Gaps in binary cyclotomic polynomials

Antonio Cafure; Eda Cesaratto

arxiv: 2507.19381 · v2 · submitted 2025-07-25 · 🧮 math.NT

Gaps in binary cyclotomic polynomials

Antonio Cafure , Eda Cesaratto This is my paper

Pith reviewed 2026-05-19 02:56 UTC · model grok-4.3

classification 🧮 math.NT

keywords cyclotomic polynomialsbinary cyclotomic polynomialscoefficient gapscircular mapodd primesnumber theorypolynomial coefficients

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The pith

The second gap in the coefficients of Φ_pq for odd primes p < q equals max(r-1, p-r-1) where r is the remainder of q divided by p.

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

This paper studies the gaps in the coefficient sequence of the cyclotomic polynomial Φ_pq when p and q are distinct odd primes. The central result gives an exact formula for the length of the second gap in terms of the remainder r when q is divided by p. The proof introduces a description of the coefficients as concatenations of words produced by iterating a circular map. A reader would care because the formula replaces direct expansion of the polynomial with a simple arithmetic check on p and q. For the special cases q congruent to plus or minus one modulo p, the paper also counts how many gaps occur at each possible length.

Core claim

We prove that the second gap of Φ_pq is the maximum of r-1 and p-r-1, where r is the remainder of q divided by p. For q congruent to ±1 modulo p, we determine the number of gaps for each possible length. To obtain these results, we develop a new approach in which the coefficients of Φ_pq are described as concatenations of words arising from iterations of a circular map.

What carries the argument

A circular map whose iterations generate words that concatenate to form the coefficient sequence of Φ_pq.

Load-bearing premise

The coefficients of Φ_pq admit a description as concatenations of words generated by iterations of a circular map.

What would settle it

Expand Φ_21 for p=3 and q=7 (where r=1) and check whether the second gap in its coefficient list has length exactly 1.

read the original abstract

For odd prime numbers $p < q$, let $\Phi_{pq} \in \mathbb{Z}[X]$ be the binary cyclotomic polynomial of order $pq$. In this paper, we prove that the second gap of $\Phi_{pq}$ is the maximum of $r-1$ and $p-r-1$, where $r$ is the remainder of $q$ divided by $p$. For $q$ congruent to $\pm 1$ modulo $p$, we determine the number of gaps for each possible length. To obtain these results, we develop a new approach in which the coefficients of $\Phi_{pq}$ are described as concatenations of words arising from iterations of a circular map.

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

This paper supplies an explicit formula for the second gap in binary cyclotomic polynomials and introduces a circular map to derive it.

read the letter

The main point to take away is that the authors give a simple closed-form expression for the second gap in the coefficient sequence of the binary cyclotomic polynomial Φ_pq. For odd primes p < q, this second gap equals the maximum of r-1 and p-r-1, where r stands for the remainder of q divided by p. They also work out the exact number of gaps of each possible length in the two cases where q is congruent to 1 or -1 modulo p. The novelty lies in their circular-map model. They represent the coefficients as concatenations of words produced by repeated applications of this map. Once the coefficients are written this way, the lengths of the zero runs become immediate to compute, which directly yields the gap sizes. This description does a good job of turning an algebraic product into a more manageable combinatorial object for the purpose of locating gaps. The resulting formulas are clean and depend only on modular arithmetic between p and q, which aligns with how these polynomials behave. A point worth checking is the fidelity of the circular map to the actual coefficients. The standard definition comes from the Möbius formula over the divisors of pq, and the map has to reproduce every coefficient exactly, including the positions and lengths of all zero stretches. The paper presents the map as the key new tool, so the proof of the gap formula stands or falls with that equivalence. I assume the full text includes the necessary verification steps for the relevant cases. The paper will interest researchers who study the detailed coefficient structure of cyclotomic polynomials, particularly those focused on runs of zeros and gap phenomena. It is a targeted contribution rather than a broad reorganization of the field. I would bring this to a reading group on explicit computations in number theory. I do not expect to cite it in my own work in the next year, but the explicit nature of the results makes it easy to reference if similar questions arise. It should go through peer review. The central claim is precise, the method is original, and the supporting arguments appear checkable.

Referee Report

2 major / 2 minor

Summary. The paper proves that for odd primes p < q, the second gap of the binary cyclotomic polynomial Φ_pq is max(r-1, p-r-1) where r = q mod p. For q ≡ ±1 mod p it also determines the number of gaps of each possible length. The proof introduces a description of the coefficients of Φ_pq as concatenations of words generated by iterations of a circular map on words.

Significance. If the circular-map description is shown to be equivalent to the standard definition of Φ_pq, the result supplies an explicit, simple formula for the second gap length in terms of the arithmetic remainder r. This would add to the literature on coefficient patterns and zero runs in cyclotomic polynomials. The new circular-map construction is a potentially reusable tool for analyzing such polynomials.

major comments (2)

[Section defining the circular map] The section introducing the circular map and the concatenation rule: the manuscript must supply a proof or explicit verification that iterations of the map, followed by the stated concatenations, reproduce exactly the coefficient sequence of Φ_pq (including all zero runs) that arises from the product formula Φ_pq(x) = ∏_{d|pq} (x^{pq/d}-1)^{μ(d)}. This equivalence is the single load-bearing step for extracting the gap lengths; without it the formula for the second gap cannot be justified.
[Proof of the second-gap formula] The derivation of the second-gap formula (presumably in the main theorem section): the argument that the second gap equals max(r-1, p-r-1) must be checked to hold uniformly for every residue class r (1 ≤ r ≤ p-1) and to account for the precise placement of non-zero coefficients produced by the map. Any omission of a non-zero coefficient in a particular residue class would alter the reported gap length.

minor comments (2)

Add a small explicit example (e.g., p=5, q=7 or q=11) showing the coefficient sequence produced by the circular map side-by-side with the known coefficients of Φ_pq to illustrate the claimed match.
Clarify the precise definition of a 'gap' (length of consecutive zero coefficients) at the beginning of the paper, including whether leading or trailing zeros are counted.

Simulated Author's Rebuttal

2 responses · 0 unresolved

Thank you for the referee's thorough review and valuable suggestions. We address the major comments point by point below and will incorporate revisions to strengthen the manuscript, particularly by providing a rigorous proof of the equivalence for the circular map construction.

read point-by-point responses

Referee: [Section defining the circular map] The section introducing the circular map and the concatenation rule: the manuscript must supply a proof or explicit verification that iterations of the map, followed by the stated concatenations, reproduce exactly the coefficient sequence of Φ_pq (including all zero runs) that arises from the product formula Φ_pq(x) = ∏_{d|pq} (x^{pq/d}-1)^{μ(d)}. This equivalence is the single load-bearing step for extracting the gap lengths; without it the formula for the second gap cannot be justified.

Authors: We agree that a clear proof of equivalence is necessary for the validity of our results. In the revised version, we will include a new subsection that proves by induction that the coefficient sequence generated by iterating the circular map and applying the concatenation rule matches exactly the one obtained from the product formula using Möbius function. The induction base case verifies the initial word, and the inductive step shows that each iteration preserves the zero runs and non-zero coefficients positions corresponding to the cyclotomic polynomial. This will justify the gap extraction. revision: yes
Referee: [Proof of the second-gap formula] The derivation of the second-gap formula (presumably in the main theorem section): the argument that the second gap equals max(r-1, p-r-1) must be checked to hold uniformly for every residue class r (1 ≤ r ≤ p-1) and to account for the precise placement of non-zero coefficients produced by the map. Any omission of a non-zero coefficient in a particular residue class would alter the reported gap length.

Authors: The derivation in the main theorem accounts for all residue classes by analyzing the word concatenations for each possible r. We will revise to make explicit that for each r, the positions of non-zero coefficients are determined uniformly by the map, leading to the second gap being max(r-1, p-r-1). A supplementary table or case analysis for representative r values will be added to confirm no omissions occur. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper develops a new description of Φ_pq coefficients via concatenations of words generated by iterations of a circular map, then extracts the second-gap formula max(r-1, p-r-1) directly from that representation, where r is the arithmetic remainder q mod p. This does not reduce to self-definition or a fitted input called prediction because the map is introduced as an auxiliary combinatorial tool whose equivalence to the standard product definition Φ_pq(x) = ∏_{d|pq} (x^{pq/d}-1)^{μ(d)} is part of the proof rather than presupposed by the target formula. No self-citations, uniqueness theorems, or ansatzes imported from prior author work appear in the given text, and the final statement is an explicit arithmetic expression independent of any gap quantities. The derivation therefore remains non-circular and self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the new combinatorial model (circular map) introduced in the paper together with standard facts about cyclotomic polynomials; no free parameters are visible from the abstract.

axioms (1)

standard math Cyclotomic polynomials Φ_n lie in ℤ[X] and their coefficients are determined by the roots of unity of order n.
Invoked implicitly when the paper refers to Φ_pq and its coefficient sequence.

invented entities (1)

circular map on words no independent evidence
purpose: To generate the coefficient sequence of Φ_pq as iterated concatenations of short words.
Introduced as the novel technical tool; no independent evidence outside the paper is supplied in the abstract.

pith-pipeline@v0.9.0 · 5640 in / 1313 out tokens · 53149 ms · 2026-05-19T02:56:18.912301+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/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We prove that the second gap of Φ_pq is the maximum of r-1 and p-r-1, where r is the remainder of q divided by p.

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

13 extracted references · 13 canonical work pages

[1]

Al-Kateeb, M

A. Al-Kateeb, M. Ambrosino, H. Hong, and E. Lee. Maximum gap in cyclotomic polynomials.J. Number Theory, 220:1–15, 2021

work page 2021
[2]

A. Cafure. A story about cyclotomic polynomials: the minimal polynomial of2 cos(2𝜋/𝑛) and its constant term.Math. Mag., 94(5):325–338, 2021

work page 2021
[3]

Cafure and E

A. Cafure and E. Cesaratto. Binary cyclotomic polynomials: representation via words and algorithms. LNCS, 124(1):65–77, 2021

work page 2021
[4]

Camburu, E

O. Camburu, E. Ciolan, F. Luca, P. Moree, and I. Shparlinski. Cyclotomic coefficients: gaps and jumps. J. Number Theory, 163:211–237, 2016

work page 2016
[5]

Habermehl, S

H. Habermehl, S. Richardson, and M. A. Szwajkos. A note on coefficients of cyclotomic polynomials. Math. Mag., 37:183–185, 1964

work page 1964
[6]

H. Hong, E. Lee, H.-S Lee, and C.-M. Park. Maximum gap in (inverse) cyclotomic polynomial.J. Number Theory, 132(10):2297–2315, 2012

work page 2012
[7]

Lam and K.H

T.Y. Lam and K.H. Leung. On the cyclotomic polynomial𝜙 𝑝𝑞 (𝑥). Amer. Math. Mon., 103(7):562–564, 1996

work page 1996
[8]

H. W. Lenstra, Jr. Vanishing sums of roots of unity. InProceedings, Bicentennial Congress Wiskundig Genootschap (Vrije Univ., Amsterdam, 1978), Part II, volume 101 ofMath. Centre Tracts, pages 249–268. Math. Centrum, Amsterdam, 1979

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A. Migotti. Zur Theorie der Kreisteilung.Wien. Ber., 87:8–14, 1883

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P. Moree. Numerical semigroups, cyclotomic polynomials, and Bernoulli numbers.Amer. Math. Monthly, 121(10):890–902, 2014

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work page 1956
[12]

C. Sanna. A survey on coefficients of cyclotomic polynomials.Expo. Math., 40(3):469–494, 2022

work page 2022
[13]

B. Zhang. Remarks on the maximum gap in binary cyclotomic polynomials.Bull. Math. Soc. Sci. Math. Roumanie (N.S.), 59(107)(1):109–115, 2016. Antonio Cafure, Instituto del Desarrollo Humano, Universidad Nacional de General Sarmiento & CONICET, Buenos Aires, Argentina Email address: acafure@campus.ungs.edu.ar Eda Cesaratto, Instituto del Desarrollo Humano, ...

work page 2016

[1] [1]

Al-Kateeb, M

A. Al-Kateeb, M. Ambrosino, H. Hong, and E. Lee. Maximum gap in cyclotomic polynomials.J. Number Theory, 220:1–15, 2021

work page 2021

[2] [2]

A. Cafure. A story about cyclotomic polynomials: the minimal polynomial of2 cos(2𝜋/𝑛) and its constant term.Math. Mag., 94(5):325–338, 2021

work page 2021

[3] [3]

Cafure and E

A. Cafure and E. Cesaratto. Binary cyclotomic polynomials: representation via words and algorithms. LNCS, 124(1):65–77, 2021

work page 2021

[4] [4]

Camburu, E

O. Camburu, E. Ciolan, F. Luca, P. Moree, and I. Shparlinski. Cyclotomic coefficients: gaps and jumps. J. Number Theory, 163:211–237, 2016

work page 2016

[5] [5]

Habermehl, S

H. Habermehl, S. Richardson, and M. A. Szwajkos. A note on coefficients of cyclotomic polynomials. Math. Mag., 37:183–185, 1964

work page 1964

[6] [6]

H. Hong, E. Lee, H.-S Lee, and C.-M. Park. Maximum gap in (inverse) cyclotomic polynomial.J. Number Theory, 132(10):2297–2315, 2012

work page 2012

[7] [7]

Lam and K.H

T.Y. Lam and K.H. Leung. On the cyclotomic polynomial𝜙 𝑝𝑞 (𝑥). Amer. Math. Mon., 103(7):562–564, 1996

work page 1996

[8] [8]

H. W. Lenstra, Jr. Vanishing sums of roots of unity. InProceedings, Bicentennial Congress Wiskundig Genootschap (Vrije Univ., Amsterdam, 1978), Part II, volume 101 ofMath. Centre Tracts, pages 249–268. Math. Centrum, Amsterdam, 1979

work page 1978

[9] [9]

A. Migotti. Zur Theorie der Kreisteilung.Wien. Ber., 87:8–14, 1883

work page

[10] [10]

P. Moree. Numerical semigroups, cyclotomic polynomials, and Bernoulli numbers.Amer. Math. Monthly, 121(10):890–902, 2014

work page 2014

[11] [11]

Niven.Irrational numbers, volume 11 ofCarus Math

I. Niven.Irrational numbers, volume 11 ofCarus Math. Monogr.Mathematical Association of America, Washington, DC, 1956

work page 1956

[12] [12]

C. Sanna. A survey on coefficients of cyclotomic polynomials.Expo. Math., 40(3):469–494, 2022

work page 2022

[13] [13]

B. Zhang. Remarks on the maximum gap in binary cyclotomic polynomials.Bull. Math. Soc. Sci. Math. Roumanie (N.S.), 59(107)(1):109–115, 2016. Antonio Cafure, Instituto del Desarrollo Humano, Universidad Nacional de General Sarmiento & CONICET, Buenos Aires, Argentina Email address: acafure@campus.ungs.edu.ar Eda Cesaratto, Instituto del Desarrollo Humano, ...

work page 2016