will searching for factors sometimes be faster than LL test?

[ixfd64]

I know that sometimes testing numbers like M<insert 15 digit prime here> takes years. Yet, some people like Ernst Mayer (Ewmayer) have been able to find factors within minutes. Would this be useful in filtering out exponents know to yield composite numbers?

[cheesehead]

Yes. Putting a certain amount of effort into trying to factor a Mersenne number, before running the L-L test on that number, has always been a part of GIMPS. The GIMPS database keeps a record of how far each Mnumber with a prime exponent has been trial-factored and how high the limits have been tried for P-1 factoring that Mnumber.

Mnumbers for which a factor is found are removed from the list of those eligible for L-L testing (first-time or double-check) assignment.

PrimeNet factoring assignments are for trial-factoring Mnumbers before they're assigned for L-L testing.

Tradeoff limits are calculated for trial factoring and P-1 factoring. Tradeoff limits are the limits at which (the time spent factoring) divided by (the probability of finding a factor) equals the time required for L-L testing. Below that limit, time spent trying to find a factor is more valuable than time spent L-L testing. Then when a Mnumber is assigned for trial factoring, Prime95 performs as much trial factoring as is required to go up to the tradeoff limit. When a Mnumber is assigned for L-L testing, Prime95 checks whether it has had P-1 factoring attempted on it, and if not then Prime95 performs P-1 factoring up to the tradeoff limit before starting the L-L test.

BTW, only Mnumbers with prime exponents are assigned by GIMPS for factoring and L-L testing, because it is known that all Mnumbers with composite expoments must have proper factors and thus cannot be prime themselves.

[Maybeso]

I know that sometimes testing numbers like M<insert 15 digit prime here> takes years.

There are also organized efforts to factor mersennes with composite exponents. These numbers often yield factors more quickly than prime exponents of similar size.

Having said that -- E.Mayer and others have logged very impressive factor lists of mersenne numbers with both prime and composite exponents. They have fine-tuned the process to an amazing level of efficiency - so that whatever catagory of target they focus on usually falls quickly.

[Prime95]

I know that sometimes testing numbers like M<insert 15 digit prime here> takes years.

You meant eons - not years, right?