I was wondering if it was common practice (ie: the norm) for P-1
to take the product of two or more factors when giving out a found
factor, if two of more factors are found?
Yes, if both factors are "smooth" enough, they could be found as their
product, rather than individually. "Smooth" is defined as one less
than the factor having no factors larger than the stage one P-1 bound
plus up to one factor between the stage 1 and stage 2 bounds.
To clarify, I was curious about how P-1 would indicate more than
one factor being found. So, I took M113 and fed it into Prime95
with the bounds of B1=200, B2=20000. Prime95 notified me that P-1
had found a factor in Stage #1, and that the factor was
9734174361238150513. This factors out to 3391 * 23279 * 65993 *
1868569, all of which are known factors of M113.
For example, all the primes factors of 3390, 23278, 65992, and 1868568
are likely smaller than 200. If not, at most one factor of each
number should be between 200 and 20,000.
Note that there are three factors of 9734174361238150512 that are
larger than 20,000; that doesn't matter.
Because of this possibility of new factors being composite, I
sometimes run a script that checks for composite factors in all my
data and my update scripts check all new factors against smaller
factors for the same exponent (to see if the smaller factor is a
factor of the new factor).
All new factors are also checked to verify that they are actually
factors and whether they are also factors of some smaller Mersenne;
the latter is not uncommon for new factors found by P-1 or ECM for
composite exponent Mersenne numbers, but cannot happen for prime
exponent Mersennes.
Will
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