On some symmetries of the base $n$ expansion of $1/m$ : Comments on Artin's Primitive root conjecture

Abstract

Suppose $m,n\ge 2$ are co prime integers. We prove certain new symmetries of the base $n$ representation of $1/m$, and in particular characterize the subgroup generated by $n$ inside $(\mathbb{Z}/m\mathbb{Z}{)}^{\times }$. As an application we give a sufficient condition for a prime $p$ such that a non square number $n$ is a primitive root modulo $p$.

Eight pages