Zuzana Masáková ; Edita Pelantová

Midy's Theorem in noninteger 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 noninteger bases and divisibility of Fibonacci
numbersArticle
Authors: Zuzana Masáková ; Edita Pelantová
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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 $10^n1$. A
number of results generalise Midy's theorem to expansions of $\frac{p}{q}$ in
different integer bases, considering nonprime 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 noninteger 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.