Structure of associated sets 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 of positive integers less than $N$ and relatively primes to 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 and such that $|b{|}_N=kd$ and $\frac{x}{N}=0.$ 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 . 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 $d$.

9 pages, final version, corrected typos, accepted for publication in the journal Matemáticas: Enseñanza Universitaria http://revistaerm.univalle.edu.co/