[QUOTE=paulunderwood;619196]The certificate has been verified by factorDB. Now [url]https://www.mersenne.ca/prp.php[/url] needs to be updated.[/QUOTE]

M43 and higher are also missing.

[QUOTE=paulunderwood;619196]The certificate has been verified by factorDB.
Now [url]https://www.mersenne.ca/prp.php[/url] needs to be updated.[/QUOTE]Sorry, I was offline for 4 days, but it seems my adminion has updated it in my absence.

[QUOTE=Mark Rose;619197]M43 and higher are also missing.[/QUOTE]Kinda-sorta... the PRP section is limited to 30M so you didn't see any larger Mersenne Primes. The limit remains in place, but I'm now always including the Primes in the list to avoid confusion.

It may now be limited to 30M, but practically that's of no importance, since if a PRP larger than that is discovered you'll find a way to include it, I imagine.

I have calculated the density of PRP cofactors that should be expected (by the 1/n law) and it agrees satisfactorily with what's actually been found: their density should exceed that expected for Mersenne primes by the ratio

log(eff. factoring level)/log(4p)

where the effective factoring limit is (logarithmically) averaged over all exponents, so increasing it is easiest by further work on _factored_ exponents, in particular, running quick P-1 on exponents that have had none is the fastest way to increase it and also (which is equivalent) the fastest way to find factors. Whether anyone should do this depends on the value placed on PRPs.

I should have clarified that my formula is fundamentally the probability of PRP after each factor found; averaging over all exponents in a range gives a density of PRPs. 'Quick' P-1 in that form of search would be nearly the Primenet standard on mersenne.ca (for B1), as it makes sense to always go to that level before a PRP; regardless of how many factors are known, the probability of another is the same.

[M]M12,588,091[/M] has been reported probably fully factored. Naegi Makoto found a 95-bit factor, the first for this exponent, and also did the PRP test on the cofactor.

This is smaller than the cofactor of [M]M12,720,787[/M] discovered last July.