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Math Help - Consecutive Twin Primes Proof

  1. #1
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    Consecutive Twin Primes

    Let p>3 be a prime. If p+2 is a prime, then p+4 is not a prime. Any way to show this?
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  2. #2
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    Quote Originally Posted by Treadstone 71
    Let p>3 be a prime. If p+2 is a prime, then p+4 is not a prime. Any way to show this?
    Theorem:
    Let p be a prime >3, then all three of p, p+2, p+4 cannot be prime, because
    at least one of them must be divisible by 3.

    Proof:
    Suppose p>3 and not divisible by 3, then there exist k>0 such that
    p=k.3+r, where r is either 1 or 2 (r cannot be 0 since then p would be
    divisible by 3, which it is not by supposition).

    If r=1 then p+2 is divisible by 3.

    If r=2 then p+4 is divisible by 3.

    QED

    Note: this is stated on page 5 of Hardy and Wright (fifth edition)
    but without the restriction that p>3, so failing to exclude 3,5,7!

    Your problem is a corollary of the theorem. For if p>3 and p and p+2
    are both prime neither of them is divisible by 3, so p+4 must be
    divisible by 3.

    RonL
    Last edited by CaptainBlack; December 18th 2005 at 05:21 AM.
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  3. #3
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    Thanks a bunch.
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