The early NFS@Home numbers were all tested extensively
by ECM (the Elliptic Curve Method) for small and medium
sized prime factors, past 55-digits. The three most
recent numbers were less tested, and of these 6, 334+
was hardly tested at all before selection (due to rapid
NFS@Home progress!). We had expected 50-digit prime
factors to have been removed, and planned on removing
prime factors up to 55-digits, with a chance of finding
larger factors, before starting sieving.

So I'm happy to report that the 59-digit prime

p59 =
37597376323754357344197406664995834047249702145969970498293

divides the 239-digit composite factor of 6,334+ we
had intended to factor. Unfortunately, there remains
a composite cofactor of 181-digits; and sieving
by snfs (with difficulty 259) is still easier than
using gnfs (with gnfs on a 181-digit number perhaps
comparable in difficulty to snfs on a number with
difficulty of 270-digits). I'm not done testing yet,
but there's most likely not another factor in ECM
range (up to 70-digit primes).

This factor was found by GMP-ECM, under the ECMNET
project, Dodson/ECMNET; which is also the software
used by the boinc project yoyo (for which Beyond of
Team Ars Technica found a recent large prime factor).
The machine used was one of 300+ 32-bit linux/xeons,
along with c. 500 pcs (1050 pcs used during 8pm to
8am) distributed under condor. The first step limit
B1 was 260,000,000 (with default B2), optimal for
finding 60-digit primes.

This p59 is the 4th largest prime found this year by
ECMNET, just below the 59-digit prime in 3rd place
found by yoyo@home earlier this year. It's nowhere
near the Top 10 on Brent's list, for which the three
smallest primes all have 62-digits. At the risk of
trying the patience of NFS@home readers, I can report
that four of those Top 10 are Dodson/ECMNET factors,
including the two largest at 67-digits and 66-digits
(just a bit larger than the 66-digit prime found by
the ECMNET founder, Paul Zimmermann). -bdodson

Thanks Bruce for doing the ECM pretesting of the numbers. It is unfortunate that the cofactor remains composite. Had it been prime, then 6,334+ would be complete.