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.
Categories: math.NT