Let's find all prime numbers $p$ for which $p^{2}+2$ is also a prime number.
$p^{2}+2=(p-1) \cdot(p+1)+3$. This is divisible by 3 if $p$ is not 3, because then, since $p$ is a prime number, it cannot be divisible by 3, so either the number before it or the number after it, which is the first term, is always divisible by 3. A prime number can only be so if it equals 3, that is, if $p=1$. However, 1 is not usually considered a prime number (it does not count as a prime factor). Thus, the only solution can be $p=3$ and in this case $p^{2}+2=11$ is indeed a prime number.