On Sat, 22 Jan 2000, George Woltman wrote:
Finding new factors of small Mersennes, so called Cunningham factors, is
getting more difficult. ECMNet and GIMPS have picked off most of the
"easy" factors. I have two CPUs running ECM full-time. The last
Cunningham factor I found was last summer. I do occasionally find new
factors of medium-sized (1200 to 10) Mersenne numbers.
If finding factors by ECM is becoming more difficult we need to start
factoring these numbers by NFS. Even if a factor is found by ECM it
may not completely factor the number. Unlike ECM, NFS is guaranteed
to factor the number.
Of the Mersenne numbers, NFSNET is doing M629 and M641. M641 has
a 148 digit composite cofactor. Enough ECM has been done to make it
unlikely to have a factor of less than 45 digits. NFS can't take
advantage of the known primitive factors, it must be used on the
whole 193 digit number. It will be the second most difficult SNFS
factorization done and is one of the Cunningham projects most wanted.
There are many "easier" 2^p+1 numbers that NFS can factor. It
would be more productive to use NFS than ECM on these numbers.
The NFSNET page is at http://orca.st.usm.edu/~cwcurry/nfs/nfs.html
We are currently working on M641 and 2,1542M (a factor of 2^1542+1).
_
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