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

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 \( p \).