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Date:         Fri, 4 Mar 2005 13:06:13 -0500
Reply-To:     Paul Zimmermann <Paul.Zimmermann@loria.fr>
Sender:       Number Theory List <NMBRTHRY@LISTSERV.NODAK.EDU>
From:         Paul Zimmermann <Paul.Zimmermann@loria.fr>
Subject:      new ECM record (59 digits) and GMP-ECM release (6.0)

During beta testing of GMP-ECM 6.0 (see below), a new ECM record prime factor was found by Bruce Dodson on an IBM/linux workstation (with 2.8Ghz xeon chips, 2Gb memory) at Lehigh University. The new record has 59-digits,

p59 = 20131492120828919814484857298874674155298711142397769181347,

and completes the factorization of the remaining 162-digit composite cofactor of the Most Wanted number 10^233-1, as p59*p103. The first step limit used was B1 = 260M, optimal for finding p60-factors; and the default step 2 limit used 1.2Gb memory. The lucky value of sigma was sigma = 4114600819, which determines an elliptic curve by the parameterization described by Brent (and recalled in Dodson's NMBRTHRY post of a previous ECM record with 57-digits, 29 July 2003). A Magma procedure written by Allan Steel was used to find the order of the elliptic curve, which gave factors

2^3 3 5 29 43 883 73327 76603 (small, < 10^6) 10038697 46168679 59527159 134939023 (medium, below B1) 7285852169 (large, between B1, B2).

This curve would have missed the p59 factor with p55-limits of B1=110M, since the 2nd largest prime factor is 134.9M; but the step 2 factor of the curve order is not especially large, with 7285M < 30*B1 << B2. An extended run on a beowulf cluster with p50-limits produced a large number of hard factors during 2004, including five of the top ten for the year, but none larger than 55-digits. Perhaps this indicates that breaking the 60-digit barrier is more likely with the larger limits (p55 or p60) for which the current GMP-ECM 6.0 substantially improves upon the previous version 5.0.3.

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GMP-ECM 6.0

We are pleased to announce the release of GMP-ECM 6.0, which may be downloaded from:

<http://home.in.tum.de/~kruppa/ecm-6.0.tar.gz> <http://www.loria.fr/~zimmerma/free/ecm-6.0.tar.gz> (MD5 e60ad7489614e94a301bfc6f3d87d1af)

This is a major release, with three new developers (Jim Fougeron, Laurent Fousse and Dave Newman), and many improvements with respect to the previous release (5.0.3):

* easier configuration and installation (configure/make); * new documentation (XML and "man" format); * new parser for symbolic input, and several new command line options; * improved code for Fermat numbers, especially in stage 2; * speed improvement in step 2, especially for large B2; * Brent-Suyama's extension for P+1; * computation of the expected number of curves and expected time to find factors of a given number of digits; * the code is now split between a library (distributed under LGPL), and a frontend (GMP-ECM, distributed under GPL), which makes it possible to call the library functions from another application.

We therefore advise all GMP-ECM users to upgrade to version 6.0.

We thank Hartmut Bock, Bruce Dodson, Torbjorn Granlund, Paul Leyland, Japke Rosink who tested preliminary versions, and Allan Steel, Karim Belabas who made useful suggestions concerning the library interface.

Jim Fougeron Laurent Fousse Alexander Kruppa Dave Newman Paul Zimmermann


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