For how many prime numbers $p$ is $p^2+2$ also a prime number?

Try $p=3:3^2+2=11$, which is prime. For any other prime $p,p$ is not a multiple of 3, so $p$ leaves remainder 1 or 2 on division by 3 and $p^2$ leaves remainder 1. Then $p^2+2$ is a multiple of 3 and bigger than 3, so it is composite — for instance $5^2+2=27$ and $7^2+2=51=3\times 17.(p=2$ gives 6, also composite.) So $p=3$ is the only one.