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Old 2011-02-20, 03:09   #1
ixfd64
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Default What's the point of factoring known composites?

I've noticed that a lot of people are ECM'ing and P-1'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 time-consuming 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?
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Old 2011-02-20, 06:38   #2
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There numbers nerds want complete factorization of numbers. It helps them sleep at night. ECM and P-1 are better at that than TF.
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Old 2011-02-20, 06:39   #3
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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 :)
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Old 2011-02-21, 10:37   #4
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Quote:
Originally Posted by ixfd64 View Post
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?
Some of us want to do exactly the latter: contribute to mathematical knowledge.
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Old 2011-02-21, 11:51   #5
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Quote:
Originally Posted by Commaster View Post
Please, tell me, there's more reason in doing it :)
Factors for small Mersenne numbers are extremely useful in producing lists of pseudoprimes, which in turn are used to make fast primality tests for small numbers.
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Old 2022-12-03, 07:45   #6
ixfd64
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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 square-free.
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Old 2022-12-06, 22:29   #7
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Quote:
Originally Posted by ixfd64 View Post
every factor found is a potential counterexample to the conjecture that all Mersenne numbers are square-free.
And of course, finding a non-square-free Mersenne numbers would mean finding a Wieferich prime.

https://primes.utm.edu/notes/proofs/SquareMerDiv.html
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