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

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.

💡 Research Summary

The paper investigates a natural extension of Midy’s theorem—originally formulated for decimal expansions of fractions with prime denominators—to numeration systems with non‑integer real bases β > 1, focusing in particular on the golden‑ratio base τ = (1+√5)/2. After recalling Rényi’s β‑expansions and the notion of the quasigreedy expansion d*β(1), the authors introduce a precise definition of the “Midy property” for a given base β: a denominator q possesses the property if there exists a coprime numerator p such that the β‑expansion of p/q is purely periodic with minimal even period 2n, and the two halves of the period, interpreted as β‑integers x and y, satisfy x + y = βⁿ − 1.

Lemma 3.1 shows that this condition is equivalent to the dynamical symmetry Tⁿ(p/q) = q − p/q and Tⁿ(q − p/q) = p/q, where T is the β‑transformation x↦βx−⌊βx⌋. The authors then translate the problem into algebraic terms: letting C be the companion matrix of the minimal polynomial of β, Theorem 3.2 proves that if q has the Midy property then there exists an integer N with Cᴺ ≡ −I (mod q). This necessary condition forces β to be an algebraic integer, and in fact a Pisot unit, because otherwise the determinant argument fails (Remark 3.3).

The core of the paper is the analysis of the golden‑ratio base τ. Its minimal polynomial X²−X−1 yields the companion matrix C =