> Sure. Not quite the same since there appears to be no certificate of > primality, but on 30 Aug 2001 there was a message on this list to > the effect that M727 (c219) = prp98.prp128. So much ECM work > was done on M727 (before the NFS people started work) that it is > highly unlikely that there are any factors < 10^50, which means > that at least the 98-digit probable prime is almost certainly a > genuine prime. (Maybe that's been proved by now. ECPP on > general numbers of around 100 digits isn't very expensive.)
Ah, but SNFS-able numbers only half-count because they are so easy ;-) You're correct: I was forgetting the Cunningham factorizations which yielded large penultimate primes. There are quite a few by now. All the factors have indeed been proved prime. > I think the 55 digit record applies to ECM. A number of much larger > factors (not counting cofactors) have been found using number field > sieve techniques. Correct. I do not know of any larger penultimate factors of hard integers with more than 200 digits. The largest I know of has 78 digits, but that is from RSA-155, which only has 155 digits. Paul _________________________________________________________________________ Unsubscribe & list info -- http://www.ndatech.com/mersenne/signup.htm Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
