Congruences for sums of powers of an integer

TL;DR

This paper investigates the minimal number of powers of an integer needed to form a sum divisible by another integer, providing bounds and analyzing cases where this number is large, especially for prime powers.

Contribution

It establishes an upper bound for the minimal sum length and explores the behavior when this length is large, with a focus on prime power moduli.

Findings

Derived an upper bound for m(q,e)

Analyzed cases where m(q,e) is large

Focused on prime power moduli

Abstract

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 $m(q.e)$ and investigate the case where $m(q,e)$ is "large". We also pay special attention to the situation where $e$ is a prime power.