# A note on the Erd\H{o}-Straus Conjecture

## TL;DR

This paper explores a fundamental reduction of the Erd ext{"o}s-Straus conjecture by assuming solutions exist for prime p, proposing a new approach to potentially prove the conjecture through variable reduction methods.

## Contribution

It introduces a novel reduction technique for the Erd ext{"o}s-Straus conjecture, linking solutions to a simplified form involving gcds, and suggests a new proof strategy.

## Key findings

- Reduction of the conjecture to fewer variables
- Explicit formula for z involving gcds and p
- Proposed method for future proof development

## Abstract

This paper makes a fundamental assertion about the Erd\H{o}s-Straus conjecture. Suppose that for a prime $p$ there exists $x,y,z \in \mathbb{N}$ with $x \leq y \leq z$ so that $$ \frac{4}{p} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}. $$ The main contribution of this paper is that, under this assumption, the Erd\H{o}s-Straus conjecture can be reduced by one variable. For example, it is necessarily true that $$ z = \frac{xyp}{\gcd(y,p) \gcd \left( xy, x+y \right)}.$$ Considering other reductions of the Erd\H{o}s-Straus conjecture, this paper suggests a method for proof.