20100826, 04:13  #1 
Mar 2003
New Zealand
2205_{8} Posts 
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: curves.txt, and input files suitable for Prime95 and GMPECM here: input.zip. 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 main results page). He also keeps a separate listing of known factors just for small m. So far I have listed only the curves I have done myself, plus the few recorded on this page. 
20100826, 06:29  #2 
"Tapio Rajala"
Feb 2010
Finland
473_{8} Posts 
A few months ago I ran some curves on 12^{2[sup]9}[/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. 
20100826, 11:40  #4  
Nov 2003
2^{2}·5·373 Posts 
Quote:
Fermat numbers are a^2^n + 1 for a > 2. The numbers you refer to should be called Homogeneous Fermat Numbers. 

20100826, 12:02  #5  
Mar 2006
Germany
5×569 Posts 
Quote:
I'm thinking of "Generalized HyperFermat" like S.Harvey did for Woodall/Cullen type (his page). And: I've made a small Summary page for W.Kellers table (1 <= a,b <= 12, 0<=n<=19) here (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. Last fiddled with by kar_bon on 20100826 at 12:23 

20100826, 14:44  #6 
(loop (#_fork))
Feb 2006
Cambridge, England
1100011010110_{2} Posts 
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 
20100826, 15:53  #7  
"William"
May 2003
New Haven
936_{16} Posts 
Quote:
http://www.asahinet.or.jp/~kc2hmsm/cn/ 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) http://wwwmaths.anu.edu.au/~brent/factors.html 

20100827, 03:06  #8  
Mar 2003
New Zealand
13×89 Posts 
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 here, discussed in Bjorn & Riesel (1998).
Quote:
(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.) 

20100827, 08:53  #9 
Mar 2010
On front of my laptop
7·17 Posts 
I reported all known factors of n^64+1, where n<=1000.
That was a time consuming work. http://factordb.com/new/index.php?qu...at=1&sent=Show 
20100902, 12:40  #10 
(loop (#_fork))
Feb 2006
Cambridge, England
2·11·17^{2} Posts 
19469280117455103039625667255411201 divides 99^128+1
111^128+1 = 2 · 769 · 2697217437953 · 125464289479028905217 . P227 1880787140548365990905051620353281 divides 106^128+1 
20100902, 12:45  #11  
Nov 2003
2^{2}·5·373 Posts 
Quote:
fivemack: I have. These factors, though small, are new. Or are there some other 'Brent's tables' which aren't http://wwwmaths.anu.edu.au/~brent/ft...ors/factors.gz Last fiddled with by fivemack on 20100902 at 15:23 

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