A note on the Erdő-Straus Conjecture

Abstract

This paper makes a fundamental assertion about the Erdős-Straus conjecture. Suppose that for a prime $p$ there exists $x,y,z\in \mathbb{N}$ with $x\le y\le z$ 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 Erdős-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 Erdős-Straus conjecture, this paper suggests a method for proof.

11 pages, 2 figures