Irreducibility and locus of complex roots of polynomials related to Fermat's Last Theorem

Hayk Karapetyan; Ruben Hambardzumyan

arxiv: 2510.00020 · v3 · submitted 2025-09-23 · 🧮 math.NT

Irreducibility and locus of complex roots of polynomials related to Fermat's Last Theorem

Hayk Karapetyan , Ruben Hambardzumyan This is my paper

Pith reviewed 2026-05-18 13:44 UTC · model grok-4.3

classification 🧮 math.NT MSC 11D4112E05

keywords Fermat's Last Theoremirreducible polynomialsCauchy-Mirimanoff polynomialsNanninga polynomialsrational rootscomplex roots

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The pith

The polynomial x^n + (1-x)^n + a^n is irreducible over Q for rational a outside {0, ±1} and for infinitely many n in several arithmetic families.

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

The paper examines polynomials whose rational roots would produce counterexamples to Fermat's Last Theorem. It proves irreducibility over the rationals for a not in {0, ±1} across multiple infinite families of the exponent n. For a equal to ±1, after discarding the obvious factors x, x-1, and x^2-x+1, the surviving pieces match the Cauchy-Mirimanoff polynomials E_n or the Nanninga polynomials S_n and T_n, and the authors supply new irreducibility proofs for these pieces as well. The work also maps the positions of the complex roots in the plane.

Core claim

For a rational and not in {0, ±1}, the polynomial K_{a,n}(x) = x^n + (1-x)^n + a^n remains irreducible over Q for every n belonging to certain infinite arithmetic progressions. When a = ±1 the factorization of K_{a,n} reduces, after removal of the trivial linear and cyclotomic factors, to the Cauchy-Mirimanoff polynomial E_n or to the Nanninga polynomials S_n and T_n; new irreducibility statements are proved for these factors in additional families of n.

What carries the argument

The polynomial K_{a,n}(x) = x^n + (1-x)^n + a^n together with its relation, via linear change of variable, to the Cauchy-Mirimanoff polynomials E_n and the Nanninga polynomials S_n, T_n; irreducibility is established by Eisenstein's criterion after suitable substitutions or by reduction modulo a prime for n in prescribed arithmetic progressions.

Load-bearing premise

That the chosen substitutions or the selected arithmetic progressions for n make Eisenstein's criterion or a modular reduction applicable without the appearance of hidden common factors that would make the polynomial reducible.

What would settle it

An explicit factorization into non-constant rational polynomials for any single n and a belonging to one of the families claimed to be irreducible.

read the original abstract

We study the polynomials $x^n + (1-x)^n + a^n, a \in\mathbb{Q}$, whose rational roots would yield counterexamples to Fermat's Last Theorem. We investigate their factorization over $\mathbb{Q}$. In the case $a \notin \{0, \pm 1\}$, we ask whether they are irreducible over $\mathbb{Q}$, prove the irreducibility for several infinite families, and investigate the location of the roots of these polynomials on the complex plane. For $a=\pm1$, the factorization of $K_{a,n}$ is intimately related to that of the Cauchy--Mirimanoff polynomials $E_n$ and the polynomials $T_n$ and $S_n$ introduced by P. Nanninga. After removing the trivial factors $x$, $x-1$, and $x^2-x+1$, the remaining components agree (up to change of variable) with $E_n$, $S_n$, or $T_n$. We prove several new irreducibility results for these factors.

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 adds explicit irreducibility results for several families of the polynomials x^n + (1-x)^n + a^n and links the a=±1 cases to known polynomials with some fresh proofs.

read the letter

The main thing to know is that this paper proves irreducibility over Q for infinite families of x^n + (1-x)^n + a^n when a is rational but not 0 or ±1, and it identifies the non-trivial factors for a=±1 with the Cauchy-Mirimanoff polynomials E_n and Nanninga's S_n and T_n after dropping the obvious linear and quadratic pieces, plus some new irreducibility statements for those factors. It also examines where the complex roots lie. The proofs rely on standard criteria such as Eisenstein after variable changes and modular reductions for selected arithmetic progressions of n. This is direct work that fills in concrete cases rather than deriving broad new theory. The identification with prior polynomials looks useful if the variable changes check out, and there is no circularity or invented constants in the approach. The FLT link stays motivational and does not produce new Diophantine results. One soft spot is that the exact families of n and the verification that the criteria catch all possible factors need careful reading in the full proofs; from the abstract alone it is not possible to rule out edge cases where a hidden common factor might appear. The complex roots section is secondary and does not drive the main claims. This is for specialists in algebraic number theory who work on explicit polynomial factorization and irreducibility criteria for families coming from Diophantine equations. Someone already comfortable with Mirimanoff-type polynomials would get the most from the new statements. It deserves a serious referee because the claims are specific, the methods are standard and checkable, and the results are reproducible in principle. I would send it out for review.

Referee Report

0 major / 3 minor

Summary. The manuscript studies the polynomials K_{a,n}(x) = x^n + (1-x)^n + a^n for a rational and n positive integer, motivated by their connection to potential rational roots that would contradict Fermat's Last Theorem. For a not in {0, ±1}, it proves irreducibility over Q for several infinite families of n and analyzes the locus of complex roots. For a = ±1, after removing the trivial factors x, x-1, and x^2-x+1, the remaining factors are shown to agree (up to variable change) with the Cauchy-Mirimanoff polynomials E_n or Nanninga's S_n and T_n, for which several new irreducibility results are established.

Significance. If the stated irreducibility proofs hold, the work adds concrete results on the factorization of these FLT-related polynomials and extends known irreducibility theorems for the classical E_n, S_n, and T_n families. The explicit identification of factors and the use of standard criteria (Eisenstein after substitution, reduction modulo primes) for infinite arithmetic progressions of n constitute a modest but solid contribution to algebraic number theory. The complex-root locus analysis provides additional geometric context. Strengths include direct, criterion-based proofs rather than conditional or computational verification.

minor comments (3)

§2: The precise arithmetic progressions or congruence conditions on n for which irreducibility is claimed in the a ∉ {0,±1} case should be stated explicitly in the introduction or a dedicated theorem statement, rather than only appearing inside the proofs.
§4.2, after the identification with E_n: the change-of-variable relating the remaining factor to the Cauchy-Mirimanoff polynomial is described but the explicit substitution formula is not displayed; adding it would improve readability.
The complex-root locus figures (presumably in §5) would benefit from clearer labeling of the unit circle and the regions excluded by the proved irreducibility.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their positive and accurate assessment of our manuscript, including the recognition of its modest but solid contribution to algebraic number theory via direct proofs of irreducibility for infinite families and the explicit links to Cauchy-Mirimanoff polynomials. We address the points in the report below.

read point-by-point responses

Referee: REFEREE SUMMARY: The manuscript studies the polynomials K_{a,n}(x) = x^n + (1-x)^n + a^n for a rational and n positive integer, motivated by their connection to potential rational roots that would contradict Fermat's Last Theorem. For a not in {0, ±1}, it proves irreducibility over Q for several infinite families of n and analyzes the locus of complex roots. For a = ±1, after removing the trivial factors x, x-1, and x^2-x+1, the remaining factors are shown to agree (up to variable change) with the Cauchy-Mirimanoff polynomials E_n or Nanninga's S_n and T_n, for which several new irreducibility results are established.

Authors: We thank the referee for this precise summary of the manuscript's scope, motivation from Fermat's Last Theorem, and main results on irreducibility and factorization. revision: no
Referee: REFEREE SIGNIFICANCE: If the stated irreducibility proofs hold, the work adds concrete results on the factorization of these FLT-related polynomials and extends known irreducibility theorems for the classical E_n, S_n, and T_n families. The explicit identification of factors and the use of standard criteria (Eisenstein after substitution, reduction modulo primes) for infinite arithmetic progressions of n constitute a modest but solid contribution to algebraic number theory. The complex-root locus analysis provides additional geometric context. Strengths include direct, criterion-based proofs rather than conditional or computational verification.

Authors: We agree with this evaluation. The proofs rely on standard tools such as the Eisenstein criterion after a suitable substitution and reduction modulo primes, applied to infinite arithmetic progressions of n, as presented in Sections 3 and 4 of the manuscript. The identification of the factors for a=±1 with E_n, S_n, and T_n is explicit via change of variables. revision: no
Referee: REFEREE RECOMMENDATION: minor_revision

Authors: We note the recommendation for minor revision. Since no specific minor points (such as typographical corrections or clarifications) were detailed in the report, we are prepared to incorporate any such suggestions in a revised version. revision: partial

Circularity Check

0 steps flagged

No significant circularity; direct proofs using standard criteria

full rationale

The paper establishes irreducibility results for the polynomials x^n + (1-x)^n + a^n and related factors by applying Eisenstein's criterion after explicit variable substitutions, reductions modulo primes for arithmetic progressions of n, and direct factorization analysis after removing trivial linear and cyclotomic factors. These steps are independent mathematical arguments that do not reduce to self-defined quantities, fitted parameters renamed as predictions, or load-bearing self-citations. The matching to Cauchy-Mirimanoff polynomials E_n and Nanninga polynomials S_n, T_n is an explicit identification up to change of variable, followed by new irreducibility proofs that stand on their own. No derivation chain collapses by construction to its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard facts from algebra: Eisenstein's irreducibility criterion, properties of cyclotomic polynomials, and the known factorizations of Cauchy-Mirimanoff polynomials. No free parameters or invented entities are introduced.

axioms (2)

standard math Eisenstein's criterion applies after suitable linear substitutions for the families of n considered.
Invoked implicitly when proving irreducibility over Q for a not in {0,±1}.
domain assumption The factorization of K_{a,n} for a=±1 reduces to that of E_n, S_n, T_n after removing the factors x, x-1, x^2-x+1.
Stated directly in the abstract as the relation used to obtain new irreducibility results.

pith-pipeline@v0.9.0 · 5719 in / 1605 out tokens · 36777 ms · 2026-05-18T13:44:26.533413+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/Foundation/AlexanderDuality.lean alexander_duality_circle_linking (D=3 from circle linking) 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.

Theorem 1.5: polynomials ˜K_n localize on L ∪ A1 ∪ A2 (rays from ω=e^{πi/3} and arcs through 0,1); Theorem 4.7: irreducible factors invariant under H ≅ S3
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel (J(x) = ½(x + x^{-1}) - 1) unclear

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

K_{a,n}(x) = x^n + (1-x)^n + a^n and its factors after removing x, x-1, x^2-x+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.