Parametrization of Pythagorean triples by a single triple of polynomials

TL;DR

This paper demonstrates that Pythagorean triples cannot be parametrized by a single polynomial triple with integer coefficients, but can be parametrized by a single triple of integer-valued polynomials, highlighting a nuanced distinction in polynomial parametrizations.

Contribution

It proves the impossibility of a single polynomial triple with integer coefficients for parametrization and establishes the existence of such a parametrization using integer-valued polynomials.

Findings

No single polynomial triple with integer coefficients suffices.

A single triple of integer-valued polynomials can parametrize Pythagorean triples.

Abstract

It is well known that Pythagorean triples can be parametrized by two triples of polynomials with integer coefficients. We show that no single triple of polynomials with integer coefficients in any number of variables is sufficient, but that there exists a parametrization of Pythagorean triples by a single triple of integer-valued polynomials.