Polynomial parametrization of Pythagorean quadruples, quintuples and sextuples

TL;DR

This paper develops polynomial parametrizations for Pythagorean quadruples, quintuples, and sextuples, providing explicit formulas and exploring their connections to Descartes quadruples, advancing the understanding of integer solutions to these classical Diophantine equations.

Contribution

It introduces explicit polynomial parametrizations for Pythagorean quadruples, quintuples, and sextuples, including a novel integer-valued polynomial for Pythagorean quintuples.

Findings

Parametrization of Pythagorean quadruples using polynomial functions.

Explicit polynomial formulas for Pythagorean quintuples and sextuples.

Connection between Pythagorean quadruples and Descartes quadruples via polynomial parametrization.

Abstract

A Pythagorean n-tuple is an integer solution of x_1^2+...+x_{n-1}^2=x_n^2. For n=4 and n=6, the Pythagorean n-tuples admit a parametrization by a single n-tuple of polynomials with integer coefficients (which is impossible for n=3). For n=5, there is an integer-valued polynomial Pythagorean 5-tuple which parametrizes Pythagorean quintuples (similar to the case n=3). Pythagorean quadruples are closely related to (integer) Descartes quadruples (solutions of 2(b_1^2+b_2^2+b_3^2+b_4^2) = (b_1+b_2+b_3+b_4)^2), which we also parametrize by a Descartes quadruple of polynomials with integer coefficients.