On December 2nd, 2003, NFSNET completed the factorization of 2^757-1.

The factors of the C213 cofactor are:

p79 = 572213702200206782424822797509585774915131282780938840696234625 \
         3182128916964593 and

p134 = 24033821640983508088736273403005965446689002356344332130565066 \
           643193813901119771090424269412054543072714914742665677774247325292327559

The number 2,757- was chosen because it's both a Cunningham number and a prime-index Mersenne number. The composite cofactor had 213 digits but because we used SNFS and couldn't exploit the known factorization, we had to factor the full 228 digits of 2^757-1.

The polynomials used were 2x^6-1 and m-x which share a common root m=2^126 modulo 2^757-1.

Sieving began on July 31st, 2003 and completed on October 13th, 2003 using 13745 work units of sieving effort.

On average there were 120 machines active daily however the daily participation levels varied over the project due to the temporary donation of a large number of cycles from one contributor. Approximately 76.7 million relations were collected.

The filtering and merging of the 76.7M relations began October 13th and finished on 28th October with the production of a matrix with slightly over 7.522 million rows and columns.

Linear algebra on that matrix took 32 days 16 hours total elapsed time, with 26d 11h recorded in the many log files along the way. The 6d 5h discrepancy is accounted for by time lost to crashes and to scheduled downtime on the cluster at Microsoft Research in Cambridge. The actual cpu time was about 7600 hours (i.e. 30 cpus at about 40% loading).

We were fortunate in that the factors were found on the first dependency, which took 157 minutes on a 2.53GHz P4 machine.

This was the hardest factorization yet completed by NFSNET, just pipping 10,227- which completed three months ago. The current project, 2,811-, is even harder but is still running along smoothly.

As always, thanks are due to all the people who contributed their machine cycles and their enthusiasm. Further, we acknowledge the contribution made by CWI in providing us with their NFS code for us to use and to redistribute in a modified form.

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

The NFSNET administrators:

Chris Card
Jeff Gilchrist
Don Leclair
Paul Leyland
Richard Wackerbarth


Contributors to the 2^757-1 factorization:

Brian Beesley, Chris Card, Robert Clark, Richard Farmbrough, Decio Luiz Gazzoni Filho, Jeff Gilchrist, Francois Grieu, Rich Harms, Sander Hoogendoorn, Simon Josefsson, Rick Lavoie, Don Leclair, Paul Leyland, Tommy McCabe, Kirk Pearson, Andreas Pipp, Nathan Russell, Igor Schein, Martin Schroeder, Team Prime Rib, Richard Wackerbarth, and 81 others who have not selected to be listed here.