20170222, 21:09  #1 
"Jeppe"
Jan 2016
Denmark
2^{5}·5 Posts 
Feasibility of testing Fermat cofactors
The site Prime factors k · 2n + 1 of Fermat numbers Fm and complete factoring status 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:
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 
20170222, 23:24  #2 
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
2^{2}×2,281 Posts 
They have already been tested. (And the similar sized GFNs, too.)
Wilfrid Keller has a conservative policy on adding asterisks. 
20170223, 11:17  #3  
"Jeppe"
Jan 2016
Denmark
2^{5}×5 Posts 
Quote:
Were even more cofactors (F28, F29, etc.) tested? Was Keller informed? It would be cool if he moved the numbers in question away from Prime factors of larger Fermat numbers and up to Factorizations known to be incomplete (which has a subsection Composite cofactors), and also added the asterisks you mention in his Summary of factoring status. /JeppeSN 

20170223, 15:34  #4 
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
2^{2}·2,281 Posts 
Maybe Keller will add the asterisk if two people submit the matching residues.
This may be a problem for GFN_{3} (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_{3}(n) is a PRP, he probably shrugs (because they are all 3PRPs). 
20170223, 16:32  #5 
Einyen
Dec 2003
Denmark
3·23·43 Posts 
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: http://mersenneforum.org/showthread.php?t=12168 Ewmayer is using his own code to test up to F29: http://www.mersenneforum.org/showthread.php?t=18748 
20170223, 23:28  #6 
"Jeppe"
Jan 2016
Denmark
2^{5}×5 Posts 
ATH, Thanks for linking these interesting threads. I see the tests were done a long time ago. /JeppeSN

20170224, 10:17  #7 
Banned
"Luigi"
Aug 2002
Team Italia
2^{5}·149 Posts 

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