Showing that if $p$ is prime, then $(p^4 + 4)$ can't be prime

Question

I want to show that if $p$ is prime, then $(p^4 + 4)$ can't be prime.

I guess Fermat's little theorem may help, but I can't figure out how to use it for the proof.

Can anyone point me in the right direction?

Answer 1

$x^4+4=(x^2-2 x+2) (x^2+2 x+2)$

$x^2-2 x+2 \ge 2$ for $x\ge 2$

$x^2+2 x+2 \ge 2$ for $x\ge 0$

So $n^4+4$ is never prime for $n\ge 2$.