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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?

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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.

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Visualization and discussion of distribution of Mersenne primes is maintained by Chris K. Caldwell on his "Where is the next Mersenne prime?" FAQ page.

Particularly, it discusses the Lenstra–Pomerance–Wagstaff conjecture:

There are asymptotically $e^\gamma\log_2 \log{x} \approx 1.78\log_2\log{x} $ Mersenne primes less than $x$.

where $\gamma$ is the Euler's constant.

(Interestingly, the formula predicts 45.9 Mersenne primes smaller than $2^{82,589,933}-1$, though this is actually 51st known Mersenne prime. So those primes currently appear more frequently than expected.)

The page also discusses the gaps between Mersenne primes.

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