$p^3 + 2$ is prime if $p$ and $p^2 + 2$ are prime?

Question

I'm self-learning number theory.
I want to prove the following statement:

$$p \text{ is prime } \land \text{ }p^2 + 2 \text{ is prime } \implies p^3 + 2 \text{ is prime }$$

I failed to do so, and I failed to find any proofs online.
My initial attempts involved using Fermat's Little Theorem:

$$\begin{align*} a^{p} &\equiv a \mod p \\ a^{p^2 + 2} &\equiv a \mod {p^2 + 2} \\ \end{align*}$$

But that form didn't really help me that much. Any hints?

Answer 1

Hint: if $n$ is not divisible by $3$, then $n^2\equiv1\pmod3$.

Hence if $p\neq3$, $3\mid p^2+2>3$ so $p^2+2$ cannot be prime. This means the implication says $\perp\implies p^3+2$ is prime. Indeed, $\perp$ implies everyting. If $p=3$, the implication appears true.