On $a^n+bn$ modulo $m$

Abstract

Let $a$ and $m>0$ be integers. We show that for any integer $b$ relatively prime to $m$, the set $\{a^n+bn:n=1,\dots ,m^2\}$ contains a complete system of residues modulo $m$. We also pose several conjectures for further research; for example, we conjecture that any integer $n>1$ can be written as $k+m$ with $2^k+m$ prime, where $k$ and $m$ are positive integers.

6 pages. Minor revision