Elemental Patterns from the Erdős Straus Conjecture

Abstract

This paper makes the following conjecture: For every prime $p$ there exists a positive integer $x$ with $\lceil \frac{p}{4}\rceil \le x\le \lceil \frac{p}{2}\rceil$ and a positive divisor $d|x^2$ so that either: (1) $d\mathrm{mod}\left(4x-p\right)\equiv -px$; or (2) $d\le x$ and $d\mathrm{mod}\left(4x-p\right)\equiv -x$. 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 Erdős Straus conjecture.