On August 23rd 2003, NFSNET completed the factorization of 10^227-1.

The factors of the C212 cofactor are:

p72 = 617431748093238554410874808506989442264783116181403206744221352101186333 and

p141 = 159061592587323425117771471330677165732172663157935428592494153340689013 \
           866064831389024866905836001165382255194784215933028413820020114354227

We used SNFS with the polynomials x^6-10 and m-x which share a root m=10^38 mod 10^227-1. We used primes up to 50 million on the rational and algebraic sides, with up to two large primes < 1 billion on each side.

The sieving stage was performed by NFSNET participants using 283 machines from 98 participants. On average there were about 90 machines active daily. Sieving began on April 28th, 2003 and finished on July 7th, 2003.

Approximately 17.6M lines were sieved and about 75 million relations were found. The filtering stage reduced these to a matrix which had 7,925,462 rows and 7,930,179 columns.

This matrix is one of the largest ever completed.

The linear algebra was performed on the MSRC cluster using 28 of its 1Ghz P-III CPUs. It began on July 24th but crashed several times. The computation was eventually completed at 20:22 GMT on August 15th, an elapsed time of 22d, 10.5 hours, or 538.5 hours. The recorded time was 168250 seconds (i.e., 467.9 hours or 19d 11.9h) with the discrepancy being time lost to crashes. The total cpu time was about 14 times this figure, or 6500 hours, as each cpu was about 50% loaded. The linear algebra found 105 dependencies in the 7,925,462 x 7,930,179 matrix. The 72-digit and 141-digit factors were found on the second dependency; each dependency took 129 minutes to process on a 2.5GHz P4 machine.

For more information visit http://www.nfsnet.org.

The NFSNET administrators:

Chris Card
Jeff Gilchrist
Don Leclair
Paul Leyland
Richard Wackerbarth

Contributors to the 10^227-1 factorization:

Brian Beesley, Andrei Belenko, Chris Card, Robert Clark, John Dilick, Richard Farmbrough, Decio Luiz Gazzoni Filho, Jeff Gilchrist, Francois Grieu, Rich Harms, Sander Hoogendoorn, Simon Josefsson, Rick Lavoie, Don Leclair, Aaron Lehmann, Paul Leyland, Tommy McCabe, Kirk Pearson, Andreas Pipp, Jeramy Ross, Nathan Russell, Igor Schein, Martin Schroeder, Team Prime Rib, Richard Wackerbarth and 73 others who have not selected to be listed here.