What's the point of factoring known composites?

ixfd64 · 2011-02-20, 03:09

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?

Uncwilly · 2011-02-20, 06:38

There numbers nerds want complete factorization of numbers. It helps them sleep at night. ECM and P-1 are better at that than TF.

Commaster · 2011-02-20, 06:39

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 :)

cheesehead · 2011-02-21, 10:37

Quote:

Originally Posted by ixfd64

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.

CRGreathouse · 2011-02-21, 11:51

Quote:

Originally Posted by Commaster

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.

ixfd64 · 2022-12-03, 07:45

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.

GP2 · 2022-12-06, 22:29

Quote:

Originally Posted by ixfd64

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

2011-02-20, 03:09	#1
ixfd64 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 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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2011-02-20, 06:38	#2
Uncwilly 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 P-1 are better at that than TF.

2011-02-20, 06:39	#3
Commaster 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 :)

2022-12-03, 07:45	#6
ixfd64 Bemusing Prompter "Danny" Dec 2002 California 100110111110₂ 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 square-free.