Elemental Patterns from the Erd\H{o}s Straus Conjecture

Abstract

This paper makes the following conjecture: For every prime $p$ there exists a positive integer $x$ with $\left\lceil \frac{p}{4} \right\rceil \leq x \leq \left\lceil \frac{p}{2} \right\rceil$ and a positive divisor $d|x^2$ so that either: (1) $ d \bmod \left( 4x - p \right) \equiv -px$; or (2) $d \leq x$ and $ d \bmod \left( 4x - p \right) \equiv -x$. Furthermore this paper proves that the solutions to these modular equations are in one-to-one correspondence with the solutions of the diophantine equation used in the Erd\H{o}s Straus conjecture.