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

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.