Zuzana Masáková ; Edita Pelantová - Midy's Theorem in non-integer bases and divisibility of Fibonacci numbers

cm:12840 - Communications in Mathematics, May 31, 2024, Volume 33 (2025), Issue 2 (Special issue: Numeration, Liège 2023, dedicated to the 75th birthday of professor Christiane Frougny) - https://doi.org/10.46298/cm.12840
Midy's Theorem in non-integer bases and divisibility of Fibonacci numbersArticle

Authors: Zuzana Masáková ; Edita Pelantová

    Fractions $\frac{p}{q} \in [0,1)$ with prime denominator $q$ written in decimal have a curious property described by Midy's Theorem, namely that two halves of their period (if it is of even length $2n$) sum up to $10^n-1$. A number of results generalise Midy's theorem to expansions of $\frac{p}{q}$ in different integer bases, considering non-prime denominators, or dividing the period into more than two parts. We show that a similar phenomena can be studied even in the context of numeration systems with non-integer bases, as introduced by Rényi. First we define the Midy property for a general real base $\beta >1$ and derive a necessary condition for validity of the Midy property. For $\beta =\frac12(1+\sqrt5)$ we characterize prime denominators $q$, which satisfy the property.


    Volume: Volume 33 (2025), Issue 2 (Special issue: Numeration, Liège 2023, dedicated to the 75th birthday of professor Christiane Frougny)
    Published on: May 31, 2024
    Accepted on: April 25, 2024
    Submitted on: January 9, 2024
    Keywords: Mathematics - Number Theory,Mathematics - Combinatorics,11A63, 11B39, 11A07, 11K16

    Consultation statistics

    This page has been seen 25 times.
    This article's PDF has been downloaded 11 times.