Pythagorean Triples, B\'{e}zout Coefficients and the Erd\H{o}s-Straus Conjeceture

TL;DR

This paper explores the connections between the Erdős-Straus conjecture, Pythagorean triples, and Bézout coefficients, proposing a new perspective on solutions for prime numbers and their polynomial roots.

Contribution

It introduces a novel relationship linking solutions to the Erdős-Straus conjecture with Pythagorean triples and Bézout coefficients, and examines polynomial roots associated with these solutions.

Findings

Identifies a relationship between conjecture solutions and Pythagorean triples.

Uses Bézout coefficients to connect solutions with polynomial roots.

Provides a preliminary framework for further exploration of the conjecture.

Abstract

This paper is a preliminary expository paper that outlines the relationship between solutions to the Erd\H{o}s-Straus conjecture for a given prime $p$ and their corresponding Pythagorean triples. This paper also uses B\'{e}zout Coefficients to find a relationship between the solutions to the conjecture and roots of a second degree polynomial.