A new complete algorithm for Irreducible Diophantine Pythagorean Triangles (IDPTs)

TL;DR

This paper introduces a novel, complete algorithm for generating all irreducible right-angled triangles with integer sides, based on the difference between the hypotenuse and the other sides, ensuring no common divisors.

Contribution

The paper presents a new, complete algorithm for generating all irreducible Pythagorean triangles, proving its completeness and exclusivity for such triangles.

Findings

Algorithm generates all irreducible Pythagorean triangles.

Proof of the algorithm's completeness.

Ensures no common divisor among side lengths.

Abstract

It is well known that a triangle with side lengths 3, 4 and 5 is right-angled. Euclid was the first to give a formula for generating other right-angled triangles with integer side lengths. In this text, I present a novel algorithm to generate all possible right-angled triangles with integer side lengths, in which the three side lengths have no common divisor. The algorithm is based on the difference in length between the hypothenuse and the largest of the two other sides. I also prove the completeness of this algorithm: it generates all possible such triangles and nothing but such triangles.