Weighted Fruit Diophantine Equations and Hyperelliptic Curves

Apeksha Sanghi; Jewel Mahajan

arxiv: 2606.27322 · v1 · pith:U3BT4RT4new · submitted 2026-06-25 · 🧮 math.NT

Weighted Fruit Diophantine Equations and Hyperelliptic Curves

Jewel Mahajan , Apeksha Sanghi This is my paper

Pith reviewed 2026-06-26 02:13 UTC · model grok-4.3

classification 🧮 math.NT

keywords Diophantine equationshyperelliptic curvesinteger solutionsinsolvabilityrational torsionweighted equationsNagell-Lutz theorem

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The pith

For primes l ≡ 3 mod 4 and b = a(2cmn)^d - l c^s t^{2q}, the equation ax^d - c(m²y² + n²z²) + xyz - b = 0 has no integer solutions except certain x residues modulo 4l, under parameter hypotheses.

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

The paper studies the weighted fruit Diophantine equation ax^d - c(m²y² + n²z²) + xyz - b = 0 and generalizes earlier results on its integer solutions. It establishes that when b incorporates a factor of a prime l congruent to 3 modulo 4, the equation admits no integer solutions outside specific residue classes of x modulo 4l. The same conclusion holds when l is replaced by an odd power of l, and some insolvability statements are given for the case l = -1. Explicit exceptional classes are computed for small such primes, yielding complete insolvability when l = 3. The work also produces bounds on positive solutions and maps the insolvability statements to the non-existence of certain rational torsion points on associated hyperelliptic curves.

Core claim

Subject to specific hypotheses on the parameters, for any prime l ≡ 3 (mod 4) and b = a (2 c m n)^d - l c^s t^{2q}, the equation ax^d - c(m²y² + n²z²) + xyz - b = 0 has no integer solutions except for certain residue classes of x modulo 4l. An analogous result holds for odd powers of l. For l = 3 this produces complete insolvability outside the listed classes. The equation is further linked to a family of hyperelliptic curves whose rational torsion points are constrained by the same insolvability statements via Grant's analogue of the Nagell–Lutz theorem.

What carries the argument

The auxiliary parameter b constructed to include the factor l c^s t^{2q} (with l ≡ 3 mod 4), which is used to obtain modular obstructions to integer solutions; the same b choice is then used to define a family of hyperelliptic curves whose torsion is controlled by the same obstructions.

If this is right

For l in {3, 7, 11, 19} the exceptional residue classes modulo 4l are determined explicitly.
When l = 3 the equation is insoluble in integers except possibly in those classes.
Weakening the hypotheses still yields non-existence provided any solutions obey stated coprimality conditions.
Positive integer solutions, when they exist, satisfy explicit bounds derived from the equation.
The associated hyperelliptic curves have no rational torsion points outside those corresponding to the exceptional classes.

Where Pith is reading between the lines

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

The modular obstruction technique may extend to other weighted Diophantine equations whose right-hand side can be adjusted by a similar prime factor.
The torsion results on the hyperelliptic curves could be combined with descent methods to bound the full set of rational points rather than only torsion.
The complete insolvability for l = 3 suggests that similar constructions might produce infinite families of Diophantine equations with no integer solutions at all.

Load-bearing premise

The parameters a, c, m, n, d, s, q, t must satisfy the specific hypotheses required for the main insolvability theorem to apply.

What would settle it

An explicit integer triple (x, y, z) satisfying the equation for l = 3, with b defined as above, and with x outside the listed residue classes modulo 12, would show the central claim false.

read the original abstract

We study the weighted fruit Diophantine equation $ax^{d} - c\bigl(m^{2}y^{2}+n^{2}z^{2}\bigr) + xyz - b = 0$, generalising previous work by Majumdar--Sury, Vaishya--Sharma, and Prakash--Chakraborty. Subject to specific hypotheses on the parameters, our main result shows that for any prime $l \equiv 3 \pmod 4$ and $b = a (2 c m n)^{d} - l\, c^{s}t^{2q}$, the above equation has no integer solutions except for certain residue classes of $x$ modulo $4l$. An analogous result also holds when $l$ is replaced by an odd power of $l$ in the definition of $b$. We prove some insolvability results for $l=-1$. By applying the main result to the small values of $l$, such as $l \in \{3, 7, 11, 19\}$, we explicitly determine the exceptional residue classes outside of which the equation has no solutions. In particular, for $l = 3$, this yields complete insolvability, and weakening these hypotheses still yields non-existence results, though with specific coprimality restrictions on any possible solutions. We also consider a more general variant of the above Diophantine equation and provide some insolvability results. Subsequently, we establish bounds for the positive solutions of the aforementioned equation. Finally, by associating a family of hyperelliptic curves with the equation under consideration and applying Grant's analogue of the Nagell--Lutz theorem, we translate these insolvability results into results about their rational torsion points.

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

Extends prior Diophantine work with a specific b parametrization for primes l ≡ 3 mod 4, gives explicit small-l cases, and translates to hyperelliptic torsion via Grant, but the parameter hypotheses remain vague.

read the letter

The paper generalizes earlier results on equations like those in Majumdar-Sury and Vaishya-Sharma by introducing a weighted form ax^d - c(m²y² + n²z²) + xyz - b = 0 and fixing b = a(2cmn)^d - l c^s t^{2q} for primes l ≡ 3 mod 4. It claims no integer solutions outside certain residue classes mod 4l, with an analogous statement for odd powers of l, plus complete non-existence for l=3 and some results for l=-1. Small cases l=3,7,11,19 get explicit residue classes, bounds on positive solutions are stated, and a more general variant is considered. The final step associates hyperelliptic curves and applies Grant's Nagell-Lutz analogue to turn the insolvability into statements about rational torsion points.

What works is the direct application to concrete small l and the clean translation to curves; those parts give explicit, usable output rather than just existence of some obstruction. The modular argument mod 4l looks like a standard descent or local check that fits the cited literature.

The soft spot is the repeated appeal to unspecified hypotheses on a, c, m, n, d, s, q, t. Without seeing how restrictive those are or whether they are chosen to make the modular arithmetic close, it is difficult to judge the actual scope. The abstract alone does not show the full derivations or error handling, so the central claims stay conditional. The curve link is a straightforward rephrasing rather than independent new machinery.

This is incremental number-theory work aimed at specialists in Diophantine equations and torsion on curves. Readers who already follow the cited papers will get concrete new examples and residue-class lists. It has enough explicit theorems and a verifiable method to deserve referee time rather than a desk reject.

Referee Report

0 major / 3 minor

Summary. The paper studies the weighted fruit Diophantine equation ax^d - c(m²y² + n²z²) + xyz - b = 0, generalizing prior work. Under specific hypotheses on the parameters, it proves that for any prime l ≡ 3 (mod 4) with b = a(2cmn)^d - l c^s t^{2q}, there are no integer solutions except certain residue classes of x modulo 4l; an analogous statement holds when l is replaced by an odd power of l. Additional insolvability results are given for l = -1, explicit exceptional classes are determined for l in {3,7,11,19} (complete insolvability for l=3), bounds on positive solutions are established, and the equation is associated to a family of hyperelliptic curves whose rational torsion is analyzed via Grant's Nagell-Lutz analogue.

Significance. If the derivations hold, the work supplies concrete generalizations of known Diophantine insolvability theorems and a direct translation of those results into statements about rational torsion on hyperelliptic curves. The explicit computations for small primes and the provision of solution bounds constitute practical additions to the literature on these equations.

minor comments (3)

The abstract states that the main theorem holds 'subject to specific hypotheses on the parameters' but does not list them; the paper should state the full list of hypotheses in the statement of the main theorem (presumably §2 or §3) so that the scope is immediately clear.
The transition from the Diophantine insolvability results to the hyperelliptic-curve torsion statements (final section) would benefit from an explicit description of the curve family and the map from solutions of the equation to points on the curve.
Notation for the exponents s, q, t in the definition of b is introduced without prior definition; a short preliminary subsection collecting all parameter assumptions would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work on the weighted fruit Diophantine equation and the recommendation of minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation relies on modular obstructions (mod 4l for l ≡ 3 mod 4) and descent applied to the weighted fruit equation under externally stated parameter hypotheses, generalizing results by Majumdar–Sury et al. (distinct authors). The hyperelliptic curve translation invokes Grant's Nagell–Lutz analogue as an independent external tool. No equation reduces to a fitted parameter renamed as prediction, no self-citation chain bears the central claim, and no ansatz or uniqueness result is smuggled via prior work by these authors. The results are conditional and falsifiable outside the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review; no free parameters or invented entities are mentioned. Axioms appear to be standard number-theoretic facts about primes and residues invoked in the proofs.

axioms (1)

standard math Standard arithmetic properties of primes l ≡ 3 mod 4 and related quadratic non-residues or modular arithmetic facts
Invoked to establish the residue-class exceptions and insolvability for the stated b.

pith-pipeline@v0.9.1-grok · 5850 in / 1292 out tokens · 25796 ms · 2026-06-26T02:13:57.513505+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

21 extracted references · 1 linked inside Pith

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Pith/arXiv arXiv 2013
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Generalized fruit diophantine equation and hyperelliptic curves

Om Prakash and Kalyan Chakraborty. Generalized fruit diophantine equation and hyperelliptic curves. Monatsh. Math., 203(3):667–676, 2024

2024
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Subburam and A

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On the diophantine equation ax3 +by+c=xyz.Funct

Sivanarayanapandian Subburam and Ravindrananathan Thangadurai. On the diophantine equation ax3 +by+c=xyz.Funct. Approx. Comment. Math., 53(1):167–175, 2015

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On the positive integral solution of the Diophantine equationx3 + by + 4 −xyz = 0.Afr

Alain Togb´ e. On the positive integral solution of the Diophantine equationx3 + by + 4 −xyz = 0.Afr. Diaspora J. Math., 8:81–89, 2009

2009
[20]

A class of fruit Diophantine equations.Monatsh

Lalit Vaishya and Richa Sharma. A class of fruit Diophantine equations.Monatsh. Math., 199(4):899–907, 2022

2022
[21]

L’arithm´ etique sur les courbes alg´ ebriques.Acta Math., 52(1):281–315, 1929

Andr´ e Weil. L’arithm´ etique sur les courbes alg´ ebriques.Acta Math., 52(1):281–315, 1929. Department of Mathematical Sciences, Indian Institute of Science Education and Research (IISER) Berhampur, Ganjam, Odisha 760003, India Email address:jewelmahajan421@gmail.com Department of Mathematics, Indian Institute of Science (IISc), CV Raman Road, Bengaluru...

1929

[1] [1]

Most hyperelliptic curves over Q have no rational points

Manjul Bhargava. Most hyperelliptic curves over Q have no rational points. Preprint, arXiv:1308.0395, 2013. WEIGHTED FRUIT DIOPHANTINE EQUA TIONS AND HYPERELLIPTIC CUR VES23

Pith/arXiv arXiv 2013

[2] [2]

Burton.Elementary number theory

David M. Burton.Elementary number theory. W. C. Brown Publishers, Dubuque, IA, second edition, 1989

1989

[3] [3]

Springer-Verlag, Berlin, 1993

Henri Cohen.A course in computational algebraic number theory, volume 138 ofGraduate Texts in Mathematics. Springer-Verlag, Berlin, 1993

1993

[4] [4]

https: //www.sagemath.org

The Sage Developers.SageMath, the Sage Mathematics Software System (Version 10.4), 2024. https: //www.sagemath.org

2024

[5] [5]

Endlichkeitss¨ atze f¨ ur abelsche Variet¨ aten ¨ uber Zahlk¨ orpern.Invent

Gerd Faltings. Endlichkeitss¨ atze f¨ ur abelsche Variet¨ aten ¨ uber Zahlk¨ orpern.Invent. Math., 73(3):349–366, 1983

1983

[6] [6]

On an analogue of the Lutz-Nagell theorem for hyperelliptic curves.J

David Grant. On an analogue of the Lutz-Nagell theorem for hyperelliptic curves.J. Number Theory, 133(3):963–969, 2013

2013

[7] [7]

Greenfield and Daniel Drucker

Gary R. Greenfield and Daniel Drucker. On the discriminant of a trinomial.Linear Algebra Appl., 62:105–112, 1984

1984

[8] [8]

On the positive integral solution of the Diophantine equation x3 + by + 1−xyz= 0.Bull

Florian Luca and Alain Togb´ e. On the positive integral solution of the Diophantine equation x3 + by + 1−xyz= 0.Bull. Malays. Math. Sci. Soc. (2), 31(2):129–134, 2008

2008

[9] [9]

Sur l’´ equation y2 = x3 −Ax−B dans les corps p-adiques.J

´Elisabeth Lutz. Sur l’´ equation y2 = x3 −Ax−B dans les corps p-adiques.J. Reine Angew. Math., 177:238–247, 1937

1937

[10] [10]

Dipramit Majumdar and B. Sury. 107.15 Fruit diophantine equation.Math. Gaz., 107(569):302–306, 2023

2023

[11] [11]

L. J. Mordell. On the rational solutions of the indeterminate equations of the third and fourth degrees. Proc. Cambridge Philos. Soc., 21:179–192, 1922

1922

[12] [12]

Solution de quelques probl` emes dans la th´ eorie arithm´ etique des cubiques planes du premier genre.Skr

Trygve Nagell. Solution de quelques probl` emes dans la th´ eorie arithm´ etique des cubiques planes du premier genre.Skr. Norske Vid.-Akad. Oslo I, 1935(1):1–25, 1935

1935

[13] [13]

Generalized fruit diophantine equation and hyperelliptic curves

Om Prakash and Kalyan Chakraborty. Generalized fruit diophantine equation and hyperelliptic curves. Monatsh. Math., 203(3):667–676, 2024

2024

[14] [14]

Silverman.The arithmetic of elliptic curves, volume 106 ofGraduate Texts in Mathematics

Joseph H. Silverman.The arithmetic of elliptic curves, volume 106 ofGraduate Texts in Mathematics. Springer, Dordrecht, second edition, 2009

2009

[15] [15]

Silverman and John T

Joseph H. Silverman and John T. Tate.Rational points on elliptic curves. Undergraduate Texts in Mathematics. Springer, Cham, second edition, 2015

2015

[16] [16]

Subburam and R

S. Subburam and R. Thangadurai. On the Diophantine equation x3 + by + 1 −xyz = 0.C. R. Math. Acad. Sci. Soc. R. Can., 36(1):15–19, 2014

2014

[17] [17]

Subburam and A

S. Subburam and A. Togb´ e. On the diophantine equationax3 + by + c = xyz.Afr. Mat., 28(1-2):9–21, 2017

2017

[18] [18]

On the diophantine equation ax3 +by+c=xyz.Funct

Sivanarayanapandian Subburam and Ravindrananathan Thangadurai. On the diophantine equation ax3 +by+c=xyz.Funct. Approx. Comment. Math., 53(1):167–175, 2015

2015

[19] [19]

On the positive integral solution of the Diophantine equationx3 + by + 4 −xyz = 0.Afr

Alain Togb´ e. On the positive integral solution of the Diophantine equationx3 + by + 4 −xyz = 0.Afr. Diaspora J. Math., 8:81–89, 2009

2009

[20] [20]

A class of fruit Diophantine equations.Monatsh

Lalit Vaishya and Richa Sharma. A class of fruit Diophantine equations.Monatsh. Math., 199(4):899–907, 2022

2022

[21] [21]

L’arithm´ etique sur les courbes alg´ ebriques.Acta Math., 52(1):281–315, 1929

Andr´ e Weil. L’arithm´ etique sur les courbes alg´ ebriques.Acta Math., 52(1):281–315, 1929. Department of Mathematical Sciences, Indian Institute of Science Education and Research (IISER) Berhampur, Ganjam, Odisha 760003, India Email address:jewelmahajan421@gmail.com Department of Mathematics, Indian Institute of Science (IISc), CV Raman Road, Bengaluru...

1929