20050412, 14:19  #1 
"William"
May 2003
New Haven
2369_{10} Posts 
P56 ECM Factor
Today Pascal Ochem reported a 56 digit ECM factor. Paul Zimmermann's Top 100 List has it as the seventh largest ECM factor ever found. Pascal has been working from the Composites Page of the notyetreadyforprimetime Search for Odd Perfect Numbers.

20050412, 15:16  #2 
"Nancy"
Aug 2002
Alexandria
2,467 Posts 
It was one of the weird cases where ECM found the larger prime factor. The input number was only a c103, the cofactor is a p48. While this is a very nice factor, it will probably not be counted on Richard Brent's record factors list  MPQS would have been a much better choice for this number than ECM with such high bounds.
Alex 
20050423, 09:38  #3 
"Nancy"
Aug 2002
Alexandria
9A3_{16} Posts 
Pascal did it again, this time a p57 of the c133 of 3719^371. This time he found the smaller prime factor, and it seems it will just barely make Brent's Top Ten list. The condition is
2. Let r = length of input (composite) / length of factor (prime), where the lengths are measured in decimal digits. To be included a factor must satisfy r >= 2.2. [...] Now 133/57=2.33, so the factor should make the list, even though the ratio log(c133)/log(p57) = 2.1. Alex 
20050423, 11:06  #4  
Nov 2003
2^{2}·5·373 Posts 
Quote:
This number would be much better done with GNFS. 

20050423, 11:41  #5 
"Nancy"
Aug 2002
Alexandria
2,467 Posts 
Oh, correct! I was looking at the SNFS difficulty column in the table on William's page, not the cofactor size column. So again no entry in Brent's table.
Alex PS: actually, at SNFS difficulty only 133, this would still have been a SNFS job. Last fiddled with by akruppa on 20050423 at 11:46 Reason: post scriptum 
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