=============================================================================== Large Factors Found By ECM ^^^^^^^^^^^^^^^^^^^^^^^^^^ This file (champs.txt) contains information on large factors found by the elliptic curve factoring method (ECM). 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 48 decimal digits. In 1999 there are nine new entries so far (one p54, one p52, one p50, one p49, three p48, two p47), but the two p47 and two p48 have already dropped off the list, leaving five new entries. The largest factor so far, found by Nik Lygeros and Michel Mizony using GMP-ECM, has 54 decimal digits. 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/champs.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 p54 ^^^^^^ ^^^^^^^ ^^^^^^^^ ^^^^ ^^^ YYYYMMDD 484061254276878368125726870789180231995964870094916937 * (6^43-1)^42+1 Note (11) 19991226 p53 ^^^ 53625112691923843508117942311516428173021903300344567 * 2,677- C. Curry (8) 19980914 p52 ^^^ 2414704785479757305369621862192236137072000732339933 96,98+ P. Montgomery (5) 19990729 p50 ^^^ 28352547568921790651747638524393003422138037231571 3,1179L P. Zimmermann (13) 19991013 p49 ^^^ 7612068647760892587567279171698469451260170146501 6,250+ M. Quercia (9) 19981112 2359597690909288137577710805751790832324948920367 (14^82-1)^81+1 Note (10) 19991028 1078825191548640568143407841173742460493739682993 * 2,1071+ P. Zimmermann (7) 19980619 p48 ^^^ 901584692755427378722839770656354167189649601351 2,695- C. Curry (12) 19990607 662926550178509475639682769961460088456141816377 * 24,121+ R. P. Brent (6) 19971009 595580447139800838293908535595226727082115962317 p(20219) B. A. Dodson (4) 19981206 [end of current top ten] [following former champions listed for historical interest] p47 ^^^ 28207978317787299519881883345010831781124600233 * 30,109- P. Montgomery (5) 19960225 12025702000065183805751513732616276516181800961 * 5,256+ P. Montgomery (5) 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) Bruce Dodson with Peter Montgomery's program. (5) Peter Montgomery on an SGI workstation. (6) Richard Brent on a Fujitsu VPP300. (7) Paul Zimmermann on an SGI Power Challenge with Montgomery's program. (8) Conrad Curry with George Woltman's mprime program using 16 Pentiums. (9) Michel Quercia with GMP-ECM. (10) Nik Lygeros and Michel Mizony with GMP-ECM. (11) Nik Lygeros and Michel Mizony with GMP-ECM. If b = 6^43-1, the input was c127 = (b^6+1)/(b^2+1)/13/733/7177. (12) Conrad Curry with George Woltman's program on a 400MHz Pentium P-II. (13) Paul Zimmermann on a Dec Alpha 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 the information given in the "Details" section below 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 ^^^^^^ ^^^^ ^^^^ ^^^ ^^^^^ p54 = 4840... 8939393 13323719 4400000 599841120 p53 = 5362... 8867563 15880351 2400000 8689346476060549 p52 = 2414... 25190591 30318953051 39000 p50 = 2835... 840299 1315315249 1700000 2009193935 p49 = 7612... 2773699 37762327 720000 1170774930 p49 = 2359... 2654917 316054537 300000 1656203751 p49 = 1078... 28393447 2700196643 16000 p48 = 9015... 6747353 35174779 210000 42254462032221 p48 = 6629... 141667 150814537 29000000 876329474 p48 = 5955... 3973447 9010207411 67000 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, A. Brown, J. Card, S. Cavallar, T. Charron, C. Curry, B. Dodson, M. Fleuren, T. Granlund, Y. Kida, H. Kuwakado, A. Lenstra, P. Leyland, N. Lygeros, A. MacLeod, J-C. Meyrignac, M. Mizony, P. Montgomery, A. Mueller, T. Nokleby, E. Prestemon, M. Quercia, D. Rusin, R. Silverman, 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, champs98.txt for 1998, etc.] Last revised 31 December 1999.