The infinitude of prime Mersenne numbers

Let $M(n)=2^n-1$ denote the $n$th Mersenne number. The sequence of prime Mersenne numbers is infinite.

Infinitude conjecture. There are infinitely many integers $n$ such that $M(n)$ is prime.

This is a central open problem concerning Mersenne numbers; the source presents it as an open question in number theory and reformulates it dynamically using the doubling map.