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Math Help - Euclid's Theorem

  1. #1
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    Euclid's Theorem

    How do you prove that there exists infinitely many primes p such that p = 2 mod 3 ?

    I know that we can use the Euclid's Theorem but I have no idea how to do this kind of question, can anyone give me some hints?

    Thanks alot
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  2. #2
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    Quote Originally Posted by knguyen2005 View Post
    How do you prove that there exists infinitely many primes p such that p = 2 mod 3 ?
    Assume there are finitely many p_1,...,p_n.
    Form N = 3p_1....p_n - 1.

    We know that N has an odd prime divisor p. If all its prime divisiors were of the form 3k+1 then N would be of the form 3k+1 and impossibility. Therefore there is a prime divisor p of the form 3k+2. Thus, p|N and it must be amongst p_1,...,p_n. But that leads to p|1 which is a contradiction.
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    thank you very much ThePerfectHacker
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