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arXiv · 1906.00561

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

Kyle Bradford
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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.

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Kyle Bradford. 2019-06-03. A note on the Erd\H{o}-Straus Conjecture. https://arxiv.org/abs/1906.00561

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