Large Factors Found By ECM ^^^^^^^^^^^^^^^^^^^^^^^^^^ This file (champs.txt) contains information on the "top ten" factors found by the elliptic curve factoring method (ECM). Factors are included if they satisfy conditions 1 and 2. 1. They are one of the largest ten factors found so far by ECM and satisfying condition 2. 2. Let r = length(composite)/length(prime), where length is measured in decimal digits. Also, let r' = ln(composite)/ln(prime). Factors with max(r,r') < 2.2 are excluded as they could probably have been found more easily by another method such as MPQS, GNFS or SNFS. For historical interest, we also list factors of at least 40 decimal digits that satisfy condition 2 and were at any time the largest factor found so far by ECM ("champions"), even if they are no longer in the "top ten". Factors which at one time were the current "champions" are indicated by an asterisk. I imposed condition 2 from 1 Jan 2004 (r' term added 18 Sept 2005). For large factors that do not necessarily satisfy condition 2, see Paul Zimmermann's "top 50" list at http://www.loria.fr/~zimmerma/records/top50.html. So far there are four new entries in 2013, three (including the new p83 champ) found by Ryan Propper and one (a p77) by Sam Wagstaff. Ryan also found three factors which were temporarily in the top ten but were pushed out by larger factors. I reserve the right to exclude factorizations that were possibly obtained by "artificial" means. See for example Research Problem 7.27 of the book "Prime Numbers: a Computational Perspective" by Crandall and Pomerance. David Broadhurst has shown that this enables us to concoct examples where ECM finds factors of over 30,000 decimal digits! This file is available from http://maths-people.anu.edu.au/~brent/ftp/champs.txt or via the link at http://maths-people.anu.edu.au/~brent/factors.html See also Paul Zimmermann's list at http://www.loria.fr/~zimmerma/records/ecmnet.html Please send corrections/updates to champs@rpbrent.com R. P. Brent http://www.rpbrent.com =============================================================================== Summary ^^^^^^^ Factor Divides Found by Date ^^^^^^ ^^^^^^^ ^^^^^^^^ ^^^^ p83 YYYYMMDD ^^^ 16559819925107279963180573885975861071762981898238616724384425798932514688349020287 * 7,337+ R. Propper (13) 20130908 p79 ^^^ 2302872188505279576573535015926441913945044975483579529517513795897664211127797 * 11,306+ S. Wagstaff (22) 20120812 p77 ^^^ 18974159366624817405627752670504479132613571595050983959444958694223874973021 B_188 S. Wagstaff (19) 20130616 p75 ^^^ 336842026814486816413712532665671525518487238461533945786937785048474675329 * 11,304+ S. Wagstaff (16) 20120803 227432689108589532754984915075774848386671439568260420754414940780761245893 EM47 R. Propper (20) 20120912 p73 ^^^ 1808422353177349564546512035512530001279481259854248860454348989451026887 * 2,1181- J. Bos et al (21) 20100307 1042816042941845750042952206680089794415014668329850393031910483526456487 2,1163- J. Bos et al (15) 20100418 p72 ^^^ 201899170515693330875562714316614363610229288846992326512304705165161183 3,713- S. Wagstaff (12) 20120101 156667357389159077765620341252173218350111435299858918599291077497025529 3^560-2 R. Propper (14) 20130102 105411789654665518987995882770574113299251148356730172611945085373681497 3^554-2 R. Propper (18) 20130102 [end of current top ten] [following former champions listed for historical interest] p68 ^^^ 42593783346150223186979443437882164324892008462850480008134130873603 * 64*10^341-1 yoyo@home/M.Thompson (17) 20091228 p67 ^^^ 4444349792156709907895752551798631908946180608768737946280238078881 * 10,381+ B. Dodson (24) 20060824 p66 ^^^ 709601635082267320966424084955776789770864725643996885415676682297 * 3,466+ B. Dodson (23) 20050406 p59 ^^^ 20131492120828919814484857298874674155298711142397769181347 * 10,233- B. Dodson (11) 20050220 p57 ^^^ 167560816514084819488737767976263150405095191554732902607 * 2,997- B. Dodson (10) 20030621 p55 ^^^ 1139151258261034615880135106860446479526482959089061629 * 93^56+56^93 P. Gaudry (9) 20021213 p54 ^^^ 484061254276878368125726870789180231995964870094916937 * (6^43-1)^42+1 Note (8) 19991226 p53 ^^^^ 53625112691923843508117942311516428173021903300344567 * 2,677- C. Curry (7) 19980914 p49 ^^^ 1078825191548640568143407841173742460493739682993 * 2,1071+ P. Zimmermann (6) 19980619 p48 ^^^ 662926550178509475639682769961460088456141816377 * 24,121+ R. P. Brent (5) 19971009 p47 ^^^ 12025702000065183805751513732616276516181800961 * 5,256+ P. Montgomery (4) 19951127 p44 ^^^ 27885873044042449777540626664487051863162949 * p(19069) Berger-Mueller (3) 19950621 p43 ^^^ 5688864305048653702791752405107044435136231 * p(19997) Berger-Mueller (3) 19930320 p42 ^^^ 184976479633092931103313037835504355363361 * 10,201- D. Rusin (2) 19920405 p40 ^^^ 1232079689567662686148201863995544247703 * p(11279) Lenstra-Dixon (1) 19911028 Notes ^^^^^ Factors divide numbers of the form a^n +- 1 (abbreviated a,n{+-}) or partition numbers (p(n) is the n-th partition number) or other numbers described in the notes. Dates are in YYYYMMDD format. (1) Arjen Lenstra and Brandon Dixon on a MasPar (the first p40 by ECM). c89 = p40*p50, r = 2.22. (2) David Rusin using Peter Montgomery's program. c111 = p42*p70, r = 2.64. (3) Franz-Dieter Berger and Andreas M\"uller on a network of workstations. c99 = p44*p55, r = 2.25 and c139 = p43*p96, r = 2.85. (4) Peter Montgomery on an SGI workstation. c134 = p47*p88, r = 2.85. (5) Richard Brent on a Fujitsu VPP300. c130 = p48*p82, r = 2.71. (6) Paul Zimmermann on an SGI Power Challenge with Montgomery's program. c132 = p49*p84, r = 2.69. (7) Conrad Curry with George Woltman's mprime program using 16 Pentiums. c150 = p53*p98, r = 2.83. (8) Nik Lygeros and Michel Mizony with GMP-ECM. The input composite was c127 = (b^6+1)/(b^2+1)/13/733/7177, b = 6^43-1. c127 = p54*p73, r = 2.37. (9) Pierrick Gaudry with GMP-ECM. c121 = p55*p67, r = 2.2. (10) Bruce Dodson using George Woltman's Prime95. c301 = p57*p244, r = 5.28. (11) Bruce Dodson with GMP-ECM, c162 = p59*c103, r = 2.75. (12) Sam Wagstaff with GMP-ECM, c175 = p72*p104, r = 2.43. (13) Ryan Propper with GMP-ECM 6.4.3, B1=7600000000, B2=324909696561468, c237 = p83.p155, r = 2.86. 7^337+1 was a Cunningham "first hole". (14) Ryan Propper with GMP-ECM 6.4.3, c255 = p72.c184, r = 3.54. c255 is a divisor of 3^560-2. (15) Joppe Bos, Thorsten Kleinjung, Arjen Lenstra, Peter Montgomery, c318 = p73.p246, r = 4.36, details similar to (21). (16) Sam Wagstaff with GMP-ECM 6.4, c181 = p75*p107, r = 2.41 (17) yoyo@home/M.Thompson with GMP-ECM, c296 = p68*p229, r = 4.35. c296 is a factor of 64*10^341-1, see http://hpcgi2.nifty.com/m_kamada/f/c.cgi?q=71111_341 (18) Ryan Propper with GMP-ECM 6.4.3, c166 = p72.p94, r = 2.31. c156 is a divisor of 3^554-2. (19) Sam Wagstaff with GMP-ECM 6.4, c193 = p77.p117, r = 2.51. B_188 is the numerator of the Bernoulli number B188. (20) Ryan Propper with GMP-ECM 6.4.2, c256 = p75.p181, r = 3.41. EM47 is related to the Euclid-Mullin sequence. (21) Joppe Bos, Thorsten Kleinjung, Arjen Lenstra, Peter Montgomery, c291 = p73.p218, r = 3.99. "Some details of this computation: We used GMP-ECM with some modifications so we can run stage 1 on a cluster of PlayStation 3 game consoles (PS3) and stage 2 on a cluster of regular processors." (22) Sam Wagstaff with GMP-ECM 6.4, c191 = p79*p113, r = 2.42 About 600 curves with B1 = 800000000. (23) Bruce Dodson with GMP-ECM 6.0 on an Opteron cluster, c180 = p66*p114, r = 2.73. (24) Bruce Dodson with GMP-ECM 2006-03-13 [using GMP 4.1] on an Opteron cluster, c214 = p67*p147, r = 3.19. ================================================================================ Information on Curves and Group Orders ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ If g is the order of the group used to find the factor, the following table gives the second-largest prime factor (g2) and the largest prime factor (g1) of g, where known. In some cases the exact values are not known, but bounds can be given from knowledge of the phase 1 and phase 2 limits. This is indicated by the "<" and ">" symbols. The values of g2 and g1 have been deduced from information supplied by various (generally reliable) people, and have not been verified independently in all cases. C = C(g1,g2) = 1/mu, where mu is an estimate of the probability that a random integer close to p/12 has largest prime factor at most g1 and second-largest prime factor at most g2. Thus, C is an estimate of the expected number of curves to find the factor with phase 1 limit g2 and phase 2 limit g1. This assumes that the curves are chosen so that the group order is divisible by 12, which was not always the case for the computation which found the factor. It also assumes that group orders behave like random integers (apart from being a multiple of 12). The larger C, the more "improbable" it is that the group order is so smooth. [Note: my C values may be inaccurate, so I have stopped listing them until a more accurate program is written.] If the elliptic curve is known to be of the form b*y^2 = x^3 + a*x^2 + x with initial point (x1, y1), where x1 = u^3, u = (sigma^2 - 5)/(4*sigma), a + 2 = (1/u - 1)^3 * (3*u + 1)/4, then the parameter sigma is given. In this case the group order is divisible by 12. Factor g2 g1 C sigma ^^^^^^ ^^^^ ^^^^ ^^^ ^^^^^ p83 = 1655... 1143896321 7843501130401 ? 3882127693 p79 = 2302... 138483313 764489238641 ? 3648110021 p77 = 1897... 677443181 10607352143 ? 366329389 p75 = 3368... 173109949 1584686398147 ? 3885593015 p75 = 2274... 662505049 240668952133 ? 2224648366 p73 = 1808... 1923401731 10801302048203 570000 4000027779 p73 = 1042... 431421191 13007798103359 ? 3000085158 p72 = 2018... 96160859 736280462591 ? 1631036890 p72 = 1566... 224874313 1993543778599 ? 2677823895 p72 = 1054... 254998619 5319885583 ? 971177432 p68 = 4259... 79037141 723922811009 4500000 1998958586 p67 = 4444... 87373729 11805290281 6400000 834412411 p66 = 7096... 13153633 249436823 170000000 1875377824 p59 = 2013... 134939023 7285852169 180000 4114600819 p57 = 1675... 33587233 78756287 1500000 6329517009540700 p55 = 1139... 16576387 8139353693 230000 556090596 p54 = 4840... 8939393 13323719 4400000 599841120 p53 = 5362... 8867563 15880351 2400000 8689346476060549 p49 = 1078... 28393447 2700196643 16000 p48 = 6629... 141667 150814537 29000000 876329474 p47 = 1202... 2459497 903335969 85000 p44 = 2788... 949159 4818400261 49000 p43 = 5688... < 139894 < 14212100 > 2300000 p42 = 1849... < 2000000 < 100000000 > 20000 p40 = 1232... < 1000000 1209269 > 110000 =============================================================================== Compiled by R.P.Brent with assistance from K.Aoki, R.Backstrom, F.Berger, A.Bhargava, J.Bos, D.Broadhurst, A.Brown, J.Card, C.Casey, S.Cavallar, T.Charron, S.Chong, C.Clavier, C.Curry, B.Dodson, D.Doligez, W.Ekkelkamp, M.Fleuren, P.Gaudry, T.Granlund, R.Hooft, S.Irvine, P.Johansson, M.Kamada, B.Kelly, Y.Kida, T.Kleinjung, T.Kobayashi, Y.Koide, A.Kruppa, H.Kuwakado, A.Lenstra, P.Leyland, W.Lipp, N.Lygeros, A.MacLeod, J-C. Meyrignac, D.Miller, I.Miyamoto, M.Mizony, P.Montgomery, A.Mueller, T.Nokleby, S.Pelissier, E.Prestemon, R.Propper, M.Quercia, G.Reynolds, D.Rusin, W.Sakai, T.Shimoyama, R.Silverman, A.Steel, I.Tetsuya, M.Ukai, S.Wagstaff, G.Wambach, M.Wiener, G.Woltman, A.Yamasaki, P.Zimmermann and J.Zylstra. [see champs12.txt for 2012, champs11.txt for 2011, etc.] Last revised 17 September 2013.