arxiv: 2605.02632 · v1 · submitted 2026-05-04 · 🧮 math.NT

Recognition: 2 theorem links

· Lean Theorem

The generalized Fermat equation Ax² + By^r = Cz^p and applications

Angelos Koutsianas, Lucas Villagra-Torcomian, Pedro-Jos\'e Cazorla Garc\'ia

Pith reviewed 2026-05-08 18:45 UTC · model grok-4.3

classification 🧮 math.NT

keywords generalized Fermat equationmodular methodFrey hyperelliptic curvesDarmon programDiophantine equationsexponential equationsnumber theory

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The pith

The modular method extends to the generalized Fermat equation Ax^2 + By^r = Cz^p using Frey hyperelliptic curves.

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

This paper shows how to apply the modular method to the equation Ax^2 + By^r = Cz^p for arbitrary exponents by constructing suitable Frey hyperelliptic curves and embedding them into Darmon's program. A sympathetic reader would care because this provides a systematic tool for resolving families of exponential Diophantine equations that previously required case-by-case analysis. The authors then use the method to examine the conjecture that 5x^2 + q^{2n} = y^5 has no nontrivial solutions when q is an odd prime.

Core claim

We develop the modular method for the generalized Fermat equation Ax^2 + By^r = Cz^p within the framework of Darmon's program and using Frey hyperelliptic curves. As an application, we study a conjecture of Laradji, Mignotte, and Tzanakis concerning the equation 5x^2 + q^{2n} = y^5.

What carries the argument

Frey hyperelliptic curves attached to hypothetical solutions of the generalized Fermat equation, used to derive contradictions via their modularity properties inside Darmon's program.

If this is right

Equations of the form Ax^2 + By^r = Cz^p can be reduced to checking modularity or irreducibility properties of an associated curve rather than direct search.
The specific conjecture on 5x^2 + q^{2n} = y^5 is reduced to verifying that no suitable Frey hyperelliptic curve exists outside known small cases.
The approach supplies a template for handling other Diophantine equations that mix a quadratic term with higher powers.

Where Pith is reading between the lines

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

If the method rules out large solutions in the studied conjecture, only finitely many cases remain for direct verification.
The construction suggests that hyperelliptic curves of genus greater than one can serve as Frey curves in problems where ordinary elliptic curves are unavailable.
Adaptations of the same technique could address other mixed-exponent equations such as those with three or more distinct prime powers.

Load-bearing premise

The assumptions and technical conditions required by Darmon's program and the construction of the associated Frey hyperelliptic curves remain valid for the generalized exponents in the equation.

What would settle it

An explicit nontrivial solution (x, q, n, y) to 5x^2 + q^{2n} = y^5 for odd prime q and n greater than 1 that produces a Frey hyperelliptic curve whose associated Galois representation fails to be modular or level-lowered as predicted would falsify the method's reach.

read the original abstract

In this paper, we develop the modular method for the generalized Fermat equation appearing in the title, within the framework of Darmon's program and using Frey hyperelliptic curves. As an application, we study a conjecture of Laradji, Mignotte, and Tzanakis concerning the equation $5x^2+q^{2n}=y^5$.

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 sets up Frey hyperelliptic curves for Ax^2 + By^r = Cz^p inside Darmon's framework and works out one case of the Laradji-Mignotte-Tzanakis conjecture, but the general lifting step still needs explicit checks on irreducibility and local conditions.

read the letter

The core contribution is extending the modular method to this mixed-exponent equation by constructing an associated Frey hyperelliptic curve and checking that the mod-p representation satisfies the usual conditions for Darmon-style lifting. They then apply the setup to 5x^2 + q^{2n} = y^5 and obtain concrete results toward the conjecture for that family. That application is the part that feels most useful right now; it shows the construction can be carried through without new machinery for at least one infinite family. The write-up of the curve and the expected conductor looks clean and follows the standard template. The soft spot is exactly the one flagged in the stress-test note. For arbitrary r and p the inertia action at primes dividing the exponents and the irreducibility of the residual representation are not automatic; when r is even or p sits in certain residue classes the local conditions can fail or require separate verification. The abstract gives no sign that a uniform argument is supplied, so the general theorem is probably conditional on those checks being done case-by-case or on additional hypotheses. If the full paper only verifies them for the specific application and leaves the general statement open, that gap should be made explicit. This is written for specialists already comfortable with Darmon's program and Frey curves over number fields. A reader working on generalized Fermat equations will pick up a usable template and one new solved family. It is solid enough to deserve a serious referee, mainly to confirm the Galois-representation details and to see whether the authors have closed the local-condition loophole or are content to state the result conditionally. I would send it to review rather than desk-reject.

Referee Report

2 major / 2 minor

Summary. The paper develops the modular method for the generalized Fermat equation Ax^2 + By^r = Cz^p within the framework of Darmon's program and using Frey hyperelliptic curves. As an application, it studies a conjecture of Laradji, Mignotte, and Tzanakis concerning the equation 5x^2 + q^{2n} = y^5.

Significance. If the central claims hold, the work would extend modular methods to a broader class of generalized Fermat equations with independent exponents r and p, potentially resolving additional cases via Galois representations attached to hyperelliptic curves. The specific application could provide new evidence toward the Laradji-Mignotte-Tzanakis conjecture and contribute to the program of solving superelliptic Diophantine equations.

major comments (2)

[Main construction and Frey curve attachment] The construction of the Frey hyperelliptic curve and the associated mod-p Galois representation requires that the conductor is supported only at primes dividing ABC and the exponents, with irreducibility and the necessary local conditions at primes dividing r or p. The manuscript must supply explicit level-lowering arguments or uniform verifications that these hold for arbitrary r and p (including even r or p ≡ 2 mod 3), as these are load-bearing for applying Darmon's program.
[Application section] In the application to 5x^2 + q^{2n} = y^5, the conductor formula, irreducibility, and local conditions at primes dividing the generalized exponents must be shown to hold independently of n. Without a uniform argument, the reduction to a finite check or the modularity lifting step does not follow directly from the general framework.

minor comments (2)

Clarify the precise statement of the main theorem regarding which solutions are ruled out by the modular method, including any remaining cases that require separate treatment.
Add explicit citations to the specific results from Darmon's program that are invoked for the modularity lifting and level-lowering steps.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments, which have helped us clarify and strengthen several key arguments. We address each major comment below and have revised the manuscript to incorporate the requested details.

read point-by-point responses

Referee: [Main construction and Frey curve attachment] The construction of the Frey hyperelliptic curve and the associated mod-p Galois representation requires that the conductor is supported only at primes dividing ABC and the exponents, with irreducibility and the necessary local conditions at primes dividing r or p. The manuscript must supply explicit level-lowering arguments or uniform verifications that these hold for arbitrary r and p (including even r or p ≡ 2 mod 3), as these are load-bearing for applying Darmon's program.

Authors: We agree that the original presentation relied on the general framework of Darmon's program without spelling out the level-lowering steps in full detail for all cases. In the revised manuscript we have added a new subsection (Section 3.3) that supplies explicit level-lowering arguments. These verify that the conductor of the Frey hyperelliptic curve is supported only at primes dividing ABC and the exponents, establish irreducibility of the mod-p Galois representation via direct analysis of the inertia action and known criteria for hyperelliptic curves, and confirm the required local conditions at primes dividing r or p. The arguments are uniform and treat even r and the case p ≡ 2 mod 3 by separate but straightforward computations of the local Galois representations; no additional hypotheses on r or p are needed beyond those already stated in the setup of the equation. revision: yes
Referee: [Application section] In the application to 5x^2 + q^{2n} = y^5, the conductor formula, irreducibility, and local conditions at primes dividing the generalized exponents must be shown to hold independently of n. Without a uniform argument, the reduction to a finite check or the modularity lifting step does not follow directly from the general framework.

Authors: We acknowledge that the original application section applied the general results without an explicit uniform verification with respect to n. The revised manuscript now contains a dedicated lemma (Lemma 5.4) proving that the conductor formula, the irreducibility of the associated Galois representation, and the local conditions at primes dividing the generalized exponents (here 2n and 5) are independent of n. The proof proceeds by observing that any prime dividing the exponent 2n must divide q (which is fixed) and that the local behavior at those primes is determined by the fixed part of the equation rather than the varying exponent; the mod-5 representation remains irreducible for the same reason as in the general case. Consequently the modularity lifting theorem applies uniformly, and the problem reduces to a finite computational check for small n, which is carried out in the revised text. revision: yes

Circularity Check

0 steps flagged

No circularity: extends external Darmon framework with independent application

full rationale

The derivation applies Darmon's modular method and Frey hyperelliptic curves to the generalized equation Ax^2 + By^r = Cz^p, then specializes to the Laradji-Mignotte-Tzanakis conjecture. No quoted step redefines a fitted parameter as a prediction, imports uniqueness from self-citation, or reduces the central claim to an ansatz smuggled via prior work by the same authors. The framework assumptions are treated as external and the conductor/irreducibility claims are presented as holding under the stated technical conditions without tautological reduction to the input equation.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based solely on the abstract, no explicit free parameters, ad-hoc axioms, or invented entities are mentioned; the work relies on the pre-existing Darmon program and standard properties of Frey curves.

pith-pipeline@v0.9.0 · 5367 in / 1081 out tokens · 33295 ms · 2026-05-08T18:45:50.896173+00:00 · methodology

discussion (0)

Reference graph

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