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.