Construction of primitive Pythagorean triples and Pythagorean triples with a common multiplier using gnomons

TL;DR

This paper introduces a novel method for constructing ordered primitive Pythagorean triples using gnomons, enabling new approaches to classical problems like Eulerian parallelepipeds and the Perfect Cuboid.

Contribution

A new construction method for primitive Pythagorean triples based on gnomons, allowing ordered data generation and facilitating solutions to complex geometric and combinatorial problems.

Findings

Allows ordered listing of primitive Pythagorean triples

Enables systematic construction of Pythagorean systems

Facilitates approaches to Eulerian parallelepipeds and the Perfect Cuboid

Abstract

The traditional construction of primitive Pythagorean triples by the formulas of two independent variables does not allow their ordering. The paper shows a new view on the construction of primitive Pythagorean triples. A method for constructing primitive Pythagorean triples, based on the use of gnomons equal in area to the squares of the legs from a primitive Pythagorean triple, using two dependent variables, is proposed. This enables the ability to build an ordered table of primitive Pythagorean triples. Working with ordered data, it is possible to approach in a new way the solution of problems that include systems of Pythagorean triples, such as the construction of Eulerian parallelepipeds, the construction of a Perfect Cuboid, or the problem of coloring natural numbers in two colors so that no Pythagorean triple is monochrome (one of the problems of Ramsey theory).