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 GMPECM 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 GMPECM. Please report any new factors to Wilfrid Keller (See address near the bottom of the [url=http://www1.unihamburg.de/RRZ/W.Keller/GFNfacs.html]main results page[/url]). He also keeps a separate listing of known factors just for [url=http://www1.unihamburg.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]. 
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. 
[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 
[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 GMPECM 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 GMPECM. Please report any new factors to Wilfrid Keller (See address near the bottom of the [url=http://www1.unihamburg.de/RRZ/W.Keller/GFNfacs.html]main results page[/url]). He also keeps a separate listing of known factors just for [url=http://www1.unihamburg.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 misnamed. 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=R.D. Silverman;227143]These are misnamed. 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 HyperFermat" 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. 
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=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.asahinet.or.jp/~kc2hmsm/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] 
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/199867221/S0025571898008916/S0025571898008916.pdf]Bjorn & Riesel (1998)[/url].
[QUOTE=R.D. Silverman;227143]These are misnamed. 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/199564209/S00255718199512706181/S00255718199512706181.pdf]Dubner & Keller (1995)[/url] refer to b^2^m + 1 as Generalized Fermat numbers, but then [url=http://www.ams.org/journals/mcom/199867221/S0025571898008916/S0025571898008916.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 GMPECM doesn't properly parse a^2^m which is why a^(2^m) is sometimes used instead.) 
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] 
19469280117455103039625667255411201 divides 99^128+1
111^128+1 = 2 · 769 · 2697217437953 · 125464289479028905217 . P227 1880787140548365990905051620353281 divides 106^128+1 
[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] 
[QUOTE=R.D. Silverman;228164]Check Brent's tables!!!![/QUOTE]
Notice that my ECM applet already checks for factors included in Brent's tables, which is old: updated 11 September 2009 according to his Web site. 
115^128+1 completed (250 curves at 1e7 were enough); result in factordb.com
101^128+1 also completed So: 400 curves at 1e7 on the notpreviouslyfullyfactored numbers in {94..115}^128+1. This took one night on 16 threads on a macpro. 35^128+1 currently finishing linalg 
35^128+1 = 2 * 769 * 77310721 * 465774259823008864434412748488285224928909076514051885029488061896961 * P118

[QUOTE=fivemack;228351]35^128+1 = 2 * 769 * 77310721 * 465774259823008864434412748488285224928909076514051885029488061896961 * P118[/QUOTE]
69 digits by ECM?!? Luigi 
[quote=ET_;228359]69 digits by ECM?!?
Luigi[/quote] It looks like this was either SNFS or GNFS (probably SNFS): [quote=fivemack;228259]35^128+1 currently finishing linalg[/quote] 
[QUOTE=mdettweiler;228361]It looks like this was either SNFS or GNFS (probably SNFS):[/QUOTE]
SNFS197 with a quintic? Or did I miss something? 
yes; x^5+35^2, x=35^26; alim=rlim=15M, Q=7.5M17.5M, 28bit large primes. 494 CPUhours sieving on 8x K10+3x core2, 48 CPUhours on 4xphenom to do the matrix step. A trivial exercise.

[QUOTE=fivemack;228259]
35^128+1 currently finishing linalg[/QUOTE] :blush: 
I found this p44 factor after about 500 curves with B1=11e6: 14571454116637488440882751359138387691414529  8^(2^8) + 5^(2^8)
If anyone wanted to finish some of these off with GNFS, I think the two smallest remaining composites (with bases <= 12) are now: 11^(2^8) + 3^(2^8) = 2 . 3430486387626057217 . 1826300595737909153428993580626433 . 132980092956629419115138154564331009 . 11765487608073254883107740674172674049 . C143 8^(2^8) + 5^(2^8) = 206102775026177 . 25083346678208656976952833 . 14571454116637488440882751359138387691414529 . C149 
[QUOTE=geoff;229528]I found this p44 factor after about 500 curves with B1=11e6: 14571454116637488440882751359138387691414529  8^(2^8) + 5^(2^8)
If anyone wanted to finish some of these off with GNFS, I think the two smallest remaining composites (with bases <= 12) are now: 11^(2^8) + 3^(2^8) = 2 . 3430486387626057217 . 1826300595737909153428993580626433 . 132980092956629419115138154564331009 . 11765487608073254883107740674172674049 . C143 8^(2^8) + 5^(2^8) = 206102775026177 . 25083346678208656976952833 . 14571454116637488440882751359138387691414529 . C149[/QUOTE] That p44 has been found by J. Becker on 20091026, see: [URL="http://www.leyland.vispa.com/numth/factorization/anbn/UPDATE.txt"]http://www.leyland.vispa.com/numth/factorization/anbn/UPDATE.txt[/URL] 
[QUOTE=R. Gerbicz;229530]That p44 has been found by J. Becker on 20091026, see: [URL="http://www.leyland.vispa.com/numth/factorization/anbn/UPDATE.txt"]http://www.leyland.vispa.com/numth/factorization/anbn/UPDATE.txt[/URL][/QUOTE]
Thanks, I didn't know about that project. So it looks like two projects have been working independently on some of these numbers :( 
[QUOTE=geoff;229539]Thanks, I didn't know about that project. So it looks like two projects have been working independently on some of these numbers :([/QUOTE]
Well yes, but not very many of them. I think 128 is too small an exponent to have any remaining composites in the Homogeneous Cunningham tables, so it's only 256 that might be an issue, given the current table limits (except for 3^512+2^512, which happens to be fully factored). And even that's too big an exponent for quite a few of the tables. So there are really only a handful of composites that would currently intersect with work being done on numbers of the form a^2^n + b^2^n (apparently 6, including the C149 from 8^256+5^256). 
I have checked the overlap between the Homogeneous Cunningham and Generalized Fermat tables, it turns out there were 11 factors found independently by both projects, 2 in the Homogeneous Cunningham tables but not in the Generalized Fermat tables (6^2^8+5^2^8, 8^2^8+5^2^8, reported to Wilfrid keller), and 1 in the Generalized Fermat tables but not in the Homogeneous Cunningham tables (9^2^8+7^2^8, reported to Paul Leyland).

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