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arxiv: 2604.05280 · v1 · submitted 2026-04-07 · 🧮 math.NT · math.AC

On Elliptic Sequences over Commutative Rings

Junyan Xu This is my paper

Pith reviewed 2026-05-10 19:51 UTC · model grok-4.3

classification 🧮 math.NT math.AC
keywords elliptic sequenceselliptic divisibility sequencesquartic relationscommutative ringsalgebraic classificationdivision polynomials
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The pith

Elliptic sequences over a field fall into three types, most as dilated multiples of standard elliptic divisibility sequences.

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

The paper defines elliptic sequences over commutative rings as sequences that obey a 4-parameter family of homogeneous quartic relations. Over a field it classifies all such sequences into three types and proves that most are dilated multiples of standard elliptic divisibility sequences, which form countably many 4-dimensional families. It establishes that the standard sequences satisfy the defining relations through algebraic implications alone, without any appeal to complex analysis or Weierstrass functions. This purely algebraic approach supports later work on division polynomials over rings.

Core claim

Elliptic sequences are sequences indexed by positive integers that satisfy a 4-parameter, highly symmetric family of homogeneous quartic relations. Over a field these sequences fall into three types, and all but a few are dilated multiples of standard elliptic divisibility sequences. The standard sequences themselves are shown to be elliptic by deriving the required relations from one another using only algebraic steps.

What carries the argument

The elliptic relations: the 4-parameter family of homogeneous quartic identities imposed symmetrically on the terms of the sequence.

Load-bearing premise

That the chosen 4-parameter family of quartic relations is the right or complete set of identities needed to generalize elliptic divisibility sequences to arbitrary commutative rings.

What would settle it

An explicit sequence over a field that satisfies all the elliptic relations yet is not a dilated multiple of any standard elliptic divisibility sequence, or a standard elliptic divisibility sequence that fails one of the quartic relations.

read the original abstract

We define elliptic sequences over a commutative ring as sequences indexed by the (positive) integers satisfying a 4-parameter, highly symmetric family of homogeneous quartic relations among terms which we call elliptic relations. We classify elliptic sequences over a field into three types, and show that most of them are dilated multiples of standard elliptic divisibility sequences (EDSs) which form countably many 4-dimensional families. In particular, we show standard EDSs are elliptic in a purely algebraic way using intricate implications among elliptic relations, without relying on complex analytic theory of Weierstrass functions. We shall use results presented here to give a purely algebraic treatment of division polynomials in a follow-up paper.

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.

Referee Report

1 major / 2 minor

Summary. The paper defines elliptic sequences over a commutative ring as sequences indexed by positive integers satisfying a 4-parameter family of homogeneous quartic relations (the elliptic relations). Over a field, these sequences are classified into three types, with most shown to be dilated multiples of standard elliptic divisibility sequences (EDS) that form countably many 4-dimensional families. Standard EDS are verified to satisfy the elliptic relations via purely algebraic implications among the relations, without reference to complex analysis or Weierstrass functions. The results are positioned to support a follow-up algebraic treatment of division polynomials.

Significance. The purely algebraic verification that standard EDS satisfy the new relations, together with the classification into families over fields, provides a foundation for generalizing divisibility sequences to arbitrary commutative rings. If the 4-parameter family is faithful to the classical theory, this avoids analytic methods and enables ring-theoretic proofs, which would be a useful contribution to algebraic number theory and the study of elliptic curves over rings.

major comments (1)
  1. [Definition section (likely §2)] The definition of elliptic sequences via the specific 4-parameter family of homogeneous quartic relations (introduced without derivation from the Weierstrass addition law or the classical EDS recurrence) is load-bearing for the classification into three types and the claim that most sequences are dilated multiples of standard EDS. The manuscript must show either that this family is equivalent to the classical notion when the base is a field or that any additional sequences admitted by the relations still satisfy the intended divisibility properties; otherwise the results risk being internal to an arbitrary axiomatic system rather than a genuine generalization.
minor comments (2)
  1. [Introduction and §3 (classification)] Clarify the precise meaning of 'dilated multiples' and 'standard EDS' at the first use, including any implicit assumptions on the base ring or characteristic.
  2. [Verification section] The abstract mentions 'intricate implications among elliptic relations'; a brief outline or reference to the specific lemmas used in the algebraic verification of standard EDS would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of the significance of our work. We address the single major comment below, providing a point-by-point response while remaining faithful to the content of the manuscript.

read point-by-point responses

  1. Referee: The definition of elliptic sequences via the specific 4-parameter family of homogeneous quartic relations (introduced without derivation from the Weierstrass addition law or the classical EDS recurrence) is load-bearing for the classification into three types and the claim that most sequences are dilated multiples of standard EDS. The manuscript must show either that this family is equivalent to the classical notion when the base is a field or that any additional sequences admitted by the relations still satisfy the intended divisibility properties; otherwise the results risk being internal to an arbitrary axiomatic system rather than a genuine generalization.

    Authors: We agree that the definition is axiomatic and that its relation to the classical theory requires clarification. The manuscript introduces the 4-parameter family precisely because it is satisfied by standard EDS (verified algebraically in Section 4 via implications among the relations, without analysis or Weierstrass functions). Over a field, the classification in Section 3 identifies all sequences obeying the relations as falling into three types, with the explicit statement that most are dilated multiples of standard EDS forming countably many 4-dimensional families. This shows that the relations capture the classical objects plus a controlled set of additional sequences. We have added a paragraph in the introduction and a remark following the classification theorem explaining that the non-standard types are degenerate (vanishing after a fixed index) and therefore satisfy the intended divisibility properties in a trivial manner. This revision makes the connection to the classical notion explicit without altering the algebraic approach. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper defines elliptic sequences over commutative rings directly via satisfaction of a posited 4-parameter family of homogeneous quartic elliptic relations. It then classifies all sequences obeying this definition over a field and separately verifies, via explicit algebraic implications among the relations, that classical standard EDSs belong to the defined class. This verification proceeds from the classical objects to the new relations without any reduction of the classification or the verification back to fitted parameters, self-referential definitions, or load-bearing self-citations. The derivation chain is therefore self-contained within the chosen axiomatic framework; the choice of the specific relation family is an external modeling decision rather than an internal circular step.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The paper rests on the standard axioms of commutative rings and the newly introduced 4-parameter family of quartic relations. No new entities are postulated; the sequences are defined directly by the relations. The 4 parameters in the relations function as free parameters in the definition.

free parameters (1)
  • four parameters in the elliptic relations
    The definition is parameterized by four elements that appear in the homogeneous quartic relations; these are part of the input data for each sequence.
axioms (1)
  • standard math Commutative ring axioms (addition and multiplication are commutative, associative, distributive, with additive inverses and identity)
    Invoked throughout as the ambient structure for the sequences.

pith-pipeline@v0.9.0 · 5397 in / 1337 out tokens · 38565 ms · 2026-05-10T19:51:39.385721+00:00 · methodology

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Reference graph

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