Prove there are infinitely many primes using Mersenne Primes

[Kitty Kat]

Homework Statement

Prove that there are infinitely many primes using Mersenne Primes, or show that it cannot be proven with Mersenne Primes.

Homework Equations

A Mersenne prime has the form: M = 2^k - 1

The Attempt at a Solution

Lemma: If k is a prime, then M = 2^k - 1 is a prime.
Proof of Lemma: Assume that k is composite, then 2^k - 1 must also be composite.
k = a * b
2^ab - 1 is then divisible by both (2^a - 1) and (2^b - 1)
Therefore k must be a prime.

Proof for problem
Assume there are finitely many primes, p₁, ... , p_r, and let p_r be the last prime.

Then by Lemma 1, 2^p_r - 1 is a Mersenne prime, which is certainly larger than p_r, therefore there are infinitely many primes.After writing all this out, I realized that the lemma is most certainly false, so I guess there isn't a way to prove infinitely many primes with Mersenne Primes, since the number of Mersenne Primes themselves are an open question.

 

[PeroK]

You appear to think you have proved that if $k$ is prime, then $2^k -1$ is prime. This is false. E.g $k = 11$

You have proved the converse: If $2^k -1$ is prime, then $k$ is prime.