I have been exploring the fascinating world of prime numbers, particularly Mersenne Primes, and have noticed an interesting pattern. It seems to me that $2^n - 1$ is prime as long as $n$ itself is a Mersenne prime.
I already know that $2^n - 1$ is not necessarily prime just because $n$ is a normal prime number, but I have not found a counterexample that to the claim that $2^n - 1$ is a prime whenever $n$ is a Mersenne prime.
Is there a proof that this claim is false?
$2^{13}-1 = 8191$ is a Mersenne prime, but $2^{8191}-1$ is composite.