Quartic Rational Diophantine Quadruples and the Euler Surface

TL;DR

This paper proves the existence of infinitely many quartic rational Diophantine quadruples, constructing them via rational points on the Euler surface and extending the method to higher exponents.

Contribution

It introduces a novel construction linking the Euler surface to quartic rational Diophantine quadruples, providing the first explicit infinite family of such quadruples.

Findings

Infinite quartic rational Diophantine quadruples exist.

A rational map from the Euler surface generates these quadruples.

The method extends to arbitrary exponents k > 1.

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.