Only prime number for $p^2+2$ to be prime as well Prove that the only prime number $p$ for $p^2+2$ to be prime as well is $p=3$
If $p\neq 3$ then it has form $3k\pm 1$ so $p^2+2=(3k\pm 1)^2+2=9k^2\pm6k+3=3(3k^2\pm 2k+1)$ so wont be prime
hint: If $q = p^2 + 2$ is a prime number, then $p = 3k+1$ or $p = 3k+2$