[We open our next scene with a hand slapping the owner's forehead, accompanied by the utterance "doh!"]
Re above: In fact it seems silly to use powerful generalmodulus factoring machinery like ECM or QS on such (p1)found factorproduct composites. Here's why: say we have some product of prime factors F = f1*f2*...*fn discovered by running p1 to stage bounds b1 and b2 on an input Mersenne M(p) (or other bigum modulus with factors of a known form, allowing p1 to be 'seeded' with a component of same). BY DEFINITION, each prime factor f1fn will be b1/b2smooth, in the sense than fj = 2*p*C + 1, where C is a composite all of whose prime factors are <= b1, save possibly one outlierprime factor > b1 and <= b2. Thus if we again run p1 to bounds b1/b2, but now with arithmetic modulo the relatively tiny factor product F, we are guaranteed to resolve all the prime factors f1fn  the only trick is that we will need to do multiple GCDs along the way in order to capture the individual prime factors f1,...,fn, rather than have this secondary p1 run modulo F again produce the same composite GCD = F which the original p1 run mod M(p) did. Again, though, since in the followup p1 run we are working mod F, all the arithmetic is trivially cheap, including the needed GCDs. And since the cost of a p1 run is effectively akin to a single supercheap ECM curve, we've reduced the work of resolving the composite F to just that equivalent.
Last fiddled with by ewmayer on 20210523 at 22:17
