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

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