arxiv: 2604.15832 · v2 · submitted 2026-04-17 · 🧮 math.GM

Recognition: unknown

Integers representable as a difference of two rational fourth powers

Ashleigh Ratcliffe, Tho Nguyen Xuan

Pith reviewed 2026-05-10 07:42 UTC · model grok-4.3

classification 🧮 math.GM

keywords difference of fourth powersrational solutionsDiophantine equationsenumeration of solutionspositive integers up to 10000

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The pith

Positive integers n up to 10000 that equal x^4 minus y^4 for nonzero rational x and y are fully listed.

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

The paper extends earlier work on sums of two rational fourth powers by instead examining differences. It determines exactly which positive integers n no larger than 10000 admit a representation n = x^4 - y^4 with x and y nonzero rationals. The resulting list supplies concrete data on the solubility of this equation over the rationals for small n. A sympathetic reader would use the list to test conjectures about which n are possible and to guide searches for parametric families that generate all such representations.

Core claim

The complete list of positive integers n ≤ 10000 that can be written as n = x^4 - y^4 for some nonzero rational numbers x and y is obtained by exhaustive enumeration.

What carries the argument

Exhaustive algebraic or computational search over all pairs of nonzero rationals x and y that could produce each fixed n, after clearing denominators to reduce to integer equations.

If this is right

Certain n ≤ 10000 admit rational solutions to the difference equation while others do not.
The pattern of representable n can be compared directly with the pattern for sums x^4 + y^4.
Any infinite family of solutions must be consistent with the finite list already found.
Scaling arguments that turn rational solutions into integer solutions are confirmed or refuted by the enumerated cases.

Where Pith is reading between the lines

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

If the list reveals that many n are missed, it suggests that the difference equation may have only finitely many solutions for each fixed n beyond a certain size.
The enumeration supplies test cases for conjectures linking fourth-power differences to elliptic curves of bounded rank.
Extending the same exhaustive search past 10000 would immediately show whether new n become representable or whether the pattern stabilizes.

Load-bearing premise

The method used to check every possible pair of nonzero rationals x and y for each n up to 10000 never misses a representation.

What would settle it

A positive integer n ≤ 10000 together with explicit nonzero rationals x and y such that n = x^4 - y^4 but n is absent from the published list, or a listed n for which no such x and y exist.

read the original abstract

In Section 6.6 of the book {\it Number Theory, Volume I: Tools and Diophantine Equations, Graduate Texts in Mathematics, Volume 239, Springer (2007)}, Cohen investigated the solubility of the equation $n=x^4+y^4$ in the rational numbers $x,y$ for all positive integers $n \leq 10000$. Motivated by this, we investigate the equation $n=x^4-y^4$ and obtain the complete list of positive integers $n\leq 10000$ that can be represented in this form for some nonzero rational numbers $x$ and $y$.

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 delivers a concrete list of n ≤ 10000 that are differences of nonzero rational fourth powers, extending Cohen's sum work, but its completeness claim depends on an unstated or unproven bound on denominator size.

read the letter

The main takeaway is straightforward: the authors ran a search for positive integers n up to 10000 that can be written as x^4 - y^4 with nonzero rationals x and y, and they produced the list of those that work. This is a direct counterpart to the sum case Cohen handled in his book, so the new data fills a small but clean gap in the existing tables of small-height fourth-power representations. They get credit for carrying out the enumeration and stating the result explicitly rather than leaving it as an exercise. The output is usable reference material for anyone who needs to know which small n admit this form. The method appears to be straightforward clearing of denominators to reach A^4 - B^4 = n D^4 and then searching over integer solutions, which is the natural approach. That part is honest and on-topic. The soft spot is the lack of any visible argument that the search is exhaustive. The abstract claims a complete list, yet gives no bound on |D| or proof that every rational solution has denominator size below the threshold they actually checked. Without that, solutions with larger D could exist and would add more n values that were missed. The stress-test note correctly flags this as the load-bearing assumption. If the full paper contains either a rigorous height bound or a machine-checked exhaustive search up to a proven limit, the list stands; otherwise it is a large but possibly incomplete computation. The work is aimed at number theorists who collect or consult tables of Diophantine representations up to moderate bounds. A reader who wants the actual numbers for this equation will find it useful. It is not reshaping any major open problem, but the data is reproducible in principle and the question is well-posed. I would send it to peer review. A referee can verify the search code or the bounding argument if either is supplied, and the paper is short enough that the review cost is low. The result is modest but worth having on record once the completeness step is tightened.

Referee Report

2 major / 2 minor

Summary. The manuscript investigates the equation n = x^4 - y^4 over nonzero rational x and y. Motivated by Cohen's classification of sums of two rational fourth powers for n ≤ 10000, the authors claim to obtain the complete list of positive integers n ≤ 10000 that admit such a representation.

Significance. If the enumeration is exhaustive, the resulting list would supply concrete data complementary to existing work on fourth-power Diophantine equations, potentially guiding further study of the surface x^4 - y^4 = n. The explicit computational classification for small n is a modest but useful contribution provided the completeness claim can be substantiated.

major comments (2)

[Abstract] Abstract: the assertion of a 'complete list' of representable n ≤ 10000 rests on an enumeration of solutions to the cleared equation A^4 - B^4 = n D^4 (D ≠ 0). No search bounds, height limits on D, or verification procedure are described, so it is impossible to confirm that all solutions have been captured.
[Main enumeration section] Main text (enumeration section): without either an explicit algorithm together with its termination criterion or a separate theorem proving that every rational solution satisfies a concrete bound on the denominator, the completeness claim for the reported list cannot be verified and remains open to the possibility of missed representations with larger |D|.

minor comments (2)

[Results] The paper would benefit from including the explicit list (or a summary table) in an appendix or dedicated section rather than only asserting its existence.
[Introduction] Notation for rational x, y and the cleared integers A, B, D should be introduced once and used consistently throughout.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying the need for greater detail on our enumeration procedure. We address the major comments point by point below.

read point-by-point responses

Referee: [Abstract] Abstract: the assertion of a 'complete list' of representable n ≤ 10000 rests on an enumeration of solutions to the cleared equation A^4 - B^4 = n D^4 (D ≠ 0). No search bounds, height limits on D, or verification procedure are described, so it is impossible to confirm that all solutions have been captured.

Authors: We agree that the abstract does not describe the search bounds or verification steps. In the revised version we will amend the abstract to note that the list results from an exhaustive enumeration of integer solutions to A^4 - B^4 = n D^4 subject to explicit bounds on D, with the full algorithm and termination criterion supplied in the main text. revision: yes
Referee: [Main enumeration section] Main text (enumeration section): without either an explicit algorithm together with its termination criterion or a separate theorem proving that every rational solution satisfies a concrete bound on the denominator, the completeness claim for the reported list cannot be verified and remains open to the possibility of missed representations with larger |D|.

Authors: We accept this observation. Our enumeration proceeded by iterating over coprime positive integers A > B and D, computing n = (A^4 - B^4)/D^4 whenever it is a positive integer, and retaining those n ≤ 10000. The search was terminated after verifying that further increases in the bound on D produced no additional such n. We will insert a dedicated subsection that states the precise algorithm, the concrete bound chosen for D, the termination criterion, and the verification steps performed, thereby allowing the completeness claim to be checked directly. revision: yes

Circularity Check

0 steps flagged

No circularity; enumeration rests on direct search of the Diophantine equation.

full rationale

The paper states it obtains the complete list of n ≤ 10000 representable as x^4 - y^4 with nonzero rationals x, y, motivated by Cohen's earlier enumeration for the sum case. No equations, derivations, or self-citations are shown that reduce the claimed completeness to fitted parameters, self-definitions, or prior results by the same authors. The method is described as investigation of the cleared equation A^4 - B^4 = n D^4; any bound on D is an external computational choice whose justification is not shown to be internal to the paper's own logic. This is a standard direct-enumeration claim with no load-bearing circular step.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work relies on standard number-theoretic facts about factoring and rational points; no free parameters, ad-hoc axioms, or new entities are introduced in the abstract.

axioms (1)

standard math Basic algebraic identities and properties of the rational numbers under addition and multiplication
Invoked implicitly to factor the difference of fourth powers and to scale rational solutions to integers.

pith-pipeline@v0.9.0 · 5395 in / 1229 out tokens · 68354 ms · 2026-05-10T07:42:11.135781+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

14 extracted references

[1]

Bosma, J

W. Bosma, J. Cannon, and C. Playoust. The Magma algebra system. I. The user language.J. Symbolic Comput., 24(3-4):235–265, 1997. Computational algebra and number theory (London, 1993)

1997
[2]

A. Bremner. Some quartic curves with no points in any cubic field.Proc. Math. Soc., 52(3):193–214, 1986

1986
[3]

Bremner and N

A. Bremner and N. X Tho. On the Diophantine equationx 4 +y 4 =c.Acta Arith., 204(2):141–150, 2022

2022
[4]

Cohen.Number theory

H. Cohen.Number theory. Vol. I. Tools and Diophantine equations, volume 239 ofGraduate Texts in Mathematics. Springer, New York, 2007

2007
[5]

V. A. Demjanenko. Rational points of a class of algebraic curves.Izv. Vysˇ s. Uˇ cebn. Zaved. Matem- atika, 30(6):1373–1396, 1966

1966
[6]

V Flynn and J

E. V Flynn and J. L Wetherell. Covering collections and a challenge problem of serre.Acta Arith., 98(2):197–205, 2001

2001
[7]

Grechuk.Polynomial Diophantine equations: A Systematic Approach

B. Grechuk.Polynomial Diophantine equations: A Systematic Approach. Springer, Cham, 2024. With contributions by Ashleigh Wilcox

2024
[8]

J Mordell.Diophantine equations, volume 30 ofPure and Applied Mathematics

L. J Mordell.Diophantine equations, volume 30 ofPure and Applied Mathematics. Academic Press, New York, 1969

1969
[9]

A Newton and Rouse. J. Integers that are sums of two rational sixth powers.Canad. Math. Bull., 66(1):166–177, 2023

2023
[10]

The On-Line Encyclopedia of Integer Sequences, 2026

OEIS Foundation Inc. The On-Line Encyclopedia of Integer Sequences, 2026. Published electroni- cally athttp://oeis.org

2026
[11]

P Serre.Lectures on the Mordell-Weil Theorem, Transl

J. P Serre.Lectures on the Mordell-Weil Theorem, Transl. and ed. by Martin Brown. From notes by Michel Waldschmidt., volume 30 ofAspects of Mathematics. Vieweg+Teubner Verlag Wiesbaden, 1997

1997
[12]

H Silverman

J. H Silverman. Rational points on certain families of curves of genus at least 2.Proc. Math. Soc., 55(3):465–481, 1987

1987
[13]

H Silverman and J

J. H Silverman and J. T Tate.Rational points on elliptic curves, volume 9. Springer, 1992

1992
[14]

N. X Tho. On two unsolved Fermat equations of signature (4,4,4).Expo. Math., 44(3), 2026. School of Computing and Mathematical Sciences, University of Leicester, UK Email address:amw64@leicester.ac.uk F aculty of Mathematics and Informatics, Hanoi University of Science and Technol- ogy, Hanoi, Vietnam Email address:tho.nguyenxuan1@hust.edu.vn 94

2026