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

1. Problem Statement

The paper investigates the generalization of Midy's Theorem from classical integer bases (specifically base 10) to non-integer bases ($\beta > 1$).

• Classical Midy's Theorem: For a fraction $p/q$ with a prime denominator $q$ in base 10, if the minimal period of the decimal expansion has an even length $2n$, the sum of the first half and the second half of the period equals $10^n - 1$ (a string of $n$ nines).
• The Challenge: The authors aim to define and characterize this "Midy property" in the context of $\beta$-expansions (introduced by Rényi), where $\beta$ is a real number greater than 1. Specifically, they seek to identify which denominators $q$ satisfy the property in the golden ratio base $\tau = \frac{1+\sqrt{5}}{2}$, and how this relates to the divisibility properties of Fibonacci numbers.

2. Methodology

The authors employ a combination of dynamical systems theory, linear algebra, and number theory.

• $\beta$-Expansions and the Greedy Algorithm: They utilize the greedy algorithm to generate $\beta$-expansions. A key concept is the quasigreedy expansion of 1, denoted $d^*_\beta(1)$, which determines the lexicographic constraints on admissible digit strings.
• The Midy Property Definition: They formally define the Midy property for a base $\beta$: A denominator $q$ has the property if there exists a fraction $p/q$ with a purely periodic expansion of even length $2n$ such that the sum of the two halves (interpreted as $\beta$-integers) equals $\beta^n - 1$.
• Dynamical Systems Approach: They analyze the $\beta$-transformation $T_\beta(x) = \beta x - \lfloor \beta x \rfloor$. They establish that the Midy property is equivalent to the condition $T^N(p/q) = (q-p)/q$ for some integer $N$.
• Linear Algebra (Companion Matrices): They link the transformation $T$ to the powers of the companion matrix $C$ of the minimal polynomial of $\beta$. They prove that a necessary condition for the Midy property is $C^N \equiv -I \pmod q$.
• Fibonacci Sequences: For the specific case of the golden ratio $\tau$, the companion matrix $C$ is directly related to the Fibonacci sequence ($F_n$). The authors leverage known divisibility properties of Fibonacci numbers to characterize the primes satisfying the condition.
• Finite Fields: For the characterization of prime denominators, they work within the finite field $\mathbb{Z}_q$, utilizing the Legendre symbol, quadratic reciprocity, and the order of eigenvalues of the matrix $C$ modulo $q$.

3. Key Contributions and Results

A. General Necessary Condition (Theorem 3.2)

The authors derive a necessary condition for any algebraic integer base $\beta > 1$. If an integer $q$ has the Midy property in base $\beta$, then there must exist an integer $N$ such that:
$$C^N \equiv -I \pmod q$$
where $C$ is the companion matrix of the minimal polynomial of $\beta$.

• Implication: This immediately rules out the Midy property for bases with odd degree and norm 1 (e.g., the Tribonacci base), as $\det(C^N) = 1$ while $\det(-I) = -1$ (for odd dimensions), creating a contradiction.

B. Sufficient Condition for the Golden Ratio (Theorem 4.1)

For the specific base $\tau = \frac{1+\sqrt{5}}{2}$, the necessary condition is also sufficient.

• Result: If $C^N \equiv -I \pmod q$ for the companion matrix of $\tau$, then $q$ has the Midy property in base $\tau$.
• Universality: In this base, every fraction $p/q$ (with $0 < p < q$) testifies to the Midy property if the condition holds.
• Periodicity: The minimal period of the expansion is always divisible by 4.

C. Characterization via Fibonacci Divisibility (Corollary 4.5 & 4.6)

The condition $C^N \equiv -I \pmod q$ is translated into conditions on Fibonacci numbers:

• If $q$ divides $F_{2n-1}$ (for $n \ge 3$), $q$ has the Midy property.
• If $q$ is a multiple of $F_{2n}$ (for $n \ge 3$), $q$ does not have the Midy property.
• Refined Criterion: $q$ has the property if the smallest index $a(q)$ such that $q \mid F_{a(q)}$ is odd.

D. Characterization of Prime Denominators (Theorems 5.3 & 5.4)

The paper provides a complete characterization of prime numbers $q$ satisfying the Midy property in base $\tau$:

1. Case $q \equiv \pm 2 \pmod 5$:

   • These primes always satisfy the Midy property.
   • This includes Mersenne primes of the form $2^s - 1$ where $s \equiv 3 \pmod 4$.
   • This includes all known Fermat primes ($2^{2^n} + 1$).

2. Case $q \equiv \pm 1 \pmod 5$:

   • The property depends on the structure of the multiplicative group $\mathbb{Z}_q^*$.
   • Let $q-1 = 2^k \ell$ (with $\ell$ odd). The prime $q$ has the Midy property if and only if the sequence of matrices $C^{2\ell}, C^{4\ell}, \dots, C^{2^{k-1}\ell}$ contains a matrix congruent to $-I \pmod q$.
   • Counter-examples: Primes where $q \equiv -1 \pmod{20}$ or $q \equiv 11 \pmod{20}$ (e.g., $q=19, 29$) do not satisfy the property.
   • Examples: $q=41$ has the property, but $q=101$ does not, despite both being $\equiv 1 \pmod{20}$.

4. Significance

• Extension of Classical Theory: The paper successfully extends a classic number-theoretic curiosity (Midy's Theorem) into the realm of non-integer bases, specifically the golden ratio, which is fundamental in the theory of $\beta$-expansions and quasicrystals.
• Connection to Fibonacci Numbers: It establishes a deep and explicit link between the periodicity of $\tau$-expansions and the divisibility properties of Fibonacci numbers, providing a new tool for analyzing these sequences.
• Algebraic Constraints: The work highlights the critical role of the norm of the base and the degree of its minimal polynomial. It demonstrates that the "Midy phenomenon" is not universal but is strictly constrained by the algebraic nature of the base (e.g., failing for odd-degree Pisot numbers with norm 1).
• Computational Insights: The results allow for the algorithmic determination of whether a specific prime denominator will exhibit the Midy property in base $\tau$ without computing the full expansion, simply by checking modular arithmetic conditions on Fibonacci numbers or matrix powers.

5. Conclusion

Mas´akov´a and Pelantov´a prove that the Midy property in the golden ratio base is governed by the condition $C^N \equiv -I \pmod q$. They fully characterize the primes satisfying this: all primes $q \equiv \pm 2 \pmod 5$ work, while primes $q \equiv \pm 1 \pmod 5$ work only under specific conditions related to the order of 5 in $\mathbb{Z}_q$. This work bridges the gap between dynamical systems, algebraic number theory, and the combinatorics of digit expansions.