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Math Help - Is this a true statement? Can you prove?

  1. #1
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    Is this a true statement? Can you prove?

    Let n be a natural number. If 3 does not divide (n2 +2), then n is not a prime number or n = 3.
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  2. #2
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    Re: Is this a true statement? Can you prove?

    In plain text, it is customary to write n^2 for n2.

    This seems true. If 3 does not divide n^2 + 2, then n^2 + 2 = 3k + 1 or n^2 + 2 = 3k + 2 for some k. In the latter case, n is divisible by 3. Suppose that n^2 = 3k - 1 and, towards contradiction, assume that n is prime. Then n = 6m + 1 or n = 6m - 1 for some integer m as described here. Try to derive a contradiction.

    Maybe there is an easier proof...
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  3. #3
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    Re: Is this a true statement? Can you prove?

    Quote Originally Posted by goalkeeper00 View Post
    Let n be a natural number. If 3 does not divide (n2 +2), then n is not a prime number or n = 3.
    Let n=p, where p is a prime number greater than 3 , then :

    p \equiv 1 \pmod 3 ~\text{or}~p\equiv 2 \pmod 3

    Hence :

    p^2 \equiv 1 \pmod 3 \Rightarrow p^2+2 \equiv 0 \pmod 3

    Contradiction .
    Last edited by princeps; April 26th 2012 at 06:12 AM.
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  4. #4
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    Re: Is this a true statement? Can you prove?

    So, you in fact prove a stronger statement: If n > 3 and 3 does not divide n^2 + 2, then 3 divides n (and not just that n is not prime).
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