11159 12577 11161 50166 11177 45677 11213 - 11239 44172 11251 02106 11257 17745 11261 21260 11279 40455 11351 22147 11369 27670 11383 17361 11393 01511 11411 74045 11423 03630 11443 65257 11447 51354 11467 55443 11483 03450 11489 57555 11549 76107 11551 47006 11593 12504 11597 07213 11657 53703 11681 13417 11689 23607 11717 06015 11743 76351 11779 14053 11789 71203 -17- 11801 41730 11807 64000 11821 64453 11839 25215 11867 70645 11887 10035 11897 42220 11927 14155 11933 25571 11941 23743 11987 33200 12043 05651 12071 40741 12097 51143 12109 20110 12143 27361 Table VB: Factors of M p , 5000 < p < 17,000 factors P factors 5003 10007.1050631 5009 36425449 5011 80177 5021 40169 5023 5039 10079 5051 10103 5059 179331433 5077 5081 5087 40697.1678711 5099 5101 8110591 5107 51071 5113 5119 82978991 5147 4185169817 5153 5167 6531089.131696497 5171 10343 5179 248593.998770151 5189 155671.3362473 5197 31183.20195543 5209 1945613591 5227 1129033.47126633 5231 10463.41849.470791 5233 994271 5237 4022017 5261 42089 5273 38134337 5279 10559 5281 24715081 5297 4449481.45566266567 5303 10607.6024209.33737687 5309 OOe*<3 2372950817 oooo 5347 310127 5351 5381 48170713 5387 35028913631 5393 32359 5399 10799.5272458239 5407 5413 5o<26393 5417 433361. 3196031.200515673 5419 5431 54311. 1401199. 6040879577 5437 2087809 5441 3656353 04rvw 5449 2517439 5471 5477 5479 5483 5501 5503 5507 5519 5521 5527 5557 5563 5569 888247 21675553 662281 88337.3533441 940271.14639904343 5581 5591 5623 5639 5641 5647 5651 5653 5659 5669 5683 5689 5693 5701 5711 5717 5741 5743 5749 5779 5783 5791 5801 5807 5813 5821 5827 5839 5843 5849 5851 5857 5861 5867 5869 5879 5881 5897 5903 534049.623057 37813511 52462455217 1135847 . 1324014073 11279.2447327 49748069257 2828212481. 3346929073 218122607 867830287 875183.2436643081 11423 34303.45737 48224401 643217.13432968889 924641 926561 34807.200204113 139369 88805177 91658711 712847.3069690167 4105999 46889 16239857 662671081 11807 . 15489473. 42182839 . 238433977 -18- factors factors 59::::i 5927 198969391 5939 617657.706741001 5953 5981 3883379567 5987 60583758167 6007 6011 288529 6029 84407.249359441 6037 6043 6047 6053 6067 206279 . 3118439 . 677222809 52969672361 7906897.807711391 2530783 81237913 12263 884593 591649 37039 297457 61991.743881 12527 13959247 37663.1163404289 5029601. 5180489 . 57186553 150959359 3647759. 68423863. 1958694583 694871.4548241 12647 6343 9045119 6353 38119.2172727 63671 6373 38239.140207.207849023.590432959 6379 4972928063 6389 191671 6397 1944689 6421 5689007 6427 10951609 . 3959918927 . 52235468209 6449 51593 6451 6469 6473 2796337 6481 622177 . 2799793 . 38678609 6491 12983 . 69440719 . 12296978807 6521 4890751 6529 731249.1201337 6547 6551 13103 6553 1454767 . 10286532433 DcDcO 13127.26777041.179209279.34128597577 6569 6621553 6571 6577 6581 6599 6607 8034113 . 26285724863 6619 471762607 6637 5203409 6653 4305755071 6659 6661 33930121529 6673 6766423. 19551502967 6679 6689 142168007 6691 321169 6701 6703 442399 6709 6719 1290049 6733 6737 741071.1106754361 6761 26340857 6763 6779 35331931073 6781 108497 6791 6793 13261715767 6803 217697 68 [J, 3 6827 108958921 6829 8286786631 6833 6841 41047 . 64962137 . 77973719 6863 1608865639 6869 257051719 6871 288583.39027281 =19= P factors P factors 6883 1885943 7411 1022719.133709263.153496633 S899 13799 7417 118673.16269026327 5907 60312932423 7433 5911 7451 95506919 6917 55337 7457 19969847 . 7785883529c 16694955311 5947 180623.333457 7459 134263 5949 6879511 7477 5959 55673 7481 5961 584208887 7487 179689 5967 1072919.448730537 7489 5971 7499 99121783 5977 41863.209311.389414279 7507 6983 13967.13630817 7517 12012167.27743142241 5991 7523 5997 2169071. 3572164417 . 67983747617 7529 105407.209170679 7001 39107810033 7537 723553 7013 54547899457 7541 9711963409 7019 990366863 7547 16980751 7027 373119647 7549 7460767289 7039 1252943.1057032553.8541573097 7559 7043 14087 7561 589759 7057 621017 7573 45439 7069 7577 7079 14159.56633.1486591 7583 5889276287 7103 14207 7589 288383 7109 7591 1897751 7121 7603 7127 7607 7129 1867799 7621 7151 14303 7639 550009.1840097599 7159 26058761 7643 15287 7177 7649 7187 48871601 7 669 131093887.40807239817 7193 5409137 7673 184153,13918823 7207 172969 7681 3994121 7211 14423.57689.475927 7687 445847 7213 7691 15383.23104702303 7219 164362193 7699 7229 7703 7237 7717 19446841,13968927551.5775617332 7243 7723 7247 7727 7253 52613263 7741 8964079 7283 102267887 7753 7297 1868033 7757 7307 42715085233 7759 139663 7309 7789 872369 7321 7793 7331 7817 7333 7823 15647 7349 42197959 7829 39555975527 7351 7841 2367983.6837353 7369 7853 47119.255002617 7393 6525239233 7867 =20= 7873 7877 7879 7883 7901 7907 7919 ''927 7933 7937 7949 7951 7963 7993 8009 8011 8017 8039 8053 8059 8069 8081 8089 8093 8101 8111 8117 8123 8147 8161 8167 8179 8191 8209 8219 B22] 8231 8233 8237 8243 8263 8269 8273 8287 8291 8293 8297 8311 8317 8329 factors 1180951.1200160121 517471639 259250617 15767.441449 47407 . 442457 . 679487 546405329 63353 46405939823 498813937 152889601 2066542193 80111 128273.585930463 968554799c 11908411871 219927431 7043567 3498814409. 362 635 68311. 569943717 67 96671999.360437591 29811811823 712889 l6223o 41917649. 25919771153 25798649 6708343 3136129.76835137 800759 . 9412993 . 1089700903 4318513 14759783 1988999 920753.9667897.18480809 214007 . 14329002199 1054337 16487.4207243687 727673.19630607 1042399 . 28062017 . 62014409 36877151 199033.6874946759 382583 50119.154146263.297834569 p factors 8369 8377 50263.134033 .787439.2596871 8387 9276023 8389 8419 488303.4188149417 8423 353767 8429 67433.455167 .927191 8431 809377 8443 8447 8461 1404527 8467 203209 . 6655063 . 5910592559 8501 680081.221842097 8513 272417 8521 8527 8537 147843767 8539 13662401. 10089972727 8543 8563 32402393 8573 12345121.12254091887.21973130527 8581 8597 481433 8599 85991 8609 8623 80504329 8627 123348847 8629 258871 8641 46880604889 8663 17327 8669 2618039.1337175913 8689 8693 8699 8707 8713 8719 8731 8741 8747 8753 8761 8779 8783 8803 8819 8821 1008827657 . 1639075583 260791 417553.43790767 1603193 2877271 2514529 681487.6640121 69929.5926399 21021997487 192743 57336421553 228359.46912673033 2834567 . 226906129 . 7563748873 228983 405767 8363 =21= factors factors 8839 6523183 8849 52368383 8861 8863 8867 70937.9284174617 8887 8893 8923 8929 196439 8933 36232249.1478072047 8941 8951 17903 8963 663263 8969 13345873.1142273903 8971 59568301217 8999 9001 1512169 9007 90071 9011 9013 9029 722321.25913231.455621399 9041 9043 37268518009 9049 28721527.28938703 9059 18119.30293297 9067 21497004703 9091 9103 2454405479 9109 4955297.49625833 9127 146033.8707159 9133 9137 1005071.2704553 9151 9157 3076753 9161 86901247.12187849367 9173 495343 9181 1377151 9187 9199 53354201 9203 9209 9221 719239.91841161.168025063.5660569039 9227 9239 800633263 9241 9257 9277 9281 9283 5941121. 29352847 . 34031479 . 41532143 9293 4598548121 9311 1415273 9319 9323 895009 9337 2614361.2838449 9341 74729 9343 149489 9349 1793044711 9371 18743.7281660583 9377 9391 93911 9397 225529.2405633 9403 2576423.5735831 9413 9419 18839 9421 8573111 9431 679033 9433 9437 9439 273523343 9461 75689 9463 9467 1340299993.4104001303 9473 9479 18959.48532481 9491 531497 9497 531833 9511 95111 9521 361799 9533 9539 19079 9547 631801367 9551 9587 9601 2285039.3513967.16974569 9613 57679.146559799.2253979337 9619 615617 . 17256487 . 53840698033 9623 9629 27152451199 9631 9643 12362327 9649 9661 9677 607618831 9679 9689 9697 9719 9721 9733 2932747561 9739 263751599 9743 34626623.564723767 9749 9767 -22- factors factors Yl 63 >78l >787 )791 )8Q3 )8H 10267 19583.536076833 63388001753 7770313 >817 20556799.26619050743 )829 707689 o 14075129 >833 4562513 >839 >85l 78809 . 3723679 >857 W59 1104209.1656313 >871 >883 158129 . 10436449 . 212582737 03 >887 >901 )907 >923 2182682927.8488689889 829 ©SI )94l #49 8098487 >967 >975 299191c 7419913 o 10591327 . 193( L0007 240169 . 60282169 . 136255313 L0009 35311753 45613381103 503351. 34439287537 20183,484369 306665713 2804693831 101111 141863 1216681.4075879 162257.136599271 1948993. 62820641017 20327 L0039 ^0061 L0067 10069 L0079 L0091 L0093 L0099 b0133 L0139 tOUl L0159 10163 t0l69 L0181 488689 k0193 691 k0211 81689.735193 L0223 1846703167 L0243 18847121 L0247 184447.246419857 L0253 1293682529 l 0£59 10273 10289 10301 10303 10313 10321 10331 10333 10337 10343 10357 10369 10391 10399 3 10453 10457 10459 3549U78951 20543 246937 4532441 61927 20663.268607.4049753 1239961 163076513 62143 19389607 35037410167 5569087 146063 62743. 313711. 31794194791 11632186031 252313.508471759 60962911. 37429900087 193546471 464292847 2033089 10499 10501 10513 10529 10531 10559 10567 10589 10597 10601 3980103047 10607 10613 10627 10631 10639 10651 10657 10663 10667 10755119113 531551. 31946920433 73981847 2216657 . 39041962123 170609 10691 10709 10711 '096169 51383. 10200069281. 11345203673 10729 10733 10739 64399.31021 3810390503 172049 6633.465812201 =23= factors factors 21599.1024263553 1798943.4115610839 71590201 65119 542951 1477097 260809 21767.7574569.22123941871 7584754729 196039 40668191 153287 . 1472136847 65839 . 153623 . 18434641 87833.270610393 197767 13433153 . 3551168 63 . 29H37 05 69 464479 2502047.3971411263 7447777 6883896127 . 21232248047 111191. 978473 . 106319879 7672987079 22343.501913031 268153 268729.33954656167 1281703 608743 13554264839 271177 11311 1470431 11317 2444473.16568089 11321 29230823 11329 445614887 11351 j1..-j"JI^, 1 '.,_,' 21935540009 11369 11383 11393 11399 1367881. 21834921289 11411 11423 11437 309142111 11443 11447 11467 11471 22943.144075761.48126649327 11483 11489 11491 7646938753 11497 528863.21315439 11503 58918367 11519 23039 11527 20587223 11549 11551 11579 23159 11587 76659593 11593 11597 11617 1858721.43494049.56778668351 11621 836713.107378041 JLJLdo