New Fermat factors

[GP2]

Originally Posted by R. Gerbicz

Just interestingly, if m=k*2^(n+2)+1 | F_n (where k can be even) and 0<k<2^(n+2) then m is prime! And this holds for the known factors in the list (surely) for say n>100.

Among the currently known factors, the largest n for which it the above condition 0<k<2^(n+2) fails (k can be even) is:

42 45 111318179143061

[ATH]

Originally Posted by GP2

Maybe you could use pfgw to verify that the known Fermat factors really are prime factors and not composite factors? But maybe not for the goal I wanted. Or am I misunderstanding?

Looking in the pfgwdoc.txt for -gos2 it does test the Fermat factors without testing primality.

-g Generalized Fermat factor testing
The complete syntax is: -g[x][o][s][q]{#,#}{#,#}[_dump_search_patterns]
Also allowed is -g[o][q]# and -gx[o][q]#,# to test single GF (or xGF) number
This will test the numbers to see if they divide any fermat number.
All prp's or primes (N-1 only) which are in the correct form of being
a Fermat factor will go through the test.

o: The o is needed when you only want to test for Fermat factors, and
NOT test the numbers for probable primality. ([O]nly perform GF
divisibility).

s: The s is an 'undocumented' feature, which saves the residue of each
b^(2^(n-1)) into a file, so that a re-run of the number will be much
quicker. (CAUTION here, this function does NOT delete the files, so
things can add up quickly).

The -g (or -go) may be followed by the GF base you wish to check for.
if no base is entered, then the default will be -g[o]{2,5}{2,12}
So -go2 will ONLY check (no prp) for Fermat divisors of base 2.
Ranges of bases can also be entered. The format is:

Originally Posted by GP2

Among the currently known factors, the largest n for which it fails is:

42 45 111318179143061

It does not fail? 111318179143061*2^45+1 = 222636358286122*2^44+1 is prime.

I just tested pfgw on this factor, you just have to write the factor as 111318179143061*2^45+1 in a text file.
111318179143061*2^45+1 is a Factor of F42!!!! (0.000000 seconds)

[GP2]

Originally Posted by ATH

It does not fail? 111318179143061*2^45+1 = 222636358286122*2^44+1 is prime.

No, I meant it fails the condition of:

0<K<2^(n+2)

where unlike the usual "k", this K can be even, i.e., K = 2^(m-(n+2)) * k


I edited my previous post to make this clear.

[GP2]

Originally Posted by Batalov

Try on the same platform and compare times:
1. download pfgw
2. put all factors in a file
3. run pfgw -N -k -l -gos2 file

I downloaded pfgw from Sourceforge, with precompiled executable pfgw64, it's version 3.8.3_20170121

I ran it on a c5d.large instance on AWS (one core, Skylake, 3.0 GHz Xeon), same platform as for the GMP program.

Here are the times:

7905*2^352281+1 is a Factor of F352279!!!! (39.253506 seconds)
vs. my GMP program = 957 sec (24 times slower)

11*2^960901+1 is a Factor of F960897!!!! (219.145612 seconds)
vs. my GMP program = 7587 sec (35 times slower)
vs. Gary pmfs = 47 minutes = 2820 sec (13 times slower) running on unknown system

193*2^3329782+1 is a Factor of F3329780!!!! (3428.316006 seconds)
vs. my GMP program = 124205 sec (36 times slower)

Very impressive.

It's amusing that pfgw gives times to the millionth of a second...

Running the precompiled statically-linked version pfgw64s made no difference.


I'm a bit alarmed for Gary though...

[ET_]

Originally Posted by GP2

I downloaded pfgw from Sourceforge, with precompiled executable pfgw64, it's version 3.8.3_20170121

I ran it on a c5d.large instance on AWS (one core, Skylake, 3.0 GHz Xeon), same platform as for the GMP program.

Here are the times:

7905*2^352281+1 is a Factor of F352279!!!! (39.253506 seconds)
vs. my GMP program = 957 sec (24 times slower)

11*2^960901+1 is a Factor of F960897!!!! (219.145612 seconds)
vs. my GMP program = 7587 sec (35 times slower)
vs. Gary pmfs = 47 minutes = 2820 sec (13 times slower) running on unknown system

193*2^3329782+1 is a Factor of F3329780!!!! (3428.316006 seconds)
vs. my GMP program = 124205 sec (36 times slower)

Very impressive.

It's amusing that pfgw gives times to the millionth of a second...

Running the precompiled statically-linked version pfgw64s made no difference.


I'm a bit alarmed for Gary though...

Don't

PMFS is quite impressive for N below 30,000.

[rogue]

Actually according to your table, the cutover from using pfms to gfndsieve+pfgw is around n = 5000.

[ET_]

Originally Posted by rogue

Actually according to your table, the cutover from using pfms to gfndsieve+pfgw is around n = 5000.

Sure, but pmfs can make great use of threads ;-)

[Gary]

I would like to report the following new Fermat factor:

***** 19,789,270,693 * 2^2148 + 1 divides F2144

This was discovered today running pmfs on my Superdome X system.

[ET_]

Originally Posted by Gary

I would like to report the following new Fermat factor:

***** 19,789,270,693 * 2^2148 + 1 divides F2144

This was discovered today running pmfs on my Superdome X system.

Congratulations Gary!

[Gary]

I am happy to report the following new Fermat factor:

***** 6,604,326,057 * 2^3347 + 1 divides F3345

This was discovered on Dec 13 running pmfs on my HPE Superdome X system.

Wishing everyone a joyous holiday season and a Happy New Year!

[ET_]

December 19th, 2018
New Fermat factor from FermatSearch!
1075441212722595 . 2¹³⁵+1 is a Factor of F₁₃₂!!!
Peter Strasser discovered the seventh Fermat factor of this year! He used George Woltman's mmff program running on his home computer.
Congratulations to Peter from FermatSearch, for his second factor!

Moderators please modify the title: Gostin factor was found on Dec. 13th.
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