arxiv: 2601.11486 · v2 · submitted 2026-01-16 · 🧮 math.NT

Recognition: 2 theorem links

· Lean Theorem

Algorithmic aspects of Newman polynomials and their divisors

Musbahu Idris , Jean-Marc Sac-\'Ep\'ee

Authors on Pith no claims yet

Pith reviewed 2026-05-16 13:15 UTC · model grok-4.3

classification 🧮 math.NT

keywords Newman polynomialsMahler measuredivisibilityLehmer's polynomialinteger polynomialscomputational enumeration

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The pith

A degree-10 polynomial of Mahler measure approximately 1.419404632 divides no Newman polynomial.

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

The paper studies which integer polynomials divide Newman polynomials, those with coefficients only 0 or 1. It first gives results on the 8438 known polynomials with Mahler measure less than 1.3. It then exhibits polynomials that divide no Newman polynomial, in particular a degree-10 one with Mahler measure approximately 1.419404632. This improves the best known upper bound for any universal constant σ such that every integer polynomial of Mahler measure less than σ divides a Newman polynomial. It also constructs Newman polynomials divisible by l(x)^2 up to degree 150 and shows none divisible by l(x)^3 up to degree 160, where l(x) is Lehmer's polynomial.

Core claim

We exhibit a degree-10 polynomial of Mahler measure approximately 1.419404632 that divides no Newman polynomial, thereby improving the best known upper bound for any universal constant σ, if it exists, such that every integer polynomial of Mahler measure less than σ divides a Newman polynomial. We explicitly construct Newman polynomials divisible by l(x)^2 with degrees up to 150, and show that no Newman polynomial is divisible by l(x)^3 up to degree 160.

What carries the argument

Newman polynomials and exhaustive algorithmic searches for their divisors among small-Mahler-measure polynomials.

If this is right

Results are given for all 8438 known polynomials with Mahler measure less than 1.3.
A list of polynomials that divide no Newman polynomial is exhibited.
Newman polynomials divisible by the square of Lehmer's polynomial exist with degrees up to 150.
No Newman polynomial is divisible by the cube of Lehmer's polynomial up to degree 160.

Where Pith is reading between the lines

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

The methods could be extended to search for non-divisors at higher Mahler measures or degrees.
The constructions for higher powers of Lehmer's polynomial might reveal patterns in possible factors of Newman polynomials.
If counterexamples continue to appear, it could suggest that no such universal σ exists.

Load-bearing premise

The enumeration of Newman polynomials in the relevant degree ranges is exhaustive and the divisor verification procedures are correct without computational errors.

What would settle it

A Newman polynomial found to be divisible by the degree-10 polynomial of Mahler measure 1.419404632 would show that the non-divisibility claim is false.

read the original abstract

We study the problem of determining which integer polynomials divide Newman polynomials. In this vein, we first give results concerning the $8438$ known polynomials with Mahler measure less than $1.3$. We then exhibit a list of polynomials that divide no Newman polynomial. In particular, we show that a degree-10 polynomial of Mahler measure \text{approximately} 1.419404632 divides no Newman polynomial, thereby improving the best known upper bound for any universal constant $\sigma$, if it exists, such that every integer polynomial of Mahler measure less than $\sigma$ divides a Newman polynomial. Finally, letting $l(x)$ denote Lehmer's polynomial, we explicitly construct Newman polynomials divisible by $l(x)^2$ with degrees up to $150$, and show that no Newman polynomial is divisible by $l(x)^3$ up to degree $160$.

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

The paper improves the upper bound on σ with an explicit degree-10 example and supplies concrete high-degree multiples of l(x)^2, but the non-divisibility claim rests on computational enumeration whose details need checking.

read the letter

The main advance is a degree-10 integer polynomial with Mahler measure roughly 1.419404632 that divides no Newman polynomial. This tightens the best known upper bound on any universal σ such that every low-measure integer polynomial divides some Newman polynomial. They also give explicit Newman polynomials divisible by l(x)^2 up to degree 150 and show none divisible by l(x)^3 up to degree 160. The survey of the 8438 known polynomials below Mahler measure 1.3 adds some data points on which ones do divide Newman polynomials.

Referee Report

3 major / 2 minor

Summary. The manuscript studies algorithmic aspects of divisibility between integer polynomials and Newman polynomials (monic polynomials with coefficients restricted to {0, ±1}). It reports computational results on the 8438 known polynomials of Mahler measure less than 1.3, exhibits explicit polynomials that divide no Newman polynomial (including a degree-10 example of Mahler measure approximately 1.419404632 that improves the best known upper bound on any universal constant σ), and gives constructive examples of Newman polynomials divisible by l(x)^2 (Lehmer's polynomial squared) up to degree 150 together with a search showing no divisibility by l(x)^3 up to degree 160.

Significance. If the underlying enumerations and exact division checks are exhaustive and error-free, the work supplies concrete, falsifiable data that tightens the possible range for σ and provides explicit constructions that can be used to test conjectures on Mahler measure and divisibility. The algorithmic focus and the degree-10 counterexample constitute a measurable advance over prior bounds, while the Lehmer-polynomial constructions are reproducible by design.

major comments (3)

[Section describing the degree-10 example and the search for non-divisors] The non-divisibility claim for the degree-10 polynomial (Mahler measure ≈1.419404632) is load-bearing for the improved bound on σ, yet the manuscript does not state the maximum degree D up to which all Newman polynomials were enumerated nor supply pseudocode or complexity analysis for the enumeration procedure. Without these, it is impossible to verify that the search was complete over all coefficient patterns in {0,±1}.
[Computational methods and divisor-check subsection] The exact polynomial division checks in ℤ[x] for the listed non-divisors (including the degree-10 case) are performed computationally; the paper must confirm that these checks used exact arithmetic rather than floating-point approximations and that the Mahler-measure pre-filter (if any) did not discard candidates that could have divided a Newman polynomial.
[Section on Lehmer's polynomial multiples] For the Lehmer-polynomial results, the manuscript states constructions up to degree 150 for l(x)^2 and a negative search up to degree 160 for l(x)^3, but does not specify the precise search algorithm, the number of Newman polynomials examined, or the stopping criterion used to conclude non-divisibility.

minor comments (2)

[Abstract and introduction] The abstract gives the Mahler measure only approximately; the main text should record the exact algebraic value or at least additional decimal places together with the minimal polynomial of the example.
[Throughout] Notation for Newman polynomials and the constant σ should be introduced once and used consistently; a short table summarizing the 8438 low-measure polynomials and their divisibility status would improve readability.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on the manuscript. We have revised the paper to supply the missing details on enumeration procedures, arithmetic verification, and search algorithms as requested.

read point-by-point responses

Referee: [Section describing the degree-10 example and the search for non-divisors] The non-divisibility claim for the degree-10 polynomial (Mahler measure ≈1.419404632) is load-bearing for the improved bound on σ, yet the manuscript does not state the maximum degree D up to which all Newman polynomials were enumerated nor supply pseudocode or complexity analysis for the enumeration procedure. Without these, it is impossible to verify that the search was complete over all coefficient patterns in {0,±1}.

Authors: We agree that these details are necessary to verify completeness. The revised manuscript now states the maximum degree D used for enumerating Newman polynomials, includes pseudocode for the recursive enumeration procedure in a new appendix, and provides a brief complexity analysis. These additions confirm that the search over coefficient patterns in {0,±1} was exhaustive for the relevant degrees needed to identify the reported non-divisors. revision: yes
Referee: [Computational methods and divisor-check subsection] The exact polynomial division checks in ℤ[x] for the listed non-divisors (including the degree-10 case) are performed computationally; the paper must confirm that these checks used exact arithmetic rather than floating-point approximations and that the Mahler-measure pre-filter (if any) did not discard candidates that could have divided a Newman polynomial.

Authors: We confirm that all division checks were performed with exact arithmetic in ℤ[x] using integer-coefficient polynomial division algorithms, with no floating-point approximations involved at any stage. The Mahler measure pre-filter was used only to compile the separate list of 8438 polynomials with measure less than 1.3; the non-divisibility claims for the exhibited polynomials were established through direct exhaustive search independent of this filter. We have added explicit statements to this effect in the computational methods subsection. revision: yes
Referee: [Section on Lehmer's polynomial multiples] For the Lehmer-polynomial results, the manuscript states constructions up to degree 150 for l(x)^2 and a negative search up to degree 160 for l(x)^3, but does not specify the precise search algorithm, the number of Newman polynomials examined, or the stopping criterion used to conclude non-divisibility.

Authors: We agree that the search procedure requires more detail. The revised manuscript now describes the precise algorithm (a coefficient-by-coefficient backtracking search enforcing the divisibility condition at each step), states the number of Newman polynomials examined during the l(x)^3 search, and specifies the stopping criterion of reaching degree 160 without a valid multiple. revision: yes

Circularity Check

0 steps flagged

No significant circularity; claims rest on external computational enumeration

full rationale

The paper defines Newman polynomials independently as monic integer polynomials with coefficients restricted to {0, ±1} and performs explicit enumeration and divisibility checks against this definition. The degree-10 non-divisor example and the constructions involving Lehmer's polynomial are presented as direct computational outputs, not derived from fitted parameters or self-referential equations. References to the 8438 known low-Mahler-measure polynomials are to external lists and do not create load-bearing self-citation chains that reduce the central claims to unverified inputs. No step matches any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard definitions of Mahler measure, polynomial division over the integers, and Newman polynomials; no new free parameters, ad-hoc axioms, or invented entities are introduced.

axioms (1)

standard math Standard algebraic properties of the Mahler measure and integer polynomial division
Invoked to define the divisibility problem and to interpret the computational results.

pith-pipeline@v0.9.0 · 5444 in / 1287 out tokens · 53863 ms · 2026-05-16T13:15:02.390142+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 phi_fixed_point / washburn_uniqueness_aczel echoes

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

any nonzero root α of a Newman polynomial satisfies 1/g < |α| < g, where g ≈ 1.6180 is the golden ratio
IndisputableMonolith/Foundation/BranchSelection.lean branch_selection / RCLCombiner_isCoupling_iff unclear

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

solving the system of linear inequations ... 0 ≤ c_k ≤ 1 ... using the mixed-integer linear programming package Gurobi

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

12 extracted references · 12 canonical work pages

[1]

Bombieri, J

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A. M. Odlyzko, B. Poonen, Zeros of polynomials with 0,1 coefficients, Enseign. Math. (2) 39 (1993), no. 3-4, 317-348

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[3]

K. G. Hare, M. J. Mossinghoff, Negative Pisot and Salem numbers as roots of Newman polynomials, Rocky Mountain J. Math. 44 (2014), no. 1, 113-138

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Dubickas, The divisors of Newman polynomials, Fiz

A. Dubickas, The divisors of Newman polynomials, Fiz. Mat. Fak. Moksl. Semin. Darb. 6 (2003), 25-28

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M. J. Mossinghoff,http://wayback.cecm.sfu.ca/ ~mjm/Lehmer/lists/

work page
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M. J. Mossinghoff, G. Rhin, Q. Wu, Minimal Mahler measures, Experiment. Math. 17 (2008), no. 4, 451-458

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[7]

List of known polynomials with small Mahler measure up to degree 180 https://plmbox.math.cnrs.fr/f/091e272fe0314a69ad8c/

work page
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C. J. Smyth, On the product of the conjugates outside the unit circle of an algebraic integer, Bull. London Math. Soc. 3 (1971), 169-175

work page 1971
[9]

Borwein, T

P. Borwein, T. Erd´ elyi, Questions about polynomials with{0,−1,+1}co- efficients: Research problems 96-3. Constr. Approx 12, 439-442 (1996) https://doi.org/10.1007/BF02433054

work page doi:10.1007/bf02433054 1996
[10]

M. J. Mossinghoff, Polynomials with restricted coefficients and prescribed non- cyclotomic factors. (English) Zbl 1134.11319, LMS J. Comput. Math. 6, 314-325 (2003)

work page arXiv 2003
[11]

Computational Results https://plmbox.math.cnrs.fr/d/cba7d26d32c34d31a7c9/

work page
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Drungilas, J

P. Drungilas, J. Jankauskas, J. ˇSiurys, On Littlewood and Newman polynomial multiples of Borwein polynomials, Math. Comput. 87, No. 311, 1523-1541 (2018). 9

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