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arxiv: 2603.29831 · v2 · pith:RYZDAD5Qnew · submitted 2026-03-31 · 🧮 math.GM

On the shortest open cubic equations

Bogdan Grechuk , Ashleigh Ratcliffe This is my paper

Pith reviewed 2026-05-19 18:04 UTC · model grok-4.3

classification 🧮 math.GM
keywords cubic equationsinteger solutionscubic reciprocityDiophantine equationsunsolvabilityshortest open equationscubic Diophantine
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The pith

The equation 7x³ + 2y³ = 3z² + 1 has no integer solutions.

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

The paper establishes that the cubic equation 7x cubed plus 2y cubed equals 3z squared plus one has no solutions in integers. It does this by applying the law of cubic reciprocity. Before this, the equation was the shortest cubic Diophantine equation whose solvability was unknown. Resolving it allows the authors to update the list of the shortest open cubic equations that still lack known solutions or proofs of insolubility. A sympathetic reader would care because it narrows the search for the minimal length of cubic equations that remain undecided.

Core claim

We use cubic reciprocity to prove that the equation 7x^3 + 2y^3 = 3z^2 + 1 has no integer solutions. Prior to this work, it was the shortest cubic equation for which the existence of integer solutions remained open. We conclude with a list of the new shortest open cubic equations.

What carries the argument

Cubic reciprocity applied to the equation 7x^3 + 2y^3 = 3z^2 + 1 to derive a contradiction from any assumed integer solution.

If this is right

  • The equation 7x^3 + 2y^3 = 3z^2 + 1 is now known to have no integer solutions.
  • The previous shortest open cubic equation is removed from the open list.
  • A new collection of shortest open cubic equations is presented for further study.

Where Pith is reading between the lines

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

  • The updated list of shortest open equations could focus future reciprocity or modular arithmetic checks on those remaining cases.
  • Small-scale computer searches for solutions in the new shortest candidates might still be worthwhile before attempting full reciprocity proofs.
  • The approach may extend to other short mixed cubic equations involving squares on the right-hand side.

Load-bearing premise

Cubic reciprocity applies directly to this equation without additional unstated conditions or case distinctions that might allow solutions.

What would settle it

Finding any integers x, y, z that satisfy 7x^3 + 2y^3 = 3z^2 + 1 would falsify the claim of no solutions.

read the original abstract

We use cubic reciprocity to prove that the equation $7x^3+2y^3=3z^2+1$ has no integer solutions. Prior to this work, it was the shortest cubic equation for which the existence of integer solutions remained open. We conclude with a list of the new shortest open cubic equations.

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 / 1 minor

Summary. The manuscript claims to prove, via an application of cubic reciprocity in the Eisenstein integers Z[ω], that the Diophantine equation 7x³ + 2y³ = 3z² + 1 has no integer solutions. It asserts that this equation was previously the shortest open cubic equation with integer coefficients and concludes by listing the new shortest open cubic equations after this resolution.

Significance. Resolving the solvability status of this particular short cubic equation would constitute a modest but concrete contribution to the classification of cubic Diophantine equations by coefficient size. The approach relies on the established cubic reciprocity theorem rather than ad-hoc or computational methods, which is a methodological strength if the application is carried through rigorously.

major comments (1)
  1. [Proof of the main theorem (application of cubic reciprocity)] The proof applies cubic reciprocity to the factorization of 7x³ + 2y³ − 3z² − 1 in Z[ω]. Because 3 is the ramified prime above 3 in this ring, the reciprocity law requires the relevant prime elements to be coprime to 3 (or separate handling of the 3-adic valuation). The manuscript does not contain an explicit case analysis, modulo-9 reduction, or descent argument ruling out solutions in which 3 divides x, y, or z; without this, the reciprocity step does not cover all possibilities.
minor comments (1)
  1. The abstract is concise, but the manuscript would benefit from numbered equations for the ideal factorizations and the precise statement of the reciprocity law being invoked.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying a point that requires clarification in the proof. We address the major comment below and will revise the manuscript to incorporate the suggested strengthening.

read point-by-point responses

  1. Referee: [Proof of the main theorem (application of cubic reciprocity)] The proof applies cubic reciprocity to the factorization of 7x³ + 2y³ − 3z² − 1 in Z[ω]. Because 3 is the ramified prime above 3 in this ring, the reciprocity law requires the relevant prime elements to be coprime to 3 (or separate handling of the 3-adic valuation). The manuscript does not contain an explicit case analysis, modulo-9 reduction, or descent argument ruling out solutions in which 3 divides x, y, or z; without this, the reciprocity step does not cover all possibilities.

    Authors: We agree that an explicit preliminary analysis is needed to justify the application of cubic reciprocity. In the revised manuscript we will insert a short subsection (new Lemma 2.1) that performs a complete modulo-9 case analysis on the equation. This analysis shows that if 3 divides any of x, y or z then either all three are divisible by 3 (leading to an infinite descent after dividing out the common factor) or a direct contradiction arises with the constant term +1. Only after establishing that any integer solution must satisfy 3 ∤ x y z do we factor in Z[ω] and invoke cubic reciprocity for the coprime prime elements. The added argument is elementary and uses only the ring of integers and the known ramification of 3; it does not alter the main reciprocity step. revision: yes

Circularity Check

0 steps flagged

No circularity; proof applies established external theorem

full rationale

The paper's central claim is a proof that 7x^3 + 2y^3 = 3z^2 + 1 has no integer solutions, obtained by direct application of the cubic reciprocity theorem. This theorem is a standard result from prior number-theoretic literature and is not derived, fitted, or justified inside the paper via self-citation chains or self-definitional steps. No equations are shown to reduce to their own inputs by construction, no parameters are fitted to data and then relabeled as predictions, and no uniqueness theorems or ansatzes are imported from the authors' own prior work. The derivation therefore remains self-contained against external benchmarks. Questions about possible missing case splits for v_3 valuations concern proof completeness or correctness, not circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard cubic reciprocity law and basic properties of integer rings; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • standard math Cubic reciprocity theorem holds for the relevant prime factors
    Invoked to derive a contradiction from assumed solutions.

pith-pipeline@v0.9.0 · 5560 in / 997 out tokens · 44196 ms · 2026-05-19T18:04:06.602965+00:00 · methodology

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Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages · 1 internal anchor

  1. [1]

    Brauer groups of certain affine cubic surfaces.arXiv preprint arXiv:2509.16042, 2025

    Abdulmuhsin Alfaraj. Brauer groups of certain affine cubic surfaces.arXiv preprint arXiv:2509.16042, 2025

    work page arXiv 2025

  2. [2]

    The decision problem for exponential Diophantine equations.Ann

    Martin Davis, Hilary Putnam, and Julia Robinson. The decision problem for exponential Diophantine equations.Ann. of Math. (2), 74:425–436, 1961

    work page 1961

  3. [3]

    A systematic approach to diophantine equations: open problems.arXiv preprint arXiv:2404.08518, 2024

    Bogdan Grechuk. A systematic approach to diophantine equations: open problems.arXiv preprint arXiv:2404.08518, 2024

    work page internal anchor Pith review arXiv 2024

  4. [4]

    Springer, Cham, [2024] ©2024

    Bogdan Grechuk.Polynomial Diophantine equations—a systematic approach. Springer, Cham, [2024] ©2024. With contributions by Ashleigh Wilcox

    work page 2024

  5. [5]

    Cubic diophantine equations: integer solutions beyond direct search.The American Mathematical Monthly, to appear, 2026

    Bogdan Grechuk. Cubic diophantine equations: integer solutions beyond direct search.The American Mathematical Monthly, to appear, 2026

    work page 2026

  6. [6]

    On the integer solutions of quadratic equations.J

    Fritz Grunewald and Dan Segal. On the integer solutions of quadratic equations.J. Reine Angew. Math., 569:13–45, 2004

    work page 2004

  7. [7]

    CRC press, 2013

    Franz Halter-Koch.Quadratic irrationals: An introduction to classical number theory. CRC press, 2013. 9

    work page 2013

  8. [8]

    Mathematical problems.Bull

    David Hilbert. Mathematical problems.Bull. Amer. Math. Soc., 8(10):437–479, 1902

    work page 1902

  9. [9]

    Springer Science & Business Media, 1990

    Kenneth Ireland and Michael Ira Rosen.A classical introduction to modern number theory, volume 84. Springer Science & Business Media, 1990

    work page 1990

  10. [10]

    Fruit Diophantine equation.The Mathematical Gazette, 107(569):302– 306, 2023

    Dipramit Majumdar and B Sury. Fruit Diophantine equation.The Mathematical Gazette, 107(569):302– 306, 2023

    work page 2023

  11. [11]

    How to solve a binary cubic equation in integers.Math

    David Masser. How to solve a binary cubic equation in integers.Math. Proc. Cambridge Philos. Soc., 176(3):609–624, 2024

    work page 2024

  12. [12]

    Ju. V . Matijasevich. The Diophantineness of enumerable sets.Dokl. Akad. Nauk SSSR, 191:279–282, 1970

    work page 1970

  13. [13]

    Generalized fruit Diophantine equation and hyperelliptic curves

    Om Prakash and Kalyan Chakraborty. Generalized fruit Diophantine equation and hyperelliptic curves. Monatshefte f¨ur Mathematik, 203(3):667–676, 2024

    work page 2024

  14. [14]

    Generalized fruit Diophantine equation over number fields.arXiv preprint arXiv:2408.12278, 2024

    Satyabrat Sahoo and Shanta Laishram. Generalized fruit Diophantine equation over number fields.arXiv preprint arXiv:2408.12278, 2024

    work page arXiv 2024

  15. [15]

    A class of fruit Diophantine equations.Monatshefte f ¨ur Mathematik, 199(4):899–907, 2022

    Lalit Vaishya and Richa Sharma. A class of fruit Diophantine equations.Monatshefte f ¨ur Mathematik, 199(4):899–907, 2022

    work page 2022

  16. [16]

    Algebraische zahlentheorie

    Stefan Wewers. Algebraische zahlentheorie. Lecture notes, Universit ¨at Ulm, 2014. Wintersemester 2014. 10

    work page 2014