prove that if $a^n+1$ is prime then $a$ is even and $n=2^k$

Question

i managed to show that $a$ is a even (suppose it is, and then show that $2|a^n+1$) for the second part, I understand it has something to do with fermat numbers but couldn't solve. please help

Answer 1

Hint. Suppose $n = 2^k m$, where $m\neq 1$ is an odd number. Then $$a^n + 1 = \left(a^{2^k}\right)^m + 1^m\text{.}$$

Answer 2

For fun, we reword the very good hint of Antoine using the language of congruences. Suppose that $a\gt 1$, and $n=2^km$, where $m\gt 1$ is odd.

Then $a^{(2^k)}\equiv -1 \pmod{a^{(2^k)}+1}$, and therefore $$a^n=(a^{(2^k)})^m \equiv -1\pmod{a^{(2^k)}+1}.$$