Prime $p$ with $p^2+8$ prime I need to prove that there is only one $p$ prime number such that $p^2+8$ is prime and find that prime.
Anyway, I just guessed and the answer is 3 but how do I prove that?

Suppose there exists a prime $p$ (not equal to 3) such that $p^2 + 8$ is prime. Since $p$ is indivisible by 3, therefore $p \equiv 1 \pmod 3$ or $p \equiv -1 \pmod 3$, therefore $p^2 \equiv 1 \pmod 3$. Thus, $p^2 + 8 \equiv 9 \equiv 0 \pmod 3$, therefore $p^2 + 8$ is a prime greater than 3 that is divisible by 3 (a contradiction).
