New parametrization of $A^2+B^2+C^2=3D^2$ and Lagrange's four-square theorem

TL;DR

This paper introduces a new parametrization for the equation A^2+B^2+C^2=3D^2, explores its links to Lagrange's four-square theorem, and applies it to generate integer-coordinate geometric figures like triangles and tetrahedrons.

Contribution

It presents a novel parametrization of a specific Diophantine equation and connects it to classical four-square theorem, enabling new geometric constructions with integer coordinates.

Findings

New parametrization for A^2+B^2+C^2=3D^2

Derived parameterizations of equilateral triangles with integer coordinates

Parameterizations of regular tetrahedrons with integer coordinates

Abstract

In this paper we provide a new parametrization for the diophantine equation $A^2+B^2+C^2=3D^2$ and give a series of corollaries. We discuss some connections with Lagrange's four-square theorem. As applications, we find new parameterizations of equilateral triangles and regular tetrahedrons having integer coordinates in three dimensions.