I have the following conjecture :
Let $a>1$ be an integer. Then, there is a positive integer $n$ such that $a^n+F_n$ is prime , where $F_n$ denotes the $n$-th fibonacci number.
For $a\le 10^4$ , there is a positive integer $n\le 594$ , that $a^n+F_n$ is not divisible by one of the first $30\ 000$ primes, so it seems that we cannot show for some $a$ (for example by a cover) that $a^n+F_n$ is composite for every positive integer $n$.
On the other hand , there are tough cases (no prime upto $n=5\ 000$) , the first few are $a=13,19,43,48,51,57$. For the case $a=13$ , no $n\le 35\ 000$ gives a prime.
Considering the growth rate, I think the conjecture should be true. Any ideas how to prove or disprove it ?