20171028, 11:27  #1 
"Yar"
Oct 2017
2^{2} Posts 
Fermat cofactors
F25
I twice ran PRP test for F25 with known factors: PRP=N/A,1,2,33554432,1,"25991531462657,204393464266227713,2170072644496392193" First: [Thu Oct 26 10:55:02 2017] {"status":"C", "k":1, "b":2, "n":33554432, "c":1, "knownfactors":"25991531462657,204393464266227713,2170072644496392193", "worktype":"PRP3", "res64":"7B6B087B84A45562", "residuetype":5, "fftlength":1966080, "errorcode":"00000000", ... , "program":{"name":"Prime95", "version":"29.4", "build":1, "port":4}, ... } second (verification): [Fri Oct 27 20:55:24 2017] {"status":"C", "k":1, "b":2, "n":33554432, "c":1, "knownfactors":"25991531462657,204393464266227713,2170072644496392193", "worktype":"PRP3", "res64":"7B6B087B84A45562", "residuetype":5, "fftlength":1966080, "errorcode":"00000000", ... , "program":{"name":"Prime95", "version":"29.4", "build":1, "port":4} ... } (is identical) This test with base 3 and returns the same residue independent of the number of known factors. The total computation time: ~24 hours (per test) on i74790T with using all 4 cores. F25 = 25991531462657 · 204393464266227713 · 2170072644496392193 · C10100842 
20171028, 13:08  #2 
Sep 2003
2^{2}·3·5·43 Posts 
I think this test for F25 was done previously, see this thread:
Feasibility of testing Fermat cofactors See also: Pépin tests of Fermat numbers beyond F24 Currently Ernst Mayer is currently working on F29. 
20171028, 13:16  #3  
Banned
"Luigi"
Aug 2002
Team Italia
2×2,383 Posts 
Quote:
While doing a search on cofactors of small Fermat numbers, I found the following: 1  On 23 Jul 2009, the Japanese user "msft" from MersenneForum wrote: "Hi, I check 4th cofactor of Fermat 25 is composite. 3^(((2^(2^25)+1)/(48413*2^29+1)/(1522849979*2^27+1)/(16168301139*2^27+1)1)*48413*2^29*1522849979*2^27*16168301139*2^27) != 1 (mod 2^(2^25)+1). Use Fermat Euler Theorem." 2  On 3 Aug 2009, Andreas Hoegund (MersenneForum user "ATH") wrote: "The remaining number for F25 is composite: UID: athath, F25/known_factors is not prime. RES64: 44BFC8D231602007. Wd1: B9307E03,00000000 Known factors used for PRP test were: 25991531462657,204393464266227713,2170072644496392193 " On 19 Sept 2009, Andreas Hoeglund wrote: "The remaining number of F26 is composite: UID: athath, F26/76861124116481 is not prime. RES64: 6C433D4E3CC9522E. Wd1: 7BD8A30F,00000000" The test was doublechecked by users testing residuals every 5M. On 4 Apr 2010, Andreas Hoglund wrote: "The remaining 40,403,531 digit factor of F27 is composite: UID: athath, F27/151413703311361/231292694251438081 is not prime. RES64: 481F26965DE16117. Wd1: AD647FF8,00000000 I'm not going any higher :) This one took 8 months on and off, roughly 190 days cpu time on a Core2duo (Conroe) E6750 2.66 Ghz. Residues every 1M iterations: http://www.hoegge.dk/mersenne/F27residues.txt 3  On 5 Apr 2010 Phil Moore told us about his exchange of messages with Wilfrid Keller about the compositeness of Fermat number cofactors. He "complained that the status of the Fermat cofactors is somewhat murky, although the smaller ones have undoubtedly been tested independently enough times that their status as composites is not in doubt. Even the composite cofactor of F22 does not meet his standard of two matching tests using different hardware and different software." A simple prp test done on two different machines using different software should verify this status as composite. Ernst Mayer's MLucas code also contain routines for doing calculations modulo Fermat numbers. 4  On 4 Apr 2010, user msft proved the compositeness of F22 comparing the residuals of mprime and genefer softwares. Keller's asterisk on the page http://www.prothsearch.com/fermat.html shows he accepted the result. 5  On 24 Oct 2013, Ernst Mayer (one of the codiscoverers of the compositeness of F24) wrote the following post on MersenneForum: http://www.mersenneforum.org/showpos...39&postcount=1 meaning that the compositeness pf F25 and F26 were proven with his own software. Throughout the post (http://www.mersenneforum.org/showthread.php?t=18748 ), he shows he completed the compositeness test for F27, F28 and F29 is in the run. Considering the data gathered here, may we say that the cofactors of F25, F26 and F27 are composite? Can we say anything fo F28? Should a prp run for F28 and the test of its residual with the result of Mayer's Mlucas be enough? Just thinking. Please let me know about your ideas of adding those composites to the Fermat numbers compositeness list. Luigi Morelli 

20171030, 16:25  #4  
"Yar"
Oct 2017
4_{8} Posts 
Thank you Luigi for the fullest information.
Also today I took PRP result for F25 in another vesion of Prime95 (v29.3) [Mon Oct 30 18:05:18 2017] UID: yorix, F25/known_factors is not prime. RES64: 44BFC8D231602007. Wf4: B9307E03,00000000 Known factors used for PRP test were: 25991531462657,204393464266227713,2170072644496392193 This version(test in this version) also with base 3 and returns residue dependent of the number of known factors. For F25: Mod[ 3 ^{F25 / 25991531462657 / 204393464266227713 / 2170072644496392193  1}, F25 / 25991531462657 / 204393464266227713 / 2170072644496392193 ] = .....44BFC8D231602007_{16} This result is identical to the result from Andreas Hoegund: Quote:


20171031, 05:19  #5  
Sep 2003
A14_{16} Posts 
Quote:
Perhaps fermatsearch.org could create a page for these, there are only a handful of them in any case. 

20171031, 17:34  #6  
Banned
"Luigi"
Aug 2002
Team Italia
2·2,383 Posts 
Quote:
Luigi  

20171101, 21:10  #7 
∂^{2}ω=0
Sep 2002
República de California
10110011111110_{2} Posts 
I have base3 pepin residues for F2429, each validated using 2 runs at slightly different FFT lengths (e.g. 30M and 32M for F29) from my own Mlucas runs over the past several years. Whether 2 runs at different FFT lengths "meets Wilfrid's standard" is honestly unimportant to me  the largernumber runs also have residue files every 10M iterations, allowing anyone to used them for a parallel verification of the kind used for e.g. F22 and F24, to verify integrity of the entire residue chain using whatever software they wish.
I need to update the abovelinked "Pépin tests of Fermat numbers beyond F24" thread with the F29 result, along with a new home for the residues  the old hogranch.com ftp server I used for many years to hots my code and various researchrelated files is no more. Will post update link later to day when the above is done. Last fiddled with by ewmayer on 20171101 at 21:10 
20171102, 02:25  #8 
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
23A7_{16} Posts 

20171114, 02:31  #9 
∂^{2}ω=0
Sep 2002
República de California
2·13·443 Posts 
Apologies for the delay  between finishing up my 4monthplus project of adding inlineasm support for ARM Neon to Mlucas, needing to massage my F24F29 file archives into a nice form, etc, I only today got around to uploading said archives to the ftp archive (hosted on the same server as this forum) originally created by Mike Vang to mirror my coderelated pages, and promoted to primary ftp site a few months ago.
I just added links to the above F2429 file archives to OP in this thread; the newest post my me in the same thread is about the completed matching pair of F29 runs, which lengthy process (including one illfated full Pepin test @30M FFT length) showed just how careful one must be in doing such lengthy chained computations. 
20171221, 13:08  #10  
"Yar"
Oct 2017
4_{8} Posts 
F26
PRP test for F26 with known factors: PRP=N/A,1,2,67108864,1,"76861124116481" Prime95 v29.4 PRP test with base 3 and returns the same residue independent of the number of known factors: [Mon Dec 04 11:18:46 2017] {"status":"C", "k":1, "b":2, "n":67108864, "c":1, "knownfactors":"76861124116481", "worktype":"PRP3", "res64":"FBB406B3A281838C", "residuetype":5, "fftlength":3932160, "errorcode":"00000000", ... , "program":{"name":"Prime95", "version":"29.4", "build":1, "port":4} ... } Prime95 v29.3 PRP also with base 3 and returns residue dependent of the number of known factors: [Thu Dec 21 14:56:26 2017] UID: yorix, F26/76861124116481 is not prime. RES64: 6C433D4E3CC9522E. Wf4: 7BD8A30F,00000000 Mod[ 3 ^{F26 / 76861124116481  1}, F26 / 76861124116481 ] = .....6C433D4E3CC9522E_{16} This result is identical to the result from Andreas Hoegund: Quote:
F26 = 76861124116481 · C20201768 

20171221, 16:54  #11  
Banned
"Luigi"
Aug 2002
Team Italia
2·2,383 Posts 
Quote:


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