A Geometric Consideration of the Erdős-Straus Conjecture

Abstract

In this paper we will explore the solutions to the diophantine equation in the Erdős-Straus conjecture. For a prime $p$ we are discussing the relationship between the values $x,y,z\in \mathbb{N}$ so that $$\frac{4}{p}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}.$$ We will separate the types of solutions into two cases. In particular we will argue that the most common relationship found is $$x=\lfloor \frac{py}{4y-p}\rfloor +1.$$ Finally, we will make a few conjectures to motivate further research in this area.