What do we know about the distribution of Mersenne primes? Mersenne primes are primes of the form $M_n = 2^n - 1$.  I'm wondering how far apart successive Mersenne primes can be.  For example, is $M_{n+1} \le O((M_n)^e)$?  Or, is $M_{n+1}$ always less than some power of $M_n$?  If not, how close together can successive Mesenne primes be in the worst case?

It is not even known whether there are infinitely many Mersenne primes! There are guesses only, based on probabilistic assumptions for which there is no proof.
For a brief survey of some conjectural answers about the distribution of Mersenne primes, please see Wikipedia article on Mersenne Conjectures. 
