Eberhart's Conjecture

If $q_n$ is the $n$th prime such that $M_{q_n}$ is a Mersenne Prime, then

$$q_n\sim (3/2)^n.$$

It was modified by Wagstaff (1983) to yield

$$q_n\sim (2^{e^{-\gamma}})^n,$$

where $\gamma$ is the Euler-Mascheroni Constant.

References

Ribenboim, P. The Book of Prime Number Records, 2nd ed. New York: Springer-Verlag, pp. 332-333, 1989.

Wagstaff, S. S. ``Divisors of Mersenne Numbers.'' Math. Comput. 40, 385-397, 1983.