Structure of associated sets to Midy's Property

TL;DR

This paper investigates the structure of sets of integers related to Midy's property, which concerns the divisibility of sums derived from the periodic expansion of fractions in a given base.

Contribution

It characterizes the set of integers for which a number exhibits Midy's property, revealing new structural insights into these associated sets.

Findings

Identifies conditions under which Midy's property holds for specific divisors

Provides a characterization of the set of all such divisors for a given N and b

Establishes properties of the associated sets related to Midy's property

Abstract

Let $b$ be a positive integer greater than 1, $N$ a positive integer relatively prime to $b$, $ |b|_{N}$ the order of $b$ in the multiplicative group $% \mathbb{U}_{N}$ of positive integers less than $N$ and relatively primes to $% N,$ and $x\in\mathbb{U}_{N}$. It is well known that when we write the fraction $\frac{x}{N}$ in base $b$, it is periodic. Let $d,\,k$ be positive integers with $% d\geq2$ and such that $|b|_{N}=kd$ and $\frac{x}{N}=0.% bar{a_{1}a_{2}...a_{|b|_{N}}}$ with the bar indicating the period and $a_{i}$ are digits in base $b$. We separate the period ${a_{1}a_{2}... a_{|b|_{N}}}$ in $d$ blocks of length $k$ and let $ A_{j}=[a_{(j-1)k+1}a_{(j-1)k+2}...a_{jk}]_{b} $ be the number represented in base $b$ by the $j-th$ block and $% S_{d}(x)=\sum\limits_{j=1}^{d}A_{j}$. If for all $x\in\mathbb{U}_{N}$, the sum $S_{d}(x)$ is a multiple of $b^{k}-1$ we say that $N$ has the Midy's property for $b$ and $d$. In this work we present some interesting properties of the set of positive integers $d$ such that $N$ has the Midy's property for $b$ and $d$.