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Math Help - Prove that there are infinitely many primes in the form 6k+5

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    Prove that there are infinitely many primes in the form 6k+5

    Prove that there are infinitely many primes in the form 6k+5 where k is a positive integer.

    thanks!
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    MHF Contributor Also sprach Zarathustra's Avatar
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    Re: Prove that there are infinitely many primes in the form 6k+5

    Quote Originally Posted by VinceW View Post
    Prove that there are infinitely many primes in the form 6k+5 where k is a positive integer.

    thanks!
    Notice that:

    (6t+1)(6k+1)=6(6tk+t+k)+1

    Now, suppose that there is finite numbers of prime of the form: 6k+1 and they are: p_1,p_2,...,p_n.

    N=6p_1p_2...p_n-1=5(mod 6)

    All the prime factor of N are different from p_1,p_2,...,p_n and they are not 2 or 3, N is also is of the from 6k+1 - it's a contradiction!
    Last edited by Also sprach Zarathustra; June 13th 2011 at 11:03 PM.
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