Probability that a Mersenne number is prime Let $p$ be a prime and let $M_p = 2^p-1$ be a (Mersenne) number. Is there any known result on the probability that $M_p$ is prime? In particular is it known whether the probability tends to $1$ as $p \to \infty$? If so, is there any known lower bound for the asymptotic rate with which this probability tends $1$? Thank you.
I'm pretty sure there is no such result: it is not even known that infinitely many $M_p$ are primes, or that infinitely many are composite.
For the already mentioned reason, this probability question can't be definitively solved right now.  But for fun you could assume one of the conjectures about the distribution of Mersenne primes (I think Wagstaff conjectured an asymptote of C*loglog(x) for some funky constant C?) and see what that brings.