Generating Pythagorean Triples of a Given Height

Abstract

The height of a Pythagorean triple $(a,b,c)$ is defined as the difference $c - b$. In this paper, building on the Height-Excess Enumeration Theorem, we show how to generate primitive Pythagorean triples of a given height using powers of a single matrix. Then, we briefly discuss other matrices that both map any Pythagorean triple to another and also preserve a triple's height. We also note how Pythagorean triples with a given leg difference can be generated using matrices.