TITLE: Suppose that both $p$ and $p^2 +2$ are prime numbers. Show that $p=3$.
QUESTION [6 upvotes]: Can someone help me out with this? I've been working on it for quite a long time but I'm not sure if I'm even getting anywhere.

REPLY [9 votes]: If $p$ is a prime other than 3, then $p \equiv 1 \bmod 3$ or $p \equiv 2 \bmod 3$. But either way $p^2 \equiv 1 \bmod 3$, which means that $p^2 + 2 \equiv 0 \bmod 3$, which means $p^2 + 2$ must be composite. But if $p = 3$, then $p^2 + 2 = 11$, which is prime.
If you're not convinced, try out a few cases of $p^2 + 2$:

$5^2 + 2 = 27 = 3^3$
$7^2 + 2 = 51 = 3 \times 17$
$11^2 + 2 = 123 = 3 \times 41$
etc.

REPLY [6 votes]: Hint: Suppose $p \neq 3$. What does that mean is true of $p \mod 3$? What does that mean for $p^2 \mod 3$?