NFSNET is pleased to announce that the large prime factors of 3^491+1 are

608926431044827575733062235676626887409859691974652906541738
42691745096014633691396823

with 86 digits and

758422573208035555889084544606175346326287871482159614800149
105610008730995857029540463694434227119503436377901095117590
7411916995516251500675474169

with 148 digits.

No factors of this number, other than the trivial factor "4", were previously known. It is one of the largest SNFS factorizations yet completed.

We used the polynomials x^6+3 and x-m, which share a root m=3^82 modulo 3^491+1. The factorbases included primes up to 50 million on each side and up to two large primes less than 1 billion were allowed in relations. We sieved 42.1 million lines of length 90 million and collected 80 million relations. Sieving began on 28th March 2004, finished on 22nd August and used 36.4 kWU of effort.

Filtering and merging of the relations was done by Richard Wackerbarth. The matrix produced had 8063681 rows, 8064612 columns and a weight of 455165498. This matrix was far too big to run on any single-cpu system available to us. Memory usage alone was well in excess of 2Gb and it would have taken most of a gigahertz-year processing time. When the project was started we believed that the cluster at Microsoft Research would perform the linear algebra but it was no longer accessible by the time the matrix was ready.

Herman te Riele very generously offered to run the matrix for us. He used 32 nodes on SARA's "teras" computer and the computation took close to three days elapsed, which is about 3 cpu-months.

The square root program ran on Richard's 2GHz G5 Macintosh. It took 10.5 hours to find the factors given above on the fifth dependency.

Many thanks to all the NFSNET sievers and, especially, to Herman te Riele for running the linear algebra and thereby getting us out of an embarrassing predicament.

Paul Leyland, for the NFSNET admin team.