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Math Help - infinitely many primes of the form 6k + 5 and 6K + 1

  1. #1
    Newbie
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    Aug 2009
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    infinitely many primes of the form 6k + 5 and 6K + 1

    hey the question is

    show that there are infinitely many primes of the form 6k + 5. does the method work for 6k + 1.

    my answer so far is

    suppose there are finite primes of the form 6k + 5
    order them such: p(1) < p(2) <....< p(n)
    let R = 6(p(1)p(2)...p(n)) + 5
    R can't be prime, if it is R > p(n)
    R can't be composite as any division will give a remainder of 5
    therefore there are infinitely many primes

    i think there's something not quite right with it and i can't use Dirichlet's Theorem

    kudos for any help!
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  2. #2
    Senior Member
    Joined
    Apr 2009
    From
    Atlanta, GA
    Posts
    408

    good work

    You have hit the nail on the head. Your proof is correct. Here are some clarifications:

    Let p_1<p_2<p_3<...<p_n be a finite list of primes of the form 6k+5. Let R=6p_1p_2p_3...p_n+5. Since R>p_n, R cannot be prime, so it is composite. When divided by any p_i, it leaves a remainder 5, therefore R is not divisible by any prime of the form 6k+5. So R must be the product of primes only of the form 6k+1. But if you take the product of numbers of the form 6k+1, the product is also of the form 6k+1. So R must leave a remainder 1 when divided by 6. But by construction, R leaves a remainder 5 when divided by 6. We have reached a contradiction, therefore our premise is false: there must be an infinite number of primes of the form 6k+5.

    This argument does not work for the 6k+1 case. Can you figure out why?
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