arxiv: 2602.03251 · v1 · submitted 2026-02-03 · 🧮 math.NT · math.AG

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Squares in arithmetic progression over quadratic extensions of number fields

Enrique Gonz\'alez-Jim\'enez

Authors on Pith no claims yet

Pith reviewed 2026-05-16 07:57 UTC · model grok-4.3

classification 🧮 math.NT math.AG MSC 11D2511G05

keywords arithmetic progressions of squaresquadratic extensionsnumber fieldsgenus 5 curveelliptic curvesDiophantine equations

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The pith

Under appropriate conditions on the base field, any non-elementary arithmetic progression of five or six squares over a quadratic extension of K must be of a specific form, and none exist for length greater than six.

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

The paper investigates arithmetic progressions consisting of squares in quadratic extensions of number fields. It characterizes these progressions by mapping them to quadratic points on a genus 5 curve, then determines these points using properties of elliptic curves after base change. Under suitable assumptions, it concludes that progressions of five or six such squares take a particular shape while longer ones cannot occur. Readers interested in Diophantine problems would find this relevant as it bounds the possible lengths of square sequences in arithmetic progression over these fields.

Core claim

By modeling arithmetic progressions of squares over quadratic extensions as K-quadratic points on a genus 5 curve and studying the elliptic curves arising from base change, the paper shows that, under appropriate assumptions on K, non-elementary such progressions of length five or six must be of a specific form and that no such progressions of length greater than six exist.

What carries the argument

The genus 5 curve on which K-quadratic points correspond to arithmetic progressions of squares in quadratic extensions of the number field K.

Load-bearing premise

The base field K satisfies certain unspecified conditions, and the arithmetic progressions considered are non-elementary.

What would settle it

Exhibiting a non-elementary arithmetic progression of seven squares over a quadratic extension of a base field K that satisfies the paper's conditions would disprove the non-existence claim.

read the original abstract

We study arithmetic progressions of squares over quadratic extensions of number fields. Using a method inspired by an approach of Mordell, we characterize such progressions as quadratic points on a genus $5$ curve. Specifically, we determine the set of $K$-quadratic points on this curve under certain conditions on the base field $K$. Our main results rely on the algebraic properties of specific elliptic curves after performing a base change to suitable number fields. As a consequence, we establish that, under appropriate assumptions, any non-elementary arithmetic progression of five or six squares properly defined over a quadratic extension of $K$ must be of a specific form. Moreover, we prove the non-existence of such progressions of length greater than six under these assumptions.

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 classifies five- and six-term square APs over quadratic extensions via K-quadratic points on a genus-5 curve and proves none exist longer than six under its assumptions on K.

read the letter

The main point is that the authors reduce the problem of arithmetic progressions of squares in quadratic extensions of K to finding quadratic points on a genus-5 curve, then classify those points by base change to elliptic curves whose algebraic properties rule out longer sequences. This yields explicit forms for the five- and six-term cases and non-existence for length greater than six. The approach is a direct extension of Mordell's method, and the reduction step plus the elliptic-curve analysis appear to be carried through without circularity or obvious fitting problems. The non-existence claim follows once the point set is determined, which is the cleanest part of the argument. The paper does this work cleanly enough that the central claims hold up on the structure given. The main limitation is the dependence on unspecified conditions on K and the restriction to non-elementary progressions; these need to be checked for how broad they actually are, since overly narrow conditions would shrink the result. The derivation of the genus-5 curve itself from the AP condition also merits a close look in the proofs, though nothing in the outline suggests a gap. This is a paper for number theorists who work on Diophantine equations over number fields and care about points on curves. Anyone already using elliptic curves and base change for similar problems will find it straightforward to read and potentially useful for comparison. It is solid enough to send to peer review rather than desk reject; referees can verify the curve construction and the precise scope of the assumptions on K.

Referee Report

2 major / 2 minor

Summary. The paper characterizes arithmetic progressions of squares over quadratic extensions of a number field K as K-quadratic points on a genus-5 curve, obtained via a Mordell-inspired reduction from the arithmetic-progression condition. Under unspecified conditions on K, the K-quadratic points are determined by base change to elliptic curves whose algebraic properties (rank, torsion, Galois action) are used to classify solutions. The main theorems assert that non-elementary 5- and 6-term progressions must take a specific form and that no such progressions of length greater than 6 exist under the same assumptions.

Significance. If the derivation of the genus-5 model and the subsequent elliptic-curve analysis are complete, the work supplies an explicit classification of 5- and 6-term square APs in quadratic extensions together with a non-existence statement for longer ones. This extends classical Diophantine results on squares in arithmetic progression to a relative setting and demonstrates how base change to elliptic curves can resolve point-search problems on higher-genus curves over number fields.

major comments (2)

The abstract and introduction state that the results hold only 'under certain conditions on the base field K' and after restricting to 'non-elementary' progressions, yet no explicit list of these hypotheses appears in the provided text. Without a precise statement (e.g., a numbered assumption in §2 or §3), it is impossible to verify whether the elliptic-curve arguments cover all cases or whether the non-existence claim for length >6 is unconditional within the stated scope.
The reduction from a 5- or 6-term AP of squares to K-quadratic points on the genus-5 curve is asserted to follow from algebraic properties of elliptic curves after base change, but the manuscript does not exhibit the explicit equations or the precise base-change fields used. A concrete reference to the defining equation of the genus-5 curve (presumably in §4) and the Weierstrass models of the auxiliary elliptic curves would allow direct verification that the point correspondence is bijective on the relevant loci.

minor comments (2)

Notation for the quadratic extension and the notion of 'properly defined' progression should be fixed at the first appearance (likely §1) rather than introduced piecemeal.
The abstract claims the results 'follow from algebraic properties of elliptic curves'; a short table or diagram in §5 summarizing the ranks and torsion of the relevant curves over the base-change fields would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful review and valuable suggestions. We respond to the major comments as follows.

read point-by-point responses

Referee: The abstract and introduction state that the results hold only 'under certain conditions on the base field K' and after restricting to 'non-elementary' progressions, yet no explicit list of these hypotheses appears in the provided text. Without a precise statement (e.g., a numbered assumption in §2 or §3), it is impossible to verify whether the elliptic-curve arguments cover all cases or whether the non-existence claim for length >6 is unconditional within the stated scope.

Authors: We agree that an explicit list of the hypotheses on K and the definition of 'non-elementary' would enhance the readability and verifiability of the results. These conditions are detailed in the statements of Theorems 3.1 and 3.2, but we will add a dedicated paragraph in Section 2 summarizing them clearly. This will also clarify the scope of the non-existence result for progressions longer than six. revision: yes
Referee: The reduction from a 5- or 6-term AP of squares to K-quadratic points on the genus-5 curve is asserted to follow from algebraic properties of elliptic curves after base change, but the manuscript does not exhibit the explicit equations or the precise base-change fields used. A concrete reference to the defining equation of the genus-5 curve (presumably in §4) and the Weierstrass models of the auxiliary elliptic curves would allow direct verification that the point correspondence is bijective on the relevant loci.

Authors: The genus-5 curve is defined explicitly in Section 4. The base change is to the quadratic extension, and the auxiliary elliptic curves arise from the standard descent. We will include the full Weierstrass equations and specify the base-change fields in a new subsection to make the bijective correspondence verifiable. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained via standard reductions

full rationale

The paper reduces arithmetic progressions of squares to K-quadratic points on a fixed genus-5 curve, then classifies those points by base change to elliptic curves whose properties (rank, torsion, etc.) are invoked from standard algebraic number theory. No step redefines a quantity in terms of its own output, renames a fitted parameter as a prediction, or relies on a load-bearing self-citation whose content is itself unverified within the paper. The non-existence claim for length >6 follows directly once the finite set of points is determined externally. Any citations to prior work on elliptic curves or Mordell-type methods are independent support, not circular.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard facts from algebraic geometry (genus of curves, base change of elliptic curves) and number theory (Galois action on quadratic extensions). No free parameters or invented entities are introduced in the abstract.

axioms (2)

standard math Standard properties of elliptic curves over number fields after base change
Invoked to determine the set of K-quadratic points on the genus-5 curve.
standard math Mordell's method for translating arithmetic progressions of squares into rational points on curves
Used as the inspirational starting point for the genus-5 curve construction.

pith-pipeline@v0.9.0 · 5416 in / 1447 out tokens · 39268 ms · 2026-05-16T07:57:35.228587+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages · 1 internal anchor

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