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-   -   ECM on small Generalised Fermat numbers (https://www.mersenneforum.org/showthread.php?t=13784)

geoff 2010-08-26 04:13

ECM on small Generalised Fermat numbers
 
I would like to invite anyone who has done ECM work on small Generalised Fermat numbers a^(2^m) + b^(2^m) to report their ECM curve counts in this thread. Discussion about this and related projects welcome too :-)

I will keep a record of the cumulative totals here: [url=http://sites.google.com/site/geoffreywalterreynolds/projects/gfnsmall/curves.txt]curves.txt[/url], and input files suitable for Prime95 and GMP-ECM here: [url=http://sites.google.com/site/geoffreywalterreynolds/projects/gfnsmall/input.zip]input.zip[/url]. There are separate counts and input for a^(2^m)+1 because these numbers can be done with Prime95, whereas the more general a^(2^m) + b^(2^m) can only be done with GMP-ECM.

Please report any new factors to Wilfrid Keller (See address near the bottom of the [url=http://www1.uni-hamburg.de/RRZ/W.Keller/GFNfacs.html]main results page[/url]). He also keeps a separate listing of known factors just for [url=http://www1.uni-hamburg.de/RRZ/W.Keller/GFNsmall.html]small m[/url].

So far I have listed only the curves I have done myself, plus the few recorded on [url=http://members.cox.net/jfoug/GFNFacts_ECMComposites.html]this page[/url].

rajula 2010-08-26 06:29

A few months ago I ran some curves on 12[sup]2[sup]9[/sup][/sup]+1.

5000 curves with B1=11000000 and B2=35133391030.
1200 curves with B1=43000000 and B2=240490660426.

I would guess that someone has already ran lots and lots of more curves on it. I just wanted to see if I could get a lucky hit.

ET_ 2010-08-26 09:54

[QUOTE=geoff;227118]
So far I have listed only the curves I have done myself, plus the few recorded on [url=http://members.cox.net/jfoug/GFNFacts_ECMComposites.html]this page[/url].[/QUOTE]

May I ask to put a sticky bit on this thread? :smile:

Luigi

R.D. Silverman 2010-08-26 11:40

[QUOTE=geoff;227118]I would like to invite anyone who has done ECM work on small Generalised Fermat numbers a^(2^m) + b^(2^m) to report their ECM curve counts in this thread. Discussion about this and related projects welcome too :-)

I will keep a record of the cumulative totals here: [url=http://sites.google.com/site/geoffreywalterreynolds/projects/gfnsmall/curves.txt]curves.txt[/url], and input files suitable for Prime95 and GMP-ECM here: [url=http://sites.google.com/site/geoffreywalterreynolds/projects/gfnsmall/input.zip]input.zip[/url]. There are separate counts and input for a^(2^m)+1 because these numbers can be done with Prime95, whereas the more general a^(2^m) + b^(2^m) can only be done with GMP-ECM.

Please report any new factors to Wilfrid Keller (See address near the bottom of the [url=http://www1.uni-hamburg.de/RRZ/W.Keller/GFNfacs.html]main results page[/url]). He also keeps a separate listing of known factors just for [url=http://www1.uni-hamburg.de/RRZ/W.Keller/GFNsmall.html]small m[/url].

So far I have listed only the curves I have done myself, plus the few recorded on [url=http://members.cox.net/jfoug/GFNFacts_ECMComposites.html]this page[/url].[/QUOTE]

These are mis-named. Fermat numbers are 2^2^n+1. Generalized
Fermat numbers are a^2^n + 1 for a > 2. The numbers you refer to should
be called Homogeneous Fermat Numbers.

kar_bon 2010-08-26 12:02

[QUOTE=R.D. Silverman;227143]These are mis-named. Fermat numbers are 2^2^n+1. Generalized
Fermat numbers are a^2^n + 1 for a > 2. The numbers you refer to should
be called Homogeneous Fermat Numbers.[/QUOTE]

Although W.Keller named them Gen.Fermat on his page given above but this should reconsidered!
I'm thinking of "Generalized Hyper-Fermat" like S.Harvey did for Woodall/Cullen type ([url=http://harvey563.tripod.com/]his page[/url]).

And:

I've made a small Summary page for W.Kellers table (1 <= a,b <= 12, 0<=n<=19) [url=www.rieselprime.de]here[/url] (see menu 'Interests').

Should a table for a^(2^m)+1, a={3,5,6,7,8,10,11,12} also be displayed there?

PS: I've named the page "Generalized Hyper Fermat" until the best fit is found.

PPS: "Homogeneous" seems better in view of a^n +/- b^n (Hom.Cunningham).

PPPS: Name changed.

fivemack 2010-08-26 14:44

119^64+1 = 2 * 40071520135245923033887866862313228717813761 * 85337460632264337064075451855088483819126291733854384530962498965366569052147859575225601

next I will do 284^64+1, which has resisted quite a lot of ECM, and 35^128+1

wblipp 2010-08-26 15:53

[QUOTE=fivemack;227153]119^64+1 = 2 * 40071520135245923033887866862313228717813761 * 85337460632264337064075451855088483819126291733854384530962498965366569052147859575225601

next I will do 284^64+1, which has resisted quite a lot of ECM, and 35^128+1[/QUOTE]

Somebody needs to do better project coordination. Hisanori's Cyclotomic Numbers project has known this factorization since 2008, and knows the 284^64+1, too.

[url]http://www.asahi-net.or.jp/~kc2h-msm/cn/[/url]

The ^128 ones should be on Morimoto's site, but that seems to be defunct.

All of these should be in Brent's list, too, although it appears he hasn't yet been notified (or perhaps hasn't yet updated)

[url]http://wwwmaths.anu.edu.au/~brent/factors.html[/url]

geoff 2010-08-27 03:06

Oh no, it looks like I accidentally volunteered for more than I planned to :-) I really only intended to keep track of curves on a^(2^m) + b^(2^m) where b < a <= 12, i.e. the same ones that Wilfrid Keller keeps track of [url=http://]here[/url], discussed in [url=http://www.ams.org/journals/mcom/1998-67-221/S0025-5718-98-00891-6/S0025-5718-98-00891-6.pdf]Bjorn & Riesel (1998)[/url].

[QUOTE=R.D. Silverman;227143]These are mis-named. Fermat numbers are 2^2^n+1. Generalized
Fermat numbers are a^2^n + 1 for a > 2. The numbers you refer to should
be called Homogeneous Fermat Numbers.[/QUOTE]

I don't really mind what they are called, I hadn't heard them called Homogeneous Fermat numbers before. [url=http://www.ams.org/journals/mcom/1995-64-209/S0025-5718-1995-1270618-1/S0025-5718-1995-1270618-1.pdf]Dubner & Keller (1995)[/url] refer to b^2^m + 1 as Generalized Fermat numbers, but then [url=http://www.ams.org/journals/mcom/1998-67-221/S0025-5718-98-00891-6/S0025-5718-98-00891-6.pdf]Bjorn & Riesel (1998)[/url] refer to a^2^n + b^2^n as "slightly more generalized" forms of the same.

(Just as a practical note, a^2^m = a^(2^m), but GMP-ECM doesn't properly parse a^2^m which is why a^(2^m) is sometimes used instead.)

Merfighters 2010-08-27 08:53

I reported all known factors of n^64+1, where n<=1000. :smile:
That was a time consuming work.

[URL]http://factordb.com/new/index.php?query=x%5E64%2B1&use=x&x=1&VP=on&VC=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=50&format=1&sent=Show[/URL]

fivemack 2010-09-02 12:40

19469280117455103039625667255411201 divides 99^128+1

111^128+1 = 2 · 769 · 2697217437953 · 125464289479028905217 . P227

1880787140548365990905051620353281 divides 106^128+1

R.D. Silverman 2010-09-02 12:45

[QUOTE=fivemack;228163]19469280117455103039625667255411201 divides 99^128+1

111^128+1 = 2 · 769 · 2697217437953 · 125464289479028905217 . P227

1880787140548365990905051620353281 divides 106^128+1[/QUOTE]

Check Brent's tables!!!!

[b]fivemack[/b]: I have. These factors, though small, are new. Or are there some other 'Brent's tables' which aren't [url]http://wwwmaths.anu.edu.au/~brent/ftp/factors/factors.gz[/url]


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