# Prove there are infinitely many primes using Mersenne Primes

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1. Feb 28, 2017

### Kitty Kat

1. The problem statement, all variables and given/known data
Prove that there are infinitely many primes using Mersenne Primes, or show that it cannot be proven with Mersenne Primes.

2. Relevant equations
A Mersenne prime has the form: M = 2k - 1

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

Proof for problem

Assume there are finitely many primes, p1, ... , pr, and let pr be the last prime.

Then by Lemma 1, 2pr - 1 is a Mersenne prime, which is certainly larger than pr, 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.

Last edited: Feb 28, 2017
2. Feb 28, 2017

### 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.