|55||7523020307427066273763456609903850552762967576510974081||10260+1||43e6||3933468262||24 Jun 2004||B. Dodson|
|55||1268413494411135671239686038243358243539607519968737801||31557+1||43e6||1073421943||18 Nov 2004||K. Aoki|
|54||477350833522476258826705274670317082147893737193497151||3577-1||43e6||1939606094||10 May 2004||K. Aoki|
|53||54093886887062230599082516017052647105427126977713417||3508+1||43e6||1988971643||06 Jun 2004||B. Dodson|
|53||42156379589779912226383108151000450094080783333678881||2845+1||43e6||2242386807||20 May 2004||B. Dodson|
|53||33961207341909494975722524207839732407744380168756907||3*2653-1||860500||879193285||07 Oct 2004||R. Backstrom|
|53||18489093999230985031269595355473053970276854216314509||7286+1||43e6||1937590108||27 Jul 2004||B. Dodson|
|53||11984437123431823969432655822865428382472038412325377||11473+1||43e6||3782281521||15 Jul 2004||K. Aoki|
|51||738313998235889565301599572531640626810525425999201||2984+1||11e6||1073040516||26 May 2004||R. Keiser|
|51||197093734279293731287445626494587381600438850930731||5665-1||43e6||2891223763||08 Nov 2004||B. Dodson|
Lehigh's Beowulf. Dodson's factors above, five of the top ten for the year, were found using Lehigh's beowulf cluster. During November 1, 2003 to December 31, 2004 there were 122 Cunningham factors found by ECMNET, of which 42 were found by Dodson using the cluster. Dodson intends a more narrowly focused effort on factors of 50 digits or more in 2005. An analysis of the ECM efforts for the year 1998, when the first factor of more than 49 digits was found, is given in Champs. The count of 40 - 49 digit factors was Dodson 22; Zimmermann 18; Montgomery 12, but the only factor that counted was Curry's 53 digit factor. (These earlier factors of Dodson were found using binaries of Montgomery's ECM/FFT program, below.) See also the all-times all-ecm-programs top-50 table, and the gmp-ecm top ten from 1998, 1999, 2000, 2001, 2002, 2003.
History. Richard Brent has predicted in 1985 in a paper entitled Some Integer Factorization Algorithms using Elliptic Curves that factors up to 50 digits could by found by the Elliptic Curve Method (ECM). Indeed, Peter Montgomery found in November 1995 a factor of 47 digits of 5^256+1, and Richard Brent set in October 1997 a new genuine record with a factor of 48 digits of 24^121+1.
Goal. The original purpose of the ECMNET project was to make Richard's prediction true, i.e. to find a factor of 50 digits or more by ECM. This goal was attained on September 14, 1998, when Conrad Curry found a 53-digit factor of 2^677-1 c150 using George Woltman's mprime program. The new goal of ECMNET is now to find other large factors by ecm, mainly by contributing to the Cunningham project, most likely the longest, ongoing computational project in history according to Bob Silverman. A new record was set by Nik Lygeros and Michel Mizony, who found in December 1999 a prime factor of 54 digits using GMP-ECM.
Free implementations of ECM.
Bibliography. To know how ECM works and the history of the factorization of Fermat numbers by ECM and other methods, look at the paper Factorization of the tenth Fermat number by Richard Brent. Looking at the old paper Some Integer Factorization Algorithms using Elliptic Curves you'll see that very few improvements were made to ECM since 1985. One of these improvements is the FFT continutation invented by Peter Montgomery, and detailed in his dissertation entitled An FFT extension of ECM. You might also look at the paper A Practical Analysis of the Elliptic Curve Factoring Algorithm, by Bob Silverman and Sam Wagstaff, Mathematics of Computation vol. 61, July 1993. People reading german may look at Franz-Dieter Berger's Diplomarbeit entitled ``ECM Faktorisieren mit elliptischen Kurven''. Finally, have a look at the FactorWorld page from Scott Contini.