Congruences for sums of powers of an integer

For coprime positive integers $q$ and $e$, let $m(q,e)$ denote the least positive integer $t$ such that there exists a sum of $t$ powers of $q$ which is divisible by $e$. We prove an upper bound for $m(q\mathrm{.}e)$ and investigate the case where $m(q,e)$ is “large”. We also pay special attention to the situation where $e$ is a prime power.