Feasibility of testing Fermat cofactors
 

The site [URL="http://www.prothsearch.com/fermat.html"]Prime factors [I]k[/I] · 2[I]n[/I] + 1 of Fermat numbers [I]Fm[/I] and complete factoring status[/URL] by Wilfrid Keller et al. is up again (on a slightly changed domain name).

I was wondering about the feasibility of testing some of the remaining cofactors for which the status is not known:[LIST][*]F25 / (p1*p2*p3) at 10.1 million digits[*]F26 / p4 at 20.2 million digits[*]F27 / (p5*p6) at 40.4 million digits[/LIST]How hard would it be to test these?

Clearly, the first one is smaller than what is being tested at GIMPS. If prime, it would come in 6th on the current list of largest known prime.

The second one would be 2nd (only beaten by last year's Mersenne record).

And of course, the last one I mention, if it could be tested and were to be prime, would easily be the world number one prime (though still much too small to win the next EFF prize of $150,000).

(I do realize that it is extremely unlikely to find a prime with just three candidate numbers. But it would still be nice to resolve these three.)

/JeppeSN

They have already been tested. (And the similar sized GFNs, too.)
Wilfrid Keller has a conservative policy on adding asterisks.

[QUOTE=Batalov;453493]They have already been tested. (And the similar sized GFNs, too.)
Wilfrid Keller has a conservative policy on adding asterisks.[/QUOTE]

Cool. Who did those tests? You?

Were even more cofactors (F28, F29, etc.) tested?

Was Keller informed? It would be cool if he moved the numbers in question away from [I]Prime factors of larger Fermat numbers[/I] and up to [I]Factorizations known to be incomplete[/I] (which has a subsection [I]Composite cofactors[/I]), and also added the asterisks you mention in his [I]Summary of factoring status[/I].

/JeppeSN

Maybe Keller will add the asterisk if two people submit the matching residues.

This may be a problem for GFN[SUB]3[/SUB] (which a.f.a.i.r. I tested using base 5; someone else may test using base 2; these will no match). And when Wilfrid gets an email (yet another email) that GFN[SUB]3[/SUB](n) is a PRP, he probably shrugs (because they are all 3-PRPs).

I tested the cofactor of F25, F26 and F27 with Prime95, and they are composite, but it is not "official" until it is tested on another hardware with another software. I still have the interrim files and residues somewhere.

Post #51, #62, #64 in this thread:
[url]http://mersenneforum.org/showthread.php?t=12168[/url]

Ewmayer is using his own code to test up to F29:
[url]http://www.mersenneforum.org/showthread.php?t=18748[/url]

ATH, Thanks for linking these interesting threads. I see the tests were done a long time ago. /JeppeSN

[QUOTE=JeppeSN;453586]ATH, Thanks for linking these interesting threads. I see the tests were done a long time ago. /JeppeSN[/QUOTE]

I recollectred the whole history and asked Prof. Keller if now we can consider Any of the cofactors of F25...F29 composite. Waiting for an answer.