I understand why primes are useful numbers and also why the product of large primes are useful such as for application in public key cryptography, but I am wondering why it is useful to continue the search for larger and larger prime numbers such as in the GIMPS project. It would seem to me since that since it already proven that there are an infinite number of primes, I am not quite sure why working to finding bigger and bigger really matters!? Is this a "Climbing Mt Everest because it is there" kind of thing, or is the search and finding bigger results somehow furthering mathematics in some kind of way?
[Math] Why the search for ever larger primes
big-picturent.number-theoryprime numbers
Best Answer
Well the M in GIMPS stands for Mersenne, and it hasn't been proven that there are infinitely many Mersenne primes. But it's widely believed to be true--in fact there is a conjectural estimate of their frequency. I think the search for Mersenne primes is mostly a "because it is there" thing, but it could provide numerical evidence for or against this conjecture.
GIMPS is probably more interesting as an experiment in massively distributed computation.