Structure of associated sets to Midy’s property

Abstract

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 UN of positive integers less than N and relatively primes to N, and x ∈ UN . It is well known that when we write the fraction x N in base b, it is periodic. Let d, k be positive integers with d ≥ 2 and such that |b|N = dk and x N = 0.a1a2 · · · a|b|N with the bar indicating the period and ai are digits in base b. We separate the period a1a2 · · · a|b|N in d blocks of length k and let Aj = [a(j−1)k+1a(j−1)k+2 · · · ajk]b be the number represented in base b by the j − th block and Sd(x) = Pdj=1 Aj . If for all x ∈ UN , the sum Sd(x) is a multiple of bk − 1 we say that N has 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 Midy’s property to for b and d.