An arithmetical excursion via Stoneham numbers

TL;DR

This paper derives an explicit formula for digit frequencies in the base $b$ expansion of reciprocals of prime powers and confirms conjectures about Stoneham numbers' expansions.

Contribution

It provides a new explicit formula for digit distribution in base $b$ expansions of $1/p^m$, and proves two recent conjectures related to Stoneham numbers.

Findings

Explicit formula for digit counts in base $b$ expansion of $1/p^m$

Proof of two conjectures on Stoneham numbers' expansions

Enhanced understanding of digit distribution in special algebraic numbers

Abstract

Let $p$ be a prime and $b$ a primitive root of $p^2$. In this paper, we give an explicit formula for the number of times a value in ${0,1,...,b-1}$ occurs in the periodic part of the base $b$ expansion of $1/p^m$. As a consequence of this result, we prove two recent conjectures of Francisco Arag\'on, Daivd Bailey, Jonathan Borwein, and Peter Borwein concerning the base $b$ expansion of Stoneham numbers.