  In this paper we will explore the solutions to the diophantine equation in
the Erd\H{o}s-Straus conjecture. For a prime $p$ we are discussing the
relationship between the values $x,y,z \in \mathbb{N}$ so that $$ \frac{4}{p} =
\frac{1}{x} + \frac{1}{y} + \frac{1}{z}.$$
  We will separate the types of solutions into two cases. In particular we will
argue that the most common relationship found is $$ x = \lfloor \frac{py}{4y-p}
\rfloor + 1.$$
  Finally, we will make a few conjectures to motivate further research in this
area.
