Relate $n$ with $2$

Question

Suppose $a, b$ and $n$ are positive integers, all greater than one. If $a^n+b^n$ is prime, what can you relate $n$ with 2?

My approach: for $a^n+b^n$ to be prime $\forall n>1$, $a$ and $b$ has to be coprimes. But how do I ascertain anything about $n?$

Answer 1

If $n$ is odd, then $a^n+b^n=(a+b)(a^{n-1}-a^{n-2}b\pm\ldots-ab^{n-2}+b^{n-1})$.

Answer 2

$n$ must be of the form $n=2^k$ for some non negative integer $k$ since otherwise, you can write $n$ as $n=kl$ for some odd $k$. This implies $$ a^n+b^n=(a^l)^k+(b^l)^k=(a^l+b^l)((a^l)^{k-1}-(a^l)^{k-2}b^l\pm\ldots-(a^l)(b^l)^{n-2}+(b^l)^{n-1}) $$ which is not prime.