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Prove there are infinitely many primes using Mersenne Primes

  1. Feb 28, 2017 #1
    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. jcsd
  3. Feb 28, 2017 #2

    PeroK

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    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.
     
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