Squares, Cubes and Prime numbers Suppose $p$  and $p^2+2$ are prime numbers, prove that $p^3+2$ is also a prime number. Actually I do not know what is the relationship among square numbers, cube numbers and prime numbers

Here's a slightly different take on Matthew's approach.  Multiply $p$ by $p^2+2$.  You get $p^3+2p$.  You know that $p^3\equiv p\bmod 3$ from Fermat's Little Theorem, therefore $p^3+2p\equiv 3p$ is a multiple of $3$.  If both of your factors $p$ and $p^2+2$ are prime factors of this multiple of $3$, then one factor must be exactly $3$.  Alas, $p^2+2=3$ implies $p=1$ which does not qualify as a prime, so there is only one more possibility to check.
