Are there any $a,x,n$ such that $ax|n$ and $ax+1$ is prime but $a^{n}-1$ is not a multiple of $ax+1$, apart from $a=x=n=1$?
I had an answer to a related question earlier: Can $x^{n}-1$ be prime if $x$ is not a power of $2$ and $n$ is odd?
$n^{a}-1=(n-1)(n^{a-1}+n^{a-2}+\cdots+1)$