20071105, 17:31  #1 
1976 Toyota Corona years forever!
"Wayne"
Nov 2006
Saskatchewan, Canada
3·1,667 Posts 
M727 ... History
Discussions recently regarding factoring M1061 and a few others seems to suggest that factoring attempts whether P1 or ECM for all practical purposes seldom go over 60 bits (18 digits). In very rare cases there have been factors reported just over 100 bits (30 digits) ... so how did someone manage to find the nearly 100 digit (over 300 bits) factor of M727?

20071105, 17:40  #2  
(loop (#_fork))
Feb 2006
Cambridge, England
2^{2}·1,613 Posts 
Quote:
On numbers of less than about 750 bits, and if you've got the resources of a major research group behind you, numbers up to just over 1000 bits, you can run something called SNFS which finds factors regardless of their size. The process is quite complicated; for M727, it would have taken about four months on a K8/2GHz. For M1039, it took roughly two hundred CPUyears, and one part of the calculation required four closelycoupled clusters each of around fifty machines. 

20071111, 12:53  #3 
"James Heinrich"
May 2004
exNorthern Ontario
13×277 Posts 
I'm not sure on the reasons behind it, but on occasion P1 will return very large factors, which are (always?) composite. For example, on my stats site you'll see the top10 P1 factors are all between 40 and 48 digits long (131159 bits), but every one of them is composite.
This is a handy quickcheck tool for primality testing of factors. 
20071111, 15:37  #4  
"Jacob"
Sep 2006
Brussels, Belgium
11011110001_{2} Posts 
Some time ago P1 returned a very big factor for a number (it is the biggest one on James page :) I thought it was a top ten factor, but it was composite :( The explanation given was :
Quote:


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