=============================================================================== Large Factors Found By ECM ^^^^^^^^^^^^^^^^^^^^^^^^^^ This file (champs.txt) contains information on the "top ten" factors found by the elliptic curve factoring method (ECM) as at 11 Nov 2005. 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 100" list at http://www.loria.fr/~zimmerma/records/top100.html and my list of factors found in 2003, champs03.txt. Congratulations to Bruce Dodson, for having six entries in the top ten, including his spectacular new p66 champion, found on 6 April 2005 using GMP-ECM on an Opteron cluster. 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://wwwmaths.anu.edu.au/~brent/ftp/champs.txt or via the link at http://wwwmaths.anu.edu.au/~brent/factors.html Please send corrections/updates to champs@rpbrent.com R. P. Brent http://www.rpbrent.com =============================================================================== Summary ^^^^^^^ Factor Divides Found by Date ^^^^^^ ^^^^^^^ ^^^^^^^^ ^^^^ p66 YYYYMMDD ^^^ 709601635082267320966424084955776789770864725643996885415676682297 * 3,466+ B. Dodson (12) 20050406 p64 ^^^ 4344673058714954477761314793437392900672885445361103905548950933 20050905 10,311- Aoki & Shimoyama (16) p63 ^^^ 516469933130631687266967194982169414626403685360388146231581267 3,533+ A. Kruppa (18) 20051010 p62 ^^^ 31069150378873790895208046895771360949463293546412105951449429 2,2034L B. Dodson (15) 20050411 p60 ^^^ 186347963875290244607785015115115186185151460498163342326993 2,1099+ B. Dodson (19) 20051003 p59 ^^^ 20131492120828919814484857298874674155298711142397769181347 * 10,233- B. Dodson (11) 20050220 14311578668849306832064176821408791945900114506618448115473 prSm(73) S. Chong (17) 20051006 p57 ^^^ 167560816514084819488737767976263150405095191554732902607 * 2,997- B. Dodson (10) 20030621 p56 ^^^ 69787377067722881486602094502761253930262932578924438539 2,827+ K. Aoki (14) 20031226 28723493629440097782059388914974037930708633570183584249 2,1678M B. Dodson (13) 20051016 [end of current top ten] [following former champions listed for historical interest] 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 (after conversion to GMT). (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) Bruce Dodson with GMP-ECM 6.0 on an Opteron cluster, c180 = p66*p114, r = 2.73. (13) Bruce Dodson with GMP-ECM, c252 = p56*c197, r = 4.50. (14) Kazumaro Aoki with GMP-ECM, c249 = p56*p193, r = 4.45. (15) Bruce Dodson with GMP-ECM 6.0 on an Opteron cluster, c187 = p62*p126, r = 3.02. (16) Kazumaro Aoki & Takeshi Shimoyama with GMP-ECM, c311 = p64*p247, r = 4.86. (17) Samuel Chong with GMP-ECM 6.0.1, c149 = p59*p91, r = 2.53. p59 is a factor of the Smarandache number prSm(73), where prSm(n) = decimal concatenation of the first n prime numbers. (18) Alexander Kruppa with GMP-ECM, c175 = p63*p112, r = 2.78. (19) Bruce Dodson with GMP-ECM 6.0, c242 = p60*c182, r = 4.03. ================================================================================ 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. 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. 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 ^^^^^^ ^^^^ ^^^^ ^^^ ^^^^^ p66 = 7096... 13153633 249436823 170000000 1875377824 p64 = 4344... 159860663 481875853859 470000 1917732841 p63 = 5164... 10810561 20946756331 15000000 3001167417 p62 = 3106... 40945981 390171012887 890000 1507467457 p60 = 1863... 95201383 476929979377 150000 2504147897 p59 = 2013... 134939023 7285852169 180000 4114600819 p59 = 1431... 35224997 84317840041 350000 2477705241 p57 = 1675... 33587233 78756287 1500000 6329517009540700 p56 = 6978... 27554129 750546089 550000 4029008539 p56 = 2872... 66718787 9186462143 110000 2403460401 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, D. Broadhurst, A. Brown, J. Card, C. Casey, S. Cavallar, T. Charron, S. Chong, C. Clavier, C. Curry, B. Dodson, D. Doligez, M. Fleuren, P. Gaudry, T. Granlund, S. Irvine, P. Johansson, M. Kamada, Y. Kida, 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, 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 champs2.txt for more details, champs04.txt for 2004, etc.] Last revised 11 November 2005.