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

TL;DR

The paper explores symmetries in the base n expansion of 1/m for coprime integers m and n, characterizes related subgroups, and provides conditions for primitive roots modulo primes.

Contribution

It introduces new symmetries of base n expansions of 1/m and characterizes the subgroup generated by n in the multiplicative group modulo m.

Findings

Characterization of the subgroup generated by n in (Z/mZ)^×

New symmetries of the base n expansion of 1/m

Sufficient conditions for a prime p where n is a primitive root

Abstract

Suppose $ m,n\geq 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 $.