Letnbe a natural number. If 3 does not divide (n2 +2), thennis not a prime number orn= 3.

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- Apr 25th 2012, 04:16 PMgoalkeeper00Is 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. - Apr 26th 2012, 04:57 AMemakarovRe: Is this a true statement? Can you prove?
In plain text, it is customary to write n^2 for n

^{2}.

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... - Apr 26th 2012, 06:09 AMprincepsRe: Is this a true statement? Can you prove?
Let $\displaystyle n=p$, where $\displaystyle p$ is a prime number greater than $\displaystyle 3$ , then :

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

Hence :

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

Contradiction . - Apr 26th 2012, 06:29 AMemakarovRe: 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).