A Note on Generating Almost Pythagorean Triples

TL;DR

This paper refines Frink's characterization of almost Pythagorean triples, providing an explicit method to generate specific triples using basic algebraic operations, thus advancing understanding of these special integer solutions.

Contribution

It extends Frink's original characterization by offering an explicit algebraic method to generate particular almost Pythagorean triples.

Findings

Provides an explicit algebraic characterization for generating triples.

Simplifies the process of finding specific almost Pythagorean triples.

Enhances the understanding of the structure of these triples.

Abstract

In 1987, Orrin Frink introduced the concept of almost Pythagorean triples. He defined them as an ordered triple $(x,y,z)$ that satisfies the equation $x^2+y^2=z^2+1$ where $x,y$ and $z$ are positive integers. In his paper, he showed that there were infinitely many almost Pythagorean triples by giving a characterization which suggests a method on generating all of them. However, this method does not explicitly and readily give a particular almost Pythagorean triple. In this note, using basic algebraic operations, we extend his result by giving a characterization that explicitly and readily give a particular almost Pythagorean triple.