Infinitude conjecture for Mersenne primes

This article states a conjecture about there existing infinitely many of the following numbers/structures: Mersenne prime
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Statement

The conjecture is that there are infinitely many Mersenne primes, i.e., primes of the form $M_{p}=2^{p}-1$, where $p$ is a natural number. (Note that if $2^{p}-1$ is prime, so is $p$).