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](http://www.mersenne.org/). 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?



Why the search for ever larger primes?
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](http://en.wikipedia.org/wiki/Lenstra%E2%80%93Pomerance%E2%80%93Wagstaff_conjecture#Lenstra.E2.80.93Pomerance.E2.80.93Wagstaff_conjecture) 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.


