A Geometric Consideration of the Erd\H{o}s-Straus Conjecture

TL;DR

This paper investigates solutions to the Erd ext{"o}s-Straus conjecture's diophantine equation, analyzing the structure of solutions and proposing conjectures to guide future research.

Contribution

It introduces a geometric perspective on the solutions, classifies solution types, and suggests a common solution pattern with new conjectures.

Findings

Most solutions follow the pattern x = ⌊py/(4y - p)⌋ + 1

Solution types are categorized into two cases

Proposes conjectures to inspire further exploration

Abstract

In this paper we will explore the solutions to the diophantine equation in the Erd\H{o}s-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.