Midy's Theorem in non-integer bases and divisibility of Fibonacci numbers

• Published in: Communications in Mathematics 2024-05, Vol.33 (2025), Issue 2...
• Main Authors: Masáková, Zuzana, Pelantová, Edita
• Format: Article
• Language: English
• ISSN: 2336-1298; 2336-1298
• DOI: 10.46298/cm.12840

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 R\'enyi. 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.