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

Edita Pelantov\'a; Zuzana Mas\'akov\'a

arxiv: 2401.03874 · v2 · pith:QMB4Y3LVnew · submitted 2024-01-08 · 🧮 math.NT · math.CO

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

Zuzana Mas\'akov\'a , Edita Pelantov\'a This is my paper

Pith reviewed 2026-05-24 04:20 UTC · model grok-4.3

classification 🧮 math.NT math.CO MSC 11A6311B39

keywords Midy's theorembeta-expansionsnon-integer basesgolden ratioFibonacci numbersdivisibilityprime denominatorsperiodic expansions

0 comments

The pith

For the golden-ratio base, Midy's property holds for prime q exactly when q divides a Fibonacci number.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper generalizes Midy's theorem, which states that in base 10 the two halves of an even-length period of p/q sum to a string of nines, to expansions in real bases beta greater than one. It first defines the analogous Midy property for arbitrary beta and derives a necessary condition on the base and the period. Specializing to the golden-ratio base beta equals one-half times one plus square root of five, the authors characterize exactly which prime denominators q make the property hold. The characterization is expressed in terms of divisibility by Fibonacci numbers. A reader cares because the result ties the arithmetic of periodic expansions in an irrational base to the classical divisibility properties of the Fibonacci sequence.

Core claim

For beta equal to one-half times one plus square root of five, a fraction p over prime q possesses the Midy property in its beta-expansion if and only if the period length is even and q divides the appropriate Fibonacci number determined by that length.

What carries the argument

The Midy property for beta-expansions: the sum of the two digit blocks of an even-length period equals the finite beta-expansion of (beta to the n minus one) over (beta minus one).

If this is right

The necessary condition on general bases forces the period length to be even whenever the Midy property is to hold.
In the golden-ratio case the property reduces directly to a divisibility statement about Fibonacci numbers.
The same reduction supplies a criterion for which primes can appear as denominators in periodic golden-ratio expansions that obey the half-period sum rule.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The link may let number theorists translate questions about beta-expansions into statements about entry points of primes in the Fibonacci sequence.
Similar characterizations could be sought for other quadratic Pisot bases whose beta-expansions are known to produce Fibonacci-like recurrences.

Load-bearing premise

The beta-expansion of p over q must possess an even period length before the Midy property can be checked or required.

What would settle it

Take any prime q that does not divide any Fibonacci number of even index; compute its beta-expansion in the golden-ratio base and verify whether the two halves of the period sum to the required string of digits.

read the original 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 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Extends Midy's theorem to beta-expansions with a Fibonacci characterization for golden ratio base, but even period lengths need verification.

read the letter

The one or two things to know: this paper defines the Midy property in the setting of Rényi beta-expansions for general bases beta greater than 1, derives a necessary condition for it, and then for the golden ratio base gives a characterization of the prime q that work, in terms of which Fibonacci numbers divide q. What the paper does well is to carry over the idea from integer bases without forcing it. The necessary condition looks like it comes from straightforward manipulation of the expansion sums, and the specialization to beta the golden ratio produces a result that ties directly into known properties of Fibonacci numbers. That's a nice, concrete outcome. The soft spot, and it is worth noting, is the reliance on even period length. The necessary condition is stated for cases where the period is even, and the characterization for the golden ratio case uses that the period is even precisely when q divides a certain Fibonacci number. But the stress-test raises a fair point that in non-integer bases, proving the parity of the period requires separate work. If the paper only observes this for the cases or assumes it without a general argument, then the characterization is more of a conditional result than a full one. That doesn't make it wrong, but it means the main claim needs careful checking in the proofs. The citation pattern seems standard for the area, drawing on beta-expansion theory and Fibonacci arithmetic. This paper is for researchers in numeration systems and ergodic theory on the line. A reader who works on generalizations of decimal properties to other bases will get something out of it. It is worth a serious referee because the topic is well-defined and the result is specific enough to be verifiable. Recommendation: yes, send it for peer review.

Referee Report

2 major / 2 minor

Summary. The manuscript defines the Midy property for the beta-expansion of a fraction p/q in a general base β > 1, derives a necessary condition for the property to hold, and for the specific base β = (1 + √5)/2 provides a characterization of those primes q for which the property is satisfied, expressed in terms of divisibility by Fibonacci numbers.

Significance. If the proofs are complete, the work extends Midy's theorem from integer bases to Rényi beta-expansions and establishes a concrete link between the parity of the period in the golden-ratio base and Fibonacci divisibility. The derivation of a necessary condition that applies for arbitrary β > 1 is a clear positive contribution that could be useful beyond the golden-ratio case.

major comments (2)

[characterization for golden-ratio base] The section characterizing primes q for β = (1 + √5)/2: the claimed characterization requires that the greedy beta-expansion of p/q be purely periodic of even length 2n, yet the manuscript provides no independent argument establishing that the period length is even precisely when q divides the relevant Fibonacci number; this parity assumption is load-bearing for both the necessary condition and the final characterization.
[necessary condition for general β] The derivation of the necessary condition for general β (prior to the golden-ratio specialization): the condition is stated in terms of the sum of the two halves of an even-length period equaling β^n - 1, but the argument does not address whether the period length is guaranteed to be even for prime denominators q; without this, the necessary condition applies only conditionally and its scope for the main theorem is unclear.

minor comments (2)

[abstract] The abstract states that a characterization is given but does not indicate the explicit form (e.g., q divides F_{2k} for certain k); adding one sentence would improve readability.
[introduction / preliminaries] Notation for the beta-expansion (e.g., the greedy map T_β and the digits) is introduced without a dedicated preliminary subsection; a short paragraph recalling the standard definition of purely periodic expansions in base β would aid readers unfamiliar with beta-numeration.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for the constructive comments. We respond to each major comment below, indicating where revisions will be made to address the concerns raised.

read point-by-point responses

Referee: [characterization for golden-ratio base] The section characterizing primes q for β = (1 + √5)/2: the claimed characterization requires that the greedy beta-expansion of p/q be purely periodic of even length 2n, yet the manuscript provides no independent argument establishing that the period length is even precisely when q divides the relevant Fibonacci number; this parity assumption is load-bearing for both the necessary condition and the final characterization.

Authors: We acknowledge that an explicit argument linking the parity of the period to the Fibonacci divisibility condition would strengthen the characterization. The manuscript connects the period length to the rank of appearance in the Fibonacci sequence for this specific base, but we agree that a dedicated lemma establishing the evenness criterion (drawing on known parity and divisibility properties of Fibonacci numbers) is warranted. We will add this supporting argument in the revised version. revision: yes
Referee: [necessary condition for general β] The derivation of the necessary condition for general β (prior to the golden-ratio specialization): the condition is stated in terms of the sum of the two halves of an even-length period equaling β^n - 1, but the argument does not address whether the period length is guaranteed to be even for prime denominators q; without this, the necessary condition applies only conditionally and its scope for the main theorem is unclear.

Authors: The necessary condition is formulated under the hypothesis that the Midy property holds, which by definition requires the period to have even length 2n. It is therefore conditional by construction and is not asserted to apply to all prime denominators. We will revise the text to state this scope more explicitly and to clarify how the general necessary condition is used in the subsequent specialization to the golden-ratio base. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation relies on standard beta-expansion periodicity and derives new characterization independently

full rationale

The paper defines the Midy property for general β>1, derives a necessary condition, and for β=(1+√5)/2 characterizes the prime q satisfying it via a link to Fibonacci divisibility. No quoted step reduces a prediction to a fitted input by construction, invokes a self-citation as the sole justification for a uniqueness claim, or renames a known result as a new derivation. The even-period restriction is part of the standard Midy setup and the known eventual periodicity of rationals in beta-expansions; the central claim remains an independent mathematical characterization rather than a tautology.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard background results about beta-expansions (Rényi) and divisibility properties of the Fibonacci sequence; no free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption Beta-expansions in base β > 1 are well-defined and eventually periodic for rational p/q.
Invoked when defining the Midy property for general β and when restricting attention to even-length periods.
standard math Standard arithmetic properties of the golden ratio and Fibonacci sequence hold.
Used to link the Midy property for β = (1+√5)/2 to divisibility of Fibonacci numbers.

pith-pipeline@v0.9.0 · 5695 in / 1343 out tokens · 19907 ms · 2026-05-24T04:20:02.657173+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Constants.lean (phi_golden_ratio, phi_fixed_point) phi_golden_ratio echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

For β = ½(1+√5) we characterize prime denominators q which satisfy the property... C^N ≡ -I mod q... a(q) odd implies Midy property
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean embed_eq_pow, LogicNat orbit echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

Powers of the companion matrix C... expressed using Fibonacci sequence... C^N = (F_{N-1} F_N ; F_N F_{N+1})

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages

[1]

Adamczewski, C

B. Adamczewski, C. Frougny, A. Siegel, and W. Steiner. Ra tional numbers with purely periodic β-expansion. Bull. Lond. Math. Soc. , 42:538-552, 2010

work page 2010
[2]

S. Akiyama. Pisot numbers and greedy algorithm. In Number theory (Eger, 1996) , pages 9-21. de Gruyter, Berlin, 1998

work page 1996
[3]

L. E. Dickson. History of the theory of numbers. Vol. I: Divisibility and pri mality. Chelsea Publishing Co., New York, 1966

work page 1966
[4]

B. D. Ginsburg. Midy’s (nearly) secret theorem - an exten sion after 165 years. College Math. J., 35(1):26-30, 2004. 16 Midy’s Theorem in non-integer bases and divisibility of Fibonacci numbe rs

work page 2004
[5]

H. Goodwyn. Curious properties of prime numbers taken as the divisors of unity. J. Natur. Philos. Chem. Arts , 1:314-316, 1802

work page
[6]

Gupta and B

A. Gupta and B. Sury. Decimal expansion of 1 /p and subgroup sums. Integers, 5(1):A19, 5, 2005

work page 2005
[7]

J. H. Halton. On the divisibility properties of Fibonacc i numbers. Fibonacci Quart., 4:217- 240, 1966

work page 1966
[8]

Hama and T

M. Hama and T. Imahashi. Periodic β-expansions for certain classes of Pisot numbers. Comment. Math. Univ. St. Pauli , 46:103-116, 1997

work page 1997
[9]

W. G. Leavitt. A theorem on repeating decimals. Amer. Math. Monthly , 74:669-673, 1967

work page 1967
[10]

Lewittes

J. Lewittes. Midy’s theorem for periodic decimals. Integers, 7:A2, 11, 2007

work page 2007
[11]

H. W. Martin. Generalizations of Midy’s theorem on repe ating decimals. Integers, 7:A3, 7, 2007

work page 2007
[12]

E. Midy. De Quelques Propri´ et´ es des Nombres et des Fractions D´ ecimales P´ eriodiques. Col- lege of Nantes, France, 1836

work page
[13]

W. Parry. On the β-expansions of real numbers. Acta Math. Hung. , 11(3-4):401-416, 1960

work page 1960
[14]

M. Renault. The period, rank, and order of the ( a, b)-Fibonacci sequence mod m. Math. Mag., 86(5):372-380, 2013

work page 2013
[15]

K. A. Ross. Repeating decimals: a period piece. Math. Mag. , 83(1):33-45, 2010

work page 2010
[16]

A. R´ enyi. Representations for real numbers and their e rgodic properties. Acta Math. Hung. , 8:477-493, 1957

work page 1957
[17]

K. Schmidt. On periodic expansions of Pisot numbers and Salem numbers. Bull. Lond. Math. Soc., 12(4):269-278, 1980

work page 1980
[18]

Shrader-Frechette

M. Shrader-Frechette. Complementary rational number s. Math. Mag. , 51(2):90-98, 1978. Received: January 9, 2024 Accepted for publication: April 25, 2024 Communicated by: Emilie Charlier, Julien Leroy and Michel Rigo 17

work page 1978

[1] [1]

Adamczewski, C

B. Adamczewski, C. Frougny, A. Siegel, and W. Steiner. Ra tional numbers with purely periodic β-expansion. Bull. Lond. Math. Soc. , 42:538-552, 2010

work page 2010

[2] [2]

S. Akiyama. Pisot numbers and greedy algorithm. In Number theory (Eger, 1996) , pages 9-21. de Gruyter, Berlin, 1998

work page 1996

[3] [3]

L. E. Dickson. History of the theory of numbers. Vol. I: Divisibility and pri mality. Chelsea Publishing Co., New York, 1966

work page 1966

[4] [4]

B. D. Ginsburg. Midy’s (nearly) secret theorem - an exten sion after 165 years. College Math. J., 35(1):26-30, 2004. 16 Midy’s Theorem in non-integer bases and divisibility of Fibonacci numbe rs

work page 2004

[5] [5]

H. Goodwyn. Curious properties of prime numbers taken as the divisors of unity. J. Natur. Philos. Chem. Arts , 1:314-316, 1802

work page

[6] [6]

Gupta and B

A. Gupta and B. Sury. Decimal expansion of 1 /p and subgroup sums. Integers, 5(1):A19, 5, 2005

work page 2005

[7] [7]

J. H. Halton. On the divisibility properties of Fibonacc i numbers. Fibonacci Quart., 4:217- 240, 1966

work page 1966

[8] [8]

Hama and T

M. Hama and T. Imahashi. Periodic β-expansions for certain classes of Pisot numbers. Comment. Math. Univ. St. Pauli , 46:103-116, 1997

work page 1997

[9] [9]

W. G. Leavitt. A theorem on repeating decimals. Amer. Math. Monthly , 74:669-673, 1967

work page 1967

[10] [10]

Lewittes

J. Lewittes. Midy’s theorem for periodic decimals. Integers, 7:A2, 11, 2007

work page 2007

[11] [11]

H. W. Martin. Generalizations of Midy’s theorem on repe ating decimals. Integers, 7:A3, 7, 2007

work page 2007

[12] [12]

E. Midy. De Quelques Propri´ et´ es des Nombres et des Fractions D´ ecimales P´ eriodiques. Col- lege of Nantes, France, 1836

work page

[13] [13]

W. Parry. On the β-expansions of real numbers. Acta Math. Hung. , 11(3-4):401-416, 1960

work page 1960

[14] [14]

M. Renault. The period, rank, and order of the ( a, b)-Fibonacci sequence mod m. Math. Mag., 86(5):372-380, 2013

work page 2013

[15] [15]

K. A. Ross. Repeating decimals: a period piece. Math. Mag. , 83(1):33-45, 2010

work page 2010

[16] [16]

A. R´ enyi. Representations for real numbers and their e rgodic properties. Acta Math. Hung. , 8:477-493, 1957

work page 1957

[17] [17]

K. Schmidt. On periodic expansions of Pisot numbers and Salem numbers. Bull. Lond. Math. Soc., 12(4):269-278, 1980

work page 1980

[18] [18]

Shrader-Frechette

M. Shrader-Frechette. Complementary rational number s. Math. Mag. , 51(2):90-98, 1978. Received: January 9, 2024 Accepted for publication: April 25, 2024 Communicated by: Emilie Charlier, Julien Leroy and Michel Rigo 17

work page 1978