Quote:
Originally Posted by R.D. Silverman
So if two different machines trial divide with two different sets of trial factors how do you know whether a true factor was actually tested??? You might be missing some.

It's my understanding that the leading gpu TF applications use not too few but too many candidate factors. Incomplete sieving of trial factoring candidates is accepted because it is higher performance, without excluding any candidate prime factors. It's faster in practice to sieve the Mersenne number with an occasional composite factor candidate also included, than to perfectly sieve the factor candidates. All the primes of the proper form in 2
^{b} to 2
^{b+1} are in the set of trial factors when doing TF from b to b+1, plus a varied set of composites. Assuming the hardware and software are operating to spec, that is, or sufficiently near it.
That's covered in points 8 and 9 of
https://www.mersenneforum.org/showpo...23&postcount=6 (my attempt to document all the concepts of mfaktc trial factoring I could identify, which had been reviewed by the author of mfaktc)
But suppose one in 10
^{6} primes were omitted somehow from the TF candidate factors. We would still get nearly all the benefits of a perfect TF process, with all or nearly all the run time of a perfect TF process, over the same bit size interval.
A usable proof of work algorithm needs to be tolerant of both some level of expected software imperfection and speed optimizations.