Are Mersenne numbers with Mersenne prime exponent always prime?

Question

A Mersenne number is a number on the form $2^n-1$. For it to be prime, the number $n$ must be prime. My question is that if $n$ is another Mersenne prime, will $2^n-1$ be always prime?

It seems so, to me anyway.

Answer 1

My own program found that $2^{2^{13} - 1} - 1$ is not a (probable) prime number, confirmed by Wolfram Alpha and https://oeis.org/A000043. Note that $M_{13} = 8191$ is prime and $M_{8191}$ is not.

Answer 2

https://oeis.org/A000043

We only know about finitely many Mersenne primes. If your conjecture was true, then we could generate infinitely many Mersenne primes.