I would like to ask the following (perhaps silly question) regarding divisibility.
Fix three positive integers $j, k, l$. My question is: is it true that for every $n \in \mathbb{N}$ there exists $m = m(n) \in \mathbb{N}$ such that $n | (j + k^m + l)$?
Not always. If $k=1$ it will often not be true and whether it is true will not depend on $m$. If $n$ is even, $k$ is even, and $j+l$ is odd it will never be true. If $k$ is a primitive root $\bmod n$ it will always be possible unless $j+l$ is a multiple of $n$.