20201228, 10:54  #1 
Dec 2020
7 Posts 
What does Cofactor work do?
Hi guys and gals.
I did a read through the forums but I´m not entirely sure what what the cofactor prp does. My understanding right now: exponents with a factor are going to the cfqueue and the factors are checked if they are prp. What happens if a factor is not prp? Will the tf going deeper until there is found another factor wich will be checked for again? If I´m not misstaken, a prptest would not find a factor? Could someone please explain in easy words the workflow of the cofactor work? (I´m neither a native english speaker nor a mathematician ) Thank you very much in advance. 
20201228, 14:23  #2 
Einyen
Dec 2003
Denmark
101111011101_{2} Posts 
Cofactor PRP is one of our side projects.
The main GIMPS project is to find Mersenne Primes by trial factoring candidates and then running a PRP test on exponents without a factor to check for probable primes followed by a certificate on the PRP test (new) or a PRP double check. After that the exponent is uninteresting for the main project, but then other people in side projects will run PRP test on the cofactor to find large PRPs and others will continue to try and find factors by trial factoring, P1 tests and ECM curves. PRP on cofactors is still fairly new ~23 years and have "only" reached ~11M exponents, but is slowly going higher and higher and whenever a new factor is found below 11M the exponent is immediately scheduled for a PRP cofactor test, so everything is kept checked below the "wavefront". All the sideproject trial factoring, P1 and ECM is not very organized instead people are working on their favorite ranges and method. There are a few people organizing their efforts in this thread, but right now the thread is "hijacked" by posts concerning testing a new Prime95 version 30.4b3 for P1: https://mersenneforum.org/showthread.php?t=22476 Last fiddled with by ATH on 20201228 at 14:24 
20201228, 15:04  #3 
Dec 2020
7 Posts 
Thank you very much. This is a very nice explanation for somebody like me.

20201228, 15:56  #4 
6809 > 6502
"""""""""""""""""""
Aug 2003
101×103 Posts
2·3^{2}·521 Posts 
Just a little clarification.
If we have a candidate number and we find a factor, the part that remains after dividing the number by the known factor is the cofactor. If we find several factors for the number, still what ever remains is the cofactor. We are running the PRP on that number to see if it is prime and thus, we known all of the prime factors for the candidate number. There are some numbers that have a PRP test run on the cofactors, but not a double check or a certificate on the PRP. (The ability to run that is recent). So there is a group of us working to get all of those double checks done. That way, going forward we should not have to do the longer (relative to the cert) DC's any more. 
20201228, 16:24  #5 
Oct 2007
Manchester, UK
3^{2}·149 Posts 

20201228, 16:33  #6 
6809 > 6502
"""""""""""""""""""
Aug 2003
101×103 Posts
2×3^{2}×521 Posts 
The residues are part of the record for the main exponent:
https://www.mersenne.org/report_expo...9735347&full=1 
20201228, 16:43  #7 
Oct 2007
Manchester, UK
3^{2}×149 Posts 
Is there anywhere they are recorded separately such that one might peruse a list of them, in the same way as the mersenne primes are recorded separately. Once the exponent gets large enough (>1E4? >1E6?) I assume they are relatively rare finds.

20201228, 17:21  #8  
"Viliam Furík"
Jul 2018
Martin, Slovakia
5×79 Posts 
Quote:
There is a list of a total of 350 fully or probably fully factored Mersenne numbers, plus the 51 Mersenne primes. You can set the page to show you cofactor state of all the exponents. 

20201228, 17:25  #9 
Apr 2010
Over the rainbow
2·5·11·23 Posts 
isn't that what the fully factored mersenne numer thread used for?
edit : this thread https://www.mersenneforum.org/showthread.php?t=19407 Last fiddled with by firejuggler on 20201228 at 17:32 
20201228, 18:08  #10  
Oct 2007
Manchester, UK
3^{2}×149 Posts 
Quote:


20201229, 16:26  #11 
Sep 2009
5×401 Posts 
M1399 shows as PRP on that page, but it's cofactor is proven prime in factordb. Although I didn't do it.
Chris 
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