20110220, 03:09  #1 
Bemusing Prompter
"Danny"
Dec 2002
California
2×29×43 Posts 
What's the point of factoring known composites?
I've noticed that a lot of people are ECM'ing and P1'ing Mersenne numbers that are known to be composite. Is there any reason for doing this?
Generally, searching for small factors helps eliminates candidates for timeconsuming primality tests. However, trying to factor composite numbers isn't going to help us find a new Mersenne prime. If we stopped assigning ECM factoring, we could reach those milestones a lot quicker. I know that many people like to find factors for the same reason we look for Mersenne primes: because they are there. However, there is no shortage of Mersenne factors, compared to, say, Fermat factors, which we know only a few hundred of. So my question is: what is the point of trying to factor numbers that are known to be composite, besides contributing to our mathematical knowledge? 
20110220, 06:38  #2 
6809 > 6502
"""""""""""""""""""
Aug 2003
101×103 Posts
10,891 Posts 
There numbers nerds want complete factorization of numbers. It helps them sleep at night. ECM and P1 are better at that than TF.

20110220, 06:39  #3 
Jun 2010
Kiev, Ukraine
3×19 Posts 
As I understand, which is almost obviously not the actual reason, we are looking for more factors to run global filter jobs and eliminate a lot of candidates in one run.
Please, tell me, there's more reason in doing it :) 
20110221, 10:37  #4 
"Richard B. Woods"
Aug 2002
Wisconsin USA
1111000001100_{2} Posts 

20110221, 11:51  #5 
Aug 2006
5,987 Posts 

20221203, 07:45  #6 
Bemusing Prompter
"Danny"
Dec 2002
California
100110111110_{2} Posts 
I'm aware this is a very old thread, but I suddenly realized there is another good reason to factor composite Mersenne numbers: every factor found is a potential counterexample to the conjecture that all Mersenne numbers are squarefree.

20221206, 22:29  #7  
Sep 2003
3×863 Posts 
Quote:
https://primes.utm.edu/notes/proofs/SquareMerDiv.html 

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