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Old 2017-10-28, 11:27   #1
yorix
 
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Default 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, "known-factors":"25991531462657,204393464266227713,2170072644496392193", "worktype":"PRP-3", "res64":"7B6B087B84A45562", "residue-type":5, "fft-length":1966080, "error-code":"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, "known-factors":"25991531462657,204393464266227713,2170072644496392193", "worktype":"PRP-3", "res64":"7B6B087B84A45562", "residue-type":5, "fft-length":1966080, "error-code":"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 i7-4790T with using all 4 cores.

F25 = 25991531462657 · 204393464266227713 · 2170072644496392193 · C10100842
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Old 2017-10-28, 13:08   #2
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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.
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Old 2017-10-28, 13:16   #3
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Quote:
Originally Posted by yorix View Post
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, "known-factors":"25991531462657,204393464266227713,2170072644496392193", "worktype":"PRP-3", "res64":"7B6B087B84A45562", "residue-type":5, "fft-length":1966080, "error-code":"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, "known-factors":"25991531462657,204393464266227713,2170072644496392193", "worktype":"PRP-3", "res64":"7B6B087B84A45562", "residue-type":5, "fft-length":1966080, "error-code":"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 i7-4790T with using all 4 cores.

F25 = 25991531462657 · 204393464266227713 · 2170072644496392193 · C10100842
Thank you yorix.

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 double-checked 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 co-factors 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 co-factor 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 co-discoverers 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
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Old 2017-10-30, 16:25   #4
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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 ] = .....44BFC8D23160200716

This result is identical to the result from Andreas Hoegund:
Quote:
Originally Posted by ET_ View Post

.....

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
"
.....

Luigi Morelli
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Old 2017-10-31, 05:19   #5
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Quote:
Originally Posted by yorix View Post
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 ] = .....44BFC8D23160200716
However, residues for Fermat cofactor testing are not recorded by Primenet.

Perhaps fermatsearch.org could create a page for these, there are only a handful of them in any case.
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Old 2017-10-31, 17:34   #6
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Quote:
Originally Posted by GP2 View Post
However, residues for Fermat cofactor testing are not recorded by Primenet.

Perhaps fermatsearch.org could create a page for these, there are only a handful of them in any case.
I will. Just let me have the file (l DOT morelli AT moreware DOT org ) or the columns of the needed table.

Luigi
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Old 2017-11-01, 21:10   #7
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I have base-3 pepin residues for F24-29, 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 larger-number 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 above-linked "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 research-related files is no more.

Will post update link later to day when the above is done.

Last fiddled with by ewmayer on 2017-11-01 at 21:10
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Old 2017-11-02, 02:25   #8
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Quote:
Originally Posted by ewmayer View Post
"meets Wilfrid's standard" ...
..."according to Hoyle"
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Old 2017-11-14, 02:31   #9
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Quote:
Originally Posted by ewmayer View Post
Will post update link later to day when the above is done.
Apologies for the delay - between finishing up my 4-month-plus project of adding inline-asm support for ARM Neon to Mlucas, needing to massage my F24-F29 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 code-related pages, and promoted to primary ftp site a few months ago.

I just added links to the above F24-29 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 ill-fated full Pepin test @30M FFT length) showed just how careful one must be in doing such lengthy chained computations.
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Old 2017-12-21, 13:08   #10
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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, "known-factors":"76861124116481", "worktype":"PRP-3", "res64":"FBB406B3A281838C", "residue-type":5, "fft-length":3932160, "error-code":"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 ] = .....6C433D4E3CC9522E16

This result is identical to the result from Andreas Hoegund:
Quote:
Originally Posted by ET_ View Post

.....

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"

.....

Luigi Morelli
The total computation time: ~5 days (per test) on i7-4790T with using all 4 cores.

F26 = 76861124116481 · C20201768
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Old 2017-12-21, 16:54   #11
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Quote:
Originally Posted by yorix View Post
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, "known-factors":"76861124116481", "worktype":"PRP-3", "res64":"FBB406B3A281838C", "residue-type":5, "fft-length":3932160, "error-code":"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 ] = .....6C433D4E3CC9522E16

This result is identical to the result from Andreas Hoegund:


The total computation time: ~5 days (per test) on i7-4790T with using all 4 cores.

F26 = 76861124116481 · C20201768
Well done. Now let's attack that C20201768...
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