What Impact Do Large Mersenne Primes Have on Mathematical Research?

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Pagan Harpoon said:
does everyone say "Yes... there it is, now let's find the next one."?

Pretty much that.

There's some interest in looking at the distribution of primes, but there's only so much you can deduce from one more prime.
 
Pretty much that. CRGreathouse

The problem begins with Euclid and the study of perfect numbers. It was brought into the 17th Century by Mersenne, who made some guesses, which interested others. Euler in 1772 proved the primality of 2^31-1.

There are some fairly simple criterion for primes that might be divisors, for example: A prime divisor of 2^p-1 must be of the form 2mp+1. 2^11 being such a composite case having both 23 and 89 as divisors. Also p is of the form 8k plus or minus 1 (which means that 2 is a quadratic residue modulo p). Of course if 2^q-1 is prime then so is q.

So it seems interest is mostly because of the historical value, the ease with which many potential factors are eliminated, and the vast size of the potential prime. Most of the very large primes discovered are Mersenne.
 
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Thank you for clarifying.