For what non trivial values of $a$ and $k$, $a^k + 1$ will be prime?

Question

For what non trivial values of $a$ and $k$, $a^k + 1$ will be prime?

If $a=1$, then $1^k+1=1+1=2$, where 2 is prime. If $k=1$ then we have$a^k+1=a+1$, but if some prime $p$ is equal to $a+1$, then only we can say $a+1$ is prime.

But for the rest part I am clueless to proceed. Any suggestion is highly appreciated.

Is it similar to the problem $2^{2^n}+1$ is prime for even $n$?

Answer 1

consider the case $k=3$ then we get $$a^3+1=(a+1)(1-a+a^2)$$