20080614, 14:54  #1 
Jun 2008
2^{3}·3^{2} Posts 
lower bounds on incomplete factorizations
I am searching for all 'small' factors p < 2^55 of Mersenne numbers M(n) up to n=1e5.
The Cunningham Project (in particular this site) is my primary source. However, I'd like to save myself the effort of trial dividing the remaining composite numbers, as that work probably has been done before. In the Mersenne database I see trial division bounds at 2^58 and sometimes more, for instance at n = 1061. That is exactly what I need. But what about the trial division performed on incompletely factored Mersenne numbers, like for instance M(1069)? Is there a way for people like me to see what range of trial division was attempted on such numbers? 
20080614, 16:23  #2  
"William"
May 2003
New Haven
2^{2}·3^{2}·5·13 Posts 
Quote:
37473613084215372416028665821312617743022228898297 Will Edgington's Mersenne Page is at http://www.garlic.com/~wedgingt/mersenne.html 

20080614, 18:42  #3  
Nov 2003
2^{6}·113 Posts 
Quote:
factor of M1069 found via ECM. There is no guarantee that there is not a smaller factor. Note that this number is 50 digits. The universe isn't old enough to have done trial division through 50 digits. 

20080614, 18:58  #4 
Jun 2008
2^{3}·3^{2} Posts 
Thanks for the remarks. Will Edgington's data is definitely something I can put to use. I'd only wish the uncertainty could be removed.
Code:
Note that some smaller trial factors may not have been attempted since my database updates presently assume that all factorers are trial factorers (see also the 'G' line below). 
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