=============================================================================== Large Factors Found By ECM ^^^^^^^^^^^^^^^^^^^^^^^^^^ This file (champs.txt) contains information on large factors found by the elliptic curve factoring method (ECM) up to the end of 2002. Factors are included if 1) they were at any time the largest factor found so far by ECM and at least 40 decimal digits ("champions"); or 2) they are one of the largest ten factors found so far (the "top ten"). New entries will be accepted if they are in the current "top ten". At present this means they will have to be at least 51 decimal digits. The largest factor so far, found by Izumi Miyamoto using GMP-ECM, has 55 decimal digits. In 2002 five new entries were submitted: a p53 and a p51, both by Alexander Kruppa (they eliminated p50s by Paul Leyland and Andy Brown); a p52 by David Broadhurst (which eliminated a p51 by Sean Irvine); a p51 by Tetsuya Kobayashi (which eliminated the p51 by Alexander Kruppa); and a p55 by Pierrick Gaudry (which eliminated the p51 by Tetsuya Kobayashi). The overall list is given below. Factors which at one time were the current "champions" are marked by an asterisk. This file is available from ftp://ftp.comlab.ox.ac.uk/pub/Documents/techpapers/Richard.Brent/champs02.txt Please send corrections/updates to rpb@comlab.ox.ac.uk R. P. Brent http://www.comlab.ox.ac.uk =============================================================================== Summary ^^^^^^^ Factor Divides Found by Date p55 ^^^^^^ ^^^^^^^ ^^^^^^^^ ^^^^ ^^^ YYYYMMDD 7230880127526821693925059508972082952702133004552346281 * 629,59- I. Miyamoto (14) 20011006 1139151258261034615880135106860446479526482959089061629 93^56+56^93 P. Gaudry (15) 20021213 p54 ^^^ 484061254276878368125726870789180231995964870094916937 * (6^43-1)^42+1 Note (8) 19991226 113944651856655107794996103150041939333993926230123191 (3^64-1)^63+1 Note (10) 20000321 p53 ^^^ 81477382431617858607629654669086224895030590860856949 HP49(84) A. Kruppa (9) 20020525 53625112691923843508117942311516428173021903300344567 * 2,677- C. Curry (7) 19980914 p52 ^^^ 9550932766611135096816626304308249655491758024828481 858,140+ D. Broadhurst (13) 20021010 7517596214490051335825344881028792801065608068512131 102!-1 S. Irvine (11) 20011101 2414704785479757305369621862192236137072000732339933 96,98+ P. Montgomery (4) 19990729 p51 ^^^ 946245326997772574146543757184074761849292544326417 875,41- I. Tetsuya (12) 20000514 [end of current top ten] [following former champions listed for historical interest] p49 ^^^ 1078825191548640568143407841173742460493739682993 * 2,1071+ P. Zimmermann (6) 19980619 p48 ^^^ 662926550178509475639682769961460088456141816377 * 24,121+ R. P. Brent (5) 19971009 p47 ^^^ 28207978317787299519881883345010831781124600233 * 30,109- P. Montgomery (4) 19960225 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{+-LM}) or partition numbers (p(n) is the n-th partition number) or Lucas numbers (Ln is the n-th Lucas number). Dates are in YYYYMMDD format. (1) Arjen Lenstra and Brandon Dixon on a MasPar (the first p40 by ECM). (2) David Rusin using Peter Montgomery's program. (3) Franz-Dieter Berger and Andreas M\"uller on a network of workstations. (4) Peter Montgomery on an SGI workstation. (5) Richard Brent on a Fujitsu VPP300. (6) Paul Zimmermann on an SGI Power Challenge with Montgomery's program. (7) Conrad Curry with George Woltman's mprime program using 16 Pentiums. (8) Nik Lygeros and Michel Mizony with GMP-ECM. The input composite was c127 = (b^6+1)/(b^2+1)/13/733/7177, where b = 6^43-1. (9) Alexander Kruppa with GMP-ECM 4c. See http://www.worldofnumbers.com/topic1.htm for the definition of HP49(n). (10) Nik Lygeros and Michel Mizony with GMP-ECM. (11) Sean Irvine with GMP-ECM. The p52 is a factor of 102! - 1 (c148). (12) Izu Tetsuya using ecmx.ub (revised) from Kida's UBASIC package, equivalent sigma = 401801850655068425861392031758571515108530228849089 (see champs2.txt for more information on curve parameterisation). (13) David Broadhurst with GMP-ECM. (14) Izumi Miyamoto with GMP-ECM 4c on 800 Mhz Pentium III under Linux. (15) Pierrick Gaudry with GMP-ECM. ================================================================================ 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 ^^^^^^ ^^^^ ^^^^ ^^^ ^^^^^ p55 = 7230... 25268183 41286269351 150000 267937500 p55 = 1139... 16576387 8139353693 232000 556090596 p54 = 4840... 8939393 13323719 4400000 599841120 p54 = 1139... 3015007 101673721 3500000 718797804 p53 = 8147... 8193599 322677167 690000 219538831 p53 = 5362... 8867563 15880351 2400000 8689346476060549 p52 = 9550... 9917191 58860449 710000 1991445260 p52 = 7517... 7044757 118052531 700000 1689607121 p52 = 2414... 25190591 30318953051 39000 p51 = 9462... 550177 7569839 38000000 [note (12)] p49 = 1078... 28393447 2700196643 16000 p48 = 6629... 141667 150814537 29000000 876329474 p47 = 2820... 1127603 209558929 370000 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 F. Berger, D. Broadhurst, A. Brown, J. Card, S. Cavallar, T. Charron, C. Curry, B. Dodson, D. Doligez, M. Fleuren, P. Gaudry, T. Granlund, S. Irvine, Y. Kida, T. Kobayashi, A. Kruppa, H. Kuwakado, A. Lenstra, P. Leyland, N. Lygeros, A. MacLeod, J-C. Meyrignac, D. Miller, I. Miyamoto, M. Mizony, P. Montgomery, A. Mueller, T. Nokleby, E. Prestemon, M. Quercia, D. Rusin, R. Silverman, A. Steel, I. Tetsuya, M. Ukai, S. Wagstaff, G. Wambach, M. Wiener, G. Woltman, A. Yamasaki and P. Zimmermann. [see champs1.txt for factors of at least 40 decimal digits found to 31 December 1998, champs2.txt for more details, champs01.txt for 2001, etc.] Last revised 31 December 2002.