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

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.