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.

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...

Quote:

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Originally Posted by goalkeeper00

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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Let , where is a prime number greater than , then :



Hence :



Contradiction .

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).