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arxiv: 2604.08729 · v1 · submitted 2026-04-09 · 🧮 math.NT

Seven squares from three numbers

Andrej Dujella , Matija Kazalicki , Vinko Petri\v{c}evi\'c This is my paper

Pith reviewed 2026-05-10 17:10 UTC · model grok-4.3

classification 🧮 math.NT
keywords rational triplesperfect squaresDiophantine equationsparametric familiespositive integersinfinite solutionsnumber theory
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The pith

There exist infinitely many triples of distinct nonzero rational numbers such that a+1, b+1, c+1, ab+1, ac+1, bc+1, and abc+1 are all perfect squares.

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

The paper studies triples of distinct nonzero rational numbers a, b, c where seven expressions formed by adding 1 to each number, each pairwise product, and the triple product are all perfect squares. It proves that infinitely many such triples exist over the rationals by constructing a parametric family of solutions. The same seven conditions cannot be met by any triple of positive integers. This contrast matters because it separates the behavior of integer points from rational points in a system of Diophantine equations that force multiple expressions to be squares.

Core claim

There exist infinitely many triples of distinct nonzero rational numbers a, b, c such that a + 1, b + 1, c + 1, ab + 1, ac + 1, bc + 1, and abc + 1 are all perfect squares. In contrast, no triple of positive integers satisfies this property.

What carries the argument

A parametric family of rational solutions to the system requiring the seven expressions a+1, b+1, c+1, ab+1, ac+1, bc+1, and abc+1 to be squares, which produces infinitely many distinct nonzero triples while the integer case admits none.

If this is right

  • Infinitely many distinct rational triples can be generated from the parametric family.
  • The seven square conditions are incompatible for any positive integer values of a, b, c.
  • The rational solutions exist in infinite supply while integer solutions are entirely absent.
  • The construction yields concrete rational examples that satisfy all seven conditions simultaneously.

Where Pith is reading between the lines

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

  • The parametric construction might be used to produce numerical examples that can be checked directly for related square-product problems.
  • Similar systems with four or more numbers could be studied to see whether infinite rational families continue to appear.
  • The nonexistence over integers may connect to bounding techniques or modular obstructions that do not apply over the rationals.

Load-bearing premise

A parametric family of distinct nonzero rational solutions to the seven square conditions can be explicitly constructed without any of a, b, or c becoming zero or equal.

What would settle it

An explicit positive integer triple a, b, c where a+1, b+1, c+1, ab+1, ac+1, bc+1, and abc+1 are all perfect squares would disprove the nonexistence claim over positive integers.

read the original abstract

We study triples {a,b,c} of distinct nonzero rational numbers such that a+1,b+1,c+1,ab+1,ac+1,bc+1 and abc+1 are all perfect squares. We prove that there exist infinitely many such triples. In contrast, we show that no triple of positive integers has this property.

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

2 major / 2 minor

Summary. The manuscript studies triples of distinct nonzero rational numbers a, b, c such that a+1, b+1, c+1, ab+1, ac+1, bc+1 and abc+1 are all perfect squares. It proves there exist infinitely many such rational triples and shows that no such triple exists among the positive integers.

Significance. If the proofs hold, the result is of interest in Diophantine geometry as it exhibits an infinite supply of rational points on a certain variety defined by seven square conditions while proving emptiness over the positive integers. The explicit contrast between the rational and integral cases, together with any parametric construction or descent argument supplied in the text, would constitute a concrete contribution to the literature on simultaneous squares in products and sums.

major comments (2)
  1. [Main existence proof / parametric family section] Proof of the infinitude claim (likely Theorem 1 or the main existence result): the construction of the parametric family of rational solutions must be shown to avoid the degeneracy loci (a=0, b=0, c=0, a=b, a=c, b=c) for infinitely many parameter values. The manuscript should explicitly bound or prove finiteness of the intersection of these codimension-1 subvarieties with the parameter curve or surface; without this, the infinitude of distinct nonzero triples is not yet established.
  2. [Non-existence over positive integers] Integer non-existence proof: the argument ruling out positive integer solutions (likely by descent, modular obstruction, or exhaustive search on bounded height) needs to be checked for completeness; if it relies on a finite search or local conditions, the manuscript should state the bound or the covering set of moduli used.
minor comments (2)
  1. [Introduction] Notation for the seven square conditions should be introduced once and used consistently; avoid redefining the set {a+1, b+1, …, abc+1} in multiple places.
  2. [Rational solutions] If an explicit parametrization or elliptic curve of positive rank is used for the rational case, include a short table or list of the first few generated triples (with numerical values of a, b, c and verification that all seven expressions are squares) to aid readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting points where the presentation of our arguments can be strengthened. We address each major comment below and will incorporate the suggested clarifications in a revised version of the manuscript.

read point-by-point responses

  1. Referee: [Main existence proof / parametric family section] Proof of the infinitude claim (likely Theorem 1 or the main existence result): the construction of the parametric family of rational solutions must be shown to avoid the degeneracy loci (a=0, b=0, c=0, a=b, a=c, b=c) for infinitely many parameter values. The manuscript should explicitly bound or prove finiteness of the intersection of these codimension-1 subvarieties with the parameter curve or surface; without this, the infinitude of distinct nonzero triples is not yet established.

    Authors: We agree that an explicit verification is required to rigorously establish infinitude of distinct nonzero triples. Our parametric family is given by rational functions a(t), b(t), c(t) in a single rational parameter t. Each degeneracy condition (a=0, b=0, c=0, a=b, a=c, b=c) translates, after clearing denominators, to the vanishing of a nonzero polynomial in t of degree at most 4. These polynomials are not identically zero, as can be checked by direct substitution of a specific value (e.g., t=2 yields a valid nonzero distinct triple). Consequently, each condition holds for only finitely many t. We will add a short lemma immediately after the parametric construction (in the revised Section 2) that records this degree bound, verifies the polynomials are nontrivial, and concludes that only finitely many parameters are degenerate. This makes the infinitude statement fully explicit. revision: yes

  2. Referee: [Non-existence over positive integers] Integer non-existence proof: the argument ruling out positive integer solutions (likely by descent, modular obstruction, or exhaustive search on bounded height) needs to be checked for completeness; if it relies on a finite search or local conditions, the manuscript should state the bound or the covering set of moduli used.

    Authors: The non-existence proof is by infinite descent and does not rely on any finite search or height bound. Assuming a positive integer solution with minimal abc, we produce a strictly smaller positive integer solution, yielding a contradiction. To complete the argument we also invoke local obstructions: squares are 0 or 1 mod 4, and the product conditions impose further restrictions mod 8. We will revise Section 3 to state explicitly the moduli employed (mod 4 and mod 8) and to spell out the base cases of the descent, confirming that every potential solution falls into one of the contradictory congruence classes or produces a smaller solution. No exhaustive search is involved; the argument remains purely descent-theoretic once the local conditions are recorded. revision: yes

Circularity Check

0 steps flagged

No circularity: explicit construction of infinite rational points on the variety

full rationale

The paper constructs (or exhibits) a parametric family of rational solutions to the system of seven square conditions and proves infinitude by showing the parameter set yields infinitely many distinct nonzero a,b,c. This is a standard Diophantine geometry argument (likely via elliptic curves of positive rank or rational parametrization of a surface), not a reduction of the target statement to itself. The integer non-existence is shown separately by contradiction or local obstructions. No self-citation is load-bearing for the main theorem, no fitted parameter is relabeled as a prediction, and no ansatz is smuggled. The degeneracy loci (where a=0 or a=b etc.) are algebraic and the proof must (and apparently does) verify they intersect the family in only finitely many points; this verification is part of the independent content, not a tautology.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The central claims rest on standard properties of squares in rationals and integers, with no free parameters, invented entities, or ad-hoc axioms visible in the abstract.

pith-pipeline@v0.9.0 · 5345 in / 1075 out tokens · 28245 ms · 2026-05-10T17:10:15.367956+00:00 · methodology

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Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages

  1. [1]

    Baker and H

    A. Baker and H. Davenport,The equations3x 2 −2 =y 2 and8x 2 −7 =z 2, Quart. J. Math. Oxford Ser. (2)20(1969), 129–137

    work page 1969

  2. [2]

    Brown, Numbers, Author’s edition, 2023

    K. Brown, Numbers, Author’s edition, 2023

    work page 2023

  3. [3]

    Dujella,An absolute bound for the size of Diophantine m-tuples, J

    A. Dujella,An absolute bound for the size of Diophantine m-tuples, J. Number Theory 89 (2001), 126-150

    work page 2001

  4. [4]

    Dujella,There are only finitely many Diophantine quintuples, J

    A. Dujella,There are only finitely many Diophantine quintuples, J. Reine Angew. Math.566 (2004), 183–214

    work page 2004

  5. [5]

    Dujella,What is

    A. Dujella,What is ... a Diophantinem-tuple?, Notices Amer. Math. Soc.63(2016), 772– 774

    work page 2016

  6. [6]

    Dujella, Diophantinem-tuples and Elliptic Curves, Springer, Cham, 2024

    A. Dujella, Diophantinem-tuples and Elliptic Curves, Springer, Cham, 2024

    work page 2024

  7. [7]

    Dujella and M

    A. Dujella and M. Kazalicki,More on Diophantine sextuples, in: Number Theory - Dio- phantine problems, uniform distribution and applications, Festschrift in honour of Robert F. Tichy’s 60th birthday (C. Elsholtz, P. Grabner, Eds.), Springer-Verlag, Berlin, 2017, pp. 227–235

    work page 2017

  8. [8]

    Dujella, M

    A. Dujella, M. Kazalicki, M. Miki´ c and M. Szikszai,There are infinitely many rational Diophantine sextuples, Int. Math. Res. Not. IMRN 2017 (2) (2017), 490–508

    work page 2017

  9. [9]

    Dujella, M

    A. Dujella, M. Kazalicki and V. Petriˇ cevi´ c,Rational Diophantine sextuples containing two regular quadruples and one regular quintuple, Acta Mathematica Spalatensia,1(2020), 19– 27

    work page 2020

  10. [10]

    Dujella, M

    A. Dujella, M. Kazalicki and V. Petriˇ cevi´ c,There are infinitely many rational Diophantine sextuples with square denominators, J. Number Theory205(2019), 340–346

    work page 2019

  11. [11]

    Dujella and V

    A. Dujella and V. Petriˇ cevi´ c,Doubly regular Diophantine quadruples, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Math. RACSAM114(2020), Article 189

    work page 2020

  12. [12]

    Dujella and L

    A. Dujella and L. Szalay,Four squares from three numbers, preprint, arXiv:2506.14013, 2025

    work page arXiv 2025

  13. [13]

    Gibbs,Some rational Diophantine sextuples, Glas

    P. Gibbs,Some rational Diophantine sextuples, Glas. Mat. Ser. III41(2006), 195–203

    work page 2006

  14. [14]

    B. He, A. Togb´ e and V. Ziegler,There is no Diophantine quintuple, Trans. Amer. Math. Soc. 371(2019), 6665–6709

    work page 2019

  15. [15]

    T. L. Heath, Diophantus of Alexandria. A Study in the History of Greek Algebra. Powell’s Bookstore, Chicago; Martino Publishing, Mansfield Center, 2003

    work page 2003

  16. [16]

    Piezas,Extending rational Diophantine triples to sextuples, http://mathoverflow.net/questions/233538/extending-rational-diophantine-triples-to-sextuples

    T. Piezas,Extending rational Diophantine triples to sextuples, http://mathoverflow.net/questions/233538/extending-rational-diophantine-triples-to-sextuples

    work page

  17. [17]

    Stoll,Diagonal genus 5 curves, elliptic curves overQ(t), and rational diophantine quin- tuples, Acta Arith.190(2019), 239–261

    M. Stoll,Diagonal genus 5 curves, elliptic curves overQ(t), and rational diophantine quin- tuples, Acta Arith.190(2019), 239–261. Department of Mathematics, Faculty of Science, University of Zagreb, Bijeni ˇcka cesta 30, 10000 Zagreb, Croatia Email address, A. Dujella:duje@math.hr Department of Mathematics, Faculty of Science, University of Zagreb, Bijeni...

    work page 2019