  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.
