Results 1 to 2 of 2

Math Help - Suppose that p=2^k-1 is prime. Prove that either k=2 or k is odd.

  1. #1
    Member
    Joined
    Oct 2009
    From
    Detroit
    Posts
    157
    Thanks
    5

    Suppose that p=2^k-1 is prime. Prove that either k=2 or k is odd.

    The latex outputs look goofy in my browser. I'm not sure whats up with that so I'm abandoning it.

    Suppose that p=2^k-1 is prime. Prove that either k=2 or k is odd.
    Proof;

    Assume p is prime and n,p,k are integers.

    p=2^k-1

    assume k=2 then p=3 which is prime. this concludes the first case.

    assume k is even. k=2n then,

    p=2^{2n}-1=(2^n+1)(2^n-1)

    Since k=2n>2, n>1 and both of the factors of p, (2^n+1) and (2^n-1) are greater then 1 for all n

    Since p is a prime it has two factors p and 1

    since neither factor of p can be 1, p can not be a prime if k is even.

    This is a contradiction with the premise that p is a prime, by reductio ad absurdum we conclude that it is not the case that k is even.

    Since k is a natural number it can only be even or odd. And as k is not even it must be odd.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Sep 2012
    From
    Australia
    Posts
    3,612
    Thanks
    591

    Re: Suppose that p=2^k-1 is prime. Prove that either k=2 or k is odd.

    Hey bkbowser.

    The proof is very good. Very simple and very elegant.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 6
    Last Post: March 2nd 2011, 12:36 AM
  2. Suppose that every prime...
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: March 21st 2010, 01:44 PM
  3. Suppose that b<=L+e for all e>0. Prove b<=L
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: May 4th 2009, 12:02 AM
  4. Replies: 1
    Last Post: March 20th 2009, 05:38 AM
  5. Suppose gcd(a,b) = 1, prove gcd(na,nb) = n
    Posted in the Discrete Math Forum
    Replies: 3
    Last Post: March 19th 2009, 05:51 PM

Search Tags


/mathhelpforum @mathhelpforum