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arXiv · 2401.03874

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

Zuzana Masáková·Edita Pelantová
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Abstract

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 $β>1$ and derive a necessary condition for validity of the Midy property. For $β=\frac12(1+\sqrt5)$ we characterize prime denominators $q$, which satisfy the property.

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Zuzana Masáková, Edita Pelantová. 2024-05-30. Midy's Theorem in non-integer bases and divisibility of Fibonacci numbers. https://doi.org/10.46298/cm.12840

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