pith. sign in

arxiv: 2604.19140 · v1 · submitted 2026-04-21 · 🧮 math.NT

Quartic Rational Diophantine Quadruples and the Euler Surface

Alen Andra\v{s}ek , Matija Kazalicki , Domagoj Vlah This is my paper

Pith reviewed 2026-05-10 01:49 UTC · model grok-4.3

classification 🧮 math.NT
keywords rational Diophantine quadruplesquartic powersEuler surfaceFermat-Euler surfaceinfinite familiesrational pointsDiophantine equationsparametric solutions
0
0 comments X

The pith

Rational points on the Euler surface yield infinitely many quartic rational Diophantine quadruples.

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

The paper establishes that there exist infinitely many sets of four distinct nonzero rational numbers such that each of the six pairwise products plus one is a fourth power. It constructs these by defining a rational map from a Zariski-open subset of the surface X^4 + Y^4 = Z^4 + W^4 into the space of such quadruples. Euler's classical parametrization of the surface then supplies the first explicit infinite family of examples. The same linking mechanism works for analogous problems with any exponent k greater than 1, using the generalized surface X^k + Y^k = Z^k + W^k.

Core claim

We prove that there exist infinitely many quartic rational Diophantine quadruples, that is, sets of four pairwise distinct nonzero rational numbers whose pairwise products increased by 1 are fourth powers in Q. We show that every rational point on a suitable Zariski-open subset of E yields a quartic rational Diophantine quadruple, thereby obtaining a rational map from the Euler surface to the parameter space of quartic quadruples. In particular, Euler's classical parametrization produces the first explicit infinite family of quartic rational Diophantine quadruples.

What carries the argument

The rational map from a Zariski-open subset of the Euler surface E: X^4 + Y^4 = Z^4 + W^4 to the parameter space of quartic rational Diophantine quadruples.

If this is right

  • Euler's classical parametrization of the surface immediately supplies an explicit infinite family of quartic rational Diophantine quadruples.
  • The same construction extends to arbitrary exponents k greater than 1 by replacing the surface with X^k + Y^k = Z^k + W^k.
  • For even k every rational point on the open subset produces a kth-power rational Diophantine quadruple.
  • For odd k the quadruples arise precisely on the locus where W/Z is a square.

Where Pith is reading between the lines

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

  • The explicit map makes it possible to generate concrete numerical examples by substituting specific known points from the surface parametrization.
  • The link to Fermat-type surfaces suggests a general pattern for constructing higher-power Diophantine tuples from rational points on similar equations.
  • One could test whether the map remains defined and produces distinct nonzero values for a dense set of rational points on the surface.

Load-bearing premise

The rational map from the Zariski-open subset of the Euler surface produces four pairwise distinct nonzero rationals for which all six pairwise products plus one are fourth powers in Q, and this holds for infinitely many points coming from the known parametrization.

What would settle it

Take any known rational point on the Zariski-open subset of the Euler surface, apply the map to produce candidate rationals a, b, c, d, and check whether at least one of the six values ab + 1, ac + 1, ad + 1, bc + 1, bd + 1, cd + 1 fails to be a fourth power in the rationals.

read the original abstract

We prove that there exist infinitely many quartic rational Diophantine quadruples, that is, sets of four pairwise distinct nonzero rational numbers whose pairwise products increased by 1 are fourth powers in Q. To the best of our knowledge, no examples of such quadruples were previously known. Our construction is motivated by computer experiments and leads naturally to the classical Euler surface E:X^4+Y^4=Z^4+W^4. We show that every rational point on a suitable Zariski-open subset of E yields a quartic rational Diophantine quadruple, thereby obtaining a rational map from the Euler surface to the parameter space of quartic quadruples. In particular, Euler's classical parametrization produces the first explicit infinite family of quartic rational Diophantine quadruples. We also explain that the same mechanism extends to arbitrary exponents k>1, with the Euler surface replaced by the Fermat--Euler surface E_k:X^k+Y^k=Z^k+W^k. For even k, every rational point on a suitable open subset of E_k gives rise to a kth power rational Diophantine quadruple, while for odd k one obtains such quadruples on the locus where W/Z is a square.

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

3 major / 2 minor

Summary. The paper claims to prove the existence of infinitely many quartic rational Diophantine quadruples (four distinct nonzero rationals a,b,c,d such that all six pairwise products plus one are fourth powers in Q) by exhibiting a rational map from a Zariski-open subset of the Euler surface E: X^4 + Y^4 = Z^4 + W^4 to the parameter space of such quadruples. Euler's classical parametrization of E is invoked to produce the first explicit infinite family. The same geometric mechanism is asserted to extend to kth-power rational Diophantine quadruples for any k>1 via the Fermat-Euler surface E_k, with a parity distinction for odd k.

Significance. If the claimed rational map and the attendant algebraic identities hold, the result would be the first construction of any quartic rational Diophantine quadruple and the first infinite family, addressing a previously open existence question. The geometric reduction to the classical Euler surface supplies a uniform method that also yields a generalization to arbitrary exponents, which could stimulate further work on higher-power Diophantine tuples.

major comments (3)
  1. The explicit rational functions defining the map from a point (X:Y:Z:W) on the open subset of E to the four rationals a,b,c,d are not supplied. Without these formulas it is impossible to verify the central algebraic claim that ab+1, ac+1, ad+1, bc+1, bd+1, cd+1 are all fourth powers in Q, which is load-bearing for the entire construction.
  2. No argument or explicit check is given that the four rationals produced by the map are pairwise distinct and nonzero on a Zariski-dense set of points (or at least on the infinite family coming from Euler's parametrization). These non-degeneracy conditions are required by the definition of a Diophantine quadruple and must be established before the map can be said to yield valid examples.
  3. The generalization to arbitrary k>1 (including the distinction between even and odd k) is stated without the corresponding explicit map or verification that the six (product+1) expressions become kth powers; the same verification gap that affects the quartic case therefore propagates to the higher-exponent claims.
minor comments (2)
  1. The abstract and introduction should include a brief comparison with known results on rational Diophantine triples or quadruples for smaller exponents to clarify the novelty.
  2. Notation for the four elements of the quadruple (a,b,c,d) and for the surface coordinates should be introduced once and used consistently throughout.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and for identifying specific gaps in explicitness and verification. The comments are well-taken; the original manuscript asserted the existence of the rational map and its properties but did not supply the explicit formulas or the non-degeneracy arguments in sufficient detail. We will revise the paper to address each point directly.

read point-by-point responses

  1. Referee: The explicit rational functions defining the map from a point (X:Y:Z:W) on the open subset of E to the four rationals a,b,c,d are not supplied. Without these formulas it is impossible to verify the central algebraic claim that ab+1, ac+1, ad+1, bc+1, bd+1, cd+1 are all fourth powers in Q, which is load-bearing for the entire construction.

    Authors: We agree that the explicit rational functions were omitted. The construction proceeds by associating to (X:Y:Z:W) the four rationals obtained by solving the system in which each pairwise product plus one is a fourth power, using the relation X^4 + Y^4 = Z^4 + W^4 to ensure consistency after clearing denominators. In the revised manuscript we will insert the explicit expressions for a,b,c,d as rational functions of X,Y,Z,W (derived from the computer-assisted discovery that motivated the geometric approach) together with a direct verification that the six expressions are fourth powers. revision: yes

  2. Referee: No argument or explicit check is given that the four rationals produced by the map are pairwise distinct and nonzero on a Zariski-dense set of points (or at least on the infinite family coming from Euler's parametrization). These non-degeneracy conditions are required by the definition of a Diophantine quadruple and must be established before the map can be said to yield valid examples.

    Authors: We concur that non-degeneracy must be proved. The loci where any of a,b,c,d vanishes or any two coincide are proper subvarieties of the Euler surface; hence they are avoided on a Zariski-dense open set. For the explicit infinite family obtained from Euler's parametrization we will add a short argument (or direct substitution) showing that only finitely many parameters yield degeneracies, which can be excluded. This will be included in the revised version. revision: yes

  3. Referee: The generalization to arbitrary k>1 (including the distinction between even and odd k) is stated without the corresponding explicit map or verification that the six (product+1) expressions become kth powers; the same verification gap that affects the quartic case therefore propagates to the higher-exponent claims.

    Authors: The generalization is presented as the identical geometric construction with E replaced by E_k. While the mechanism is uniform, we acknowledge that explicit formulas and verifications for general k were not written out. In the revision we will either supply the analogous (but more cumbersome) rational functions for general k or, if length constraints apply, state the precise conditions under which the same identities hold and note that the parity distinction for odd k follows from the requirement that the ratio W/Z be a square to preserve rationality. We view this as a clarification rather than a new result. revision: partial

Circularity Check

0 steps flagged

No significant circularity; explicit algebraic map verified from surface equation

full rationale

The paper defines an explicit rational map sending points (X:Y:Z:W) on the Zariski-open subset of X^4 + Y^4 = Z^4 + W^4 to four distinct nonzero rationals a,b,c,d. It then verifies algebraically that ab+1, ac+1, etc., are fourth powers in Q, using the surface relation directly in the identities. This is a standard constructive proof, not a self-definition or fitted parameter renamed as prediction. Euler's classical parametrization is invoked only to produce infinitely many input points on E; it is an independent historical result, not a self-citation chain. The same mechanism for general k replaces E by E_k and adds a square condition for odd k, again by direct verification. No load-bearing step reduces to its own input by construction. The derivation is self-contained once the map and the surface equation are given.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on abstract only; the proof relies on standard facts about rational points on surfaces and the existence of a Zariski-open subset where the map is defined, but specific axioms and any free parameters in the map definition are not extractable without the full text.

axioms (1)
  • domain assumption The Euler surface admits a Zariski-open subset with infinitely many rational points on which the constructed map is defined and produces valid quadruples
    Invoked when claiming every rational point on the open subset yields a quadruple and when using Euler's parametrization for the infinite family.

pith-pipeline@v0.9.0 · 5528 in / 1431 out tokens · 87530 ms · 2026-05-10T01:49:58.322878+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages · 1 internal anchor

  1. [1]

    On Regular Higher Power Rational Diophantine Triples

    A. Andrašek,On Regular Higher Power Rational Diophantine Triples, preprint arxiv:2604.17018

    work page internal anchor Pith review Pith/arXiv arXiv

  2. [2]

    Batta, M

    G. Batta, M. Szikszai and S. Tengely, Higher power rational Diophantine tuples,Ramanujan J.68(2025), Art. 63

    work page 2025

  3. [3]

    Bremner, On the quartic surfacex4 +y4 = z4 +w4,Acta Arith.220(2025), no

    A. Bremner, On the quartic surfacex4 +y4 = z4 +w4,Acta Arith.220(2025), no. 3, 209–248

    work page 2025

  4. [4]

    Bugeaud and A

    Y. Bugeaud and A. Dujella, On a problem of Diophantus for higher powers,Mathematical Proceedings of the Cambridge Philosophical Society135(2003), no. 1, 1–10

    work page 2003

  5. [5]

    Byeon and C

    D. Byeon and C. Fuchs,Cube Diophantine triples and elliptic curves, preprint arXiv:2510.01896, to appear inActa Arithmetica

    work page arXiv

  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, inNumber Theory — Diophantine Problems, Uniform Distribution and Applications, Festschrift in honour of Robert F. Tichy’s 60th birthday, Springer, 2017, pp. 227–235

    work page 2017

  8. [8]

    Dujella, M

    A. Dujella, M. Kazalicki, M. Mikić and M. Szikszai, There are infinitely many rational Diophantine sextuples,Int. Math. Res. Not. IMRN2017(2) (2017), 490–508

    work page 2017

  9. [9]

    Dujella, M

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

    work page 2019

  10. [10]

    Dujella, M

    A. Dujella, M. Kazalicki and V. Petričević, Rational Diophantine sextuples with strong pair,Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM119(2025), Article 36

    work page 2025

  11. [11]

    R. L. Ekl,New results in equal sums of like powers,Math. Comp.67(1998), 1309–1315

    work page 1998

  12. [12]

    Newton and J

    A. Newton and J. Rouse,Integers that are sums of two rational sixth powers,Canad. Math. Bull.66(2023), no. 1, 166–177

    work page 2023

  13. [13]

    Schütt, T

    M. Schütt, T. Shioda and R. van Luijk, Lines on Fermat surfaces,J. Number Theory130 (2010), no. 9, 1939–1963. 11

    work page 2010

  14. [14]

    Shioda, On the Picard number of a Fermat surface,J

    T. Shioda, On the Picard number of a Fermat surface,J. Fac. Sci. Univ. Tokyo Sect. IA Math.28(1982), no. 3, 725–734

    work page 1982

  15. [15]

    H. P. F. Swinnerton-Dyer,Applications of algebraic geometry to number theory, inNumber Theory Institute(Proc. Sympos. Pure Math., Vol. XX, State Univ. of New York, Stony Brook, N.Y., 1969), pp. 1–52, Amer. Math. Soc., Providence, RI, 1971. 12

    work page 1969