Why have ECM testing for known non-prime Mersenne numbers?

[sd235]

Out of curiosity, I requested that my computer get ECM assignments. I got assignments for numbers like M48437, M65413, etc. These are relatively small Mersenne numbers that we already know are not prime. Why is GIMPS asking my computer to test them further?

[Uncwilly]

Part of the GIMPS entire project is finding factors for exponents. There are folks looking at them. Lower numbers like the ones you mentioned are now best served by ECM, not TF or other methods.

[Prime95]

If your computer is rather slow or not running many hours a day, then the server may have determined it would take too long to run a double-check test, instead giving you the quicker ECM assignment.

You can force the work type you prefer in the Test / Worker Windows dialog box.

[sd235]

Originally Posted by Uncwilly

Part of the GIMPS entire project is finding factors for exponents. There are folks looking at them. Lower numbers like the ones you mentioned are now best served by ECM, not TF or other methods.

Got it, thanks for the clarification!

[GP2]

You yourself requested ECM testing for your computer. In practice, ECM is only done for Mersenne exponents that already have known factors.

ECM is not cost-effective for exponents that haven't been first-time tested (in the 70 million range) or exponents that haven't been double-checked (in the 40 million range). Using ECM to look for factors of such exponents would take longer than just doing Lucas-Lehmer (LL) tests.

It is cost-effective to do trial-factoring (TF) or P−1 testing to search for factors of such exponents, in order to save time and skip some Lucas-Lehmer tests. However, as mentioned above, it isn't cost-effective to do ECM if your goal is solely to look for Mersenne primes.

However, ECM is capable of finding factors that TF and P−1 aren't able to find, and some consider it interesting to search for additional factors just for fun. This is usually done only for fairly small exponents.

Another reason that the software provides ECM is that some older computers simply can't do LL testing fast enough for exponents in the ranges currently being tested. They might take a year or more to do a single exponent, and in that time their assignment would probably expire, and the exponent would be reassigned to another user. However, those older computers can still do interesting work by searching for additional factors using ECM.

[lycorn]

Originally Posted by GP2

In practice, ECM is only done for Mersenne exponents that already have known factors.

Well, not quite. A considerable amount of the global ECM effort, I would say the largest part of it, is geared towards finding factors for numbers that, although already LL tested and DCed, thus being necessarily not prime, have not yet any known factor.

[GP2]

Originally Posted by lycorn

Well, not quite. A considerable amount of the global ECM effort, I would say the largest part of it, is geared towards finding factors for numbers that, although already LL tested and DCed, thus being necessarily not prime, have not yet any known factor.

Yes, I should have written "already known to be composite" rather than "already have known factors". Just got muddled there for some reason. I've even run a few ECM tests myself on exponents with no known factors.

[section31]

Originally Posted by GP2

However, ECM is capable of finding factors that TF and P−1 aren't able to find, and some consider it interesting to search for additional factors just for fun. This is usually done only for fairly small exponents.

I was recently assigned ECM work via the "Whatever makes sense" option (for the first time that I can recall), had this same question, and found this thread. Finding factors through ECM just for fun is fine... I'm wondering though if there's anything it's useful for otherwise... i.e. they'll end world hunger or get us to Mars faster.

[chalsall]

Originally Posted by section31

i.e. they'll end world hunger or get us to Mars faster.

No. We do this kind of work because we enjoy it.

It consumes energy. There is no upside beyond the enjoyment.

[irowiki]

Originally Posted by GP2

You yourself requested ECM testing for your computer. I
Another reason that the software provides ECM is that some older computers simply can't do LL testing fast enough for exponents in the ranges currently being tested. They might take a year or more to do a single exponent, and in that time their assignment would probably expire, and the exponent would be reassigned to another user. However, those older computers can still do interesting work by searching for additional factors using ECM.

Originally Posted by section31

I was recently assigned ECM work via the "Whatever makes sense" option (for the first time that I can recall), had this same question, and found this thread. Finding factors through ECM just for fun is fine... I'm wondering though if there's anything it's useful for otherwise... i.e. they'll end world hunger or get us to Mars faster.

Also, sometimes GIMPS, via "Whatever makes sense", will assign ECM because for whatever reason, on a new install, the computer (client) reports as a 100mhz.

After crunching ECM for a few days the server realizes it is fine and then assigns a DC instead.

[Dr Sardonicus]

I suppose it is also possible that, TF and P-1 being done, and the large cofactor still being composite, ECM might split off another factor, after which the remaining cofactor might be a very large PRP. That would be a nice feather in someone's cap...