A remark on a conjecture of Erdős and Straus

Abstract

The aim of this note is to show that given a positive integer $n\ge 5$, the positive integral solutions of the diophantine equation $4/n=1/x+1/y+1/z$ cannot have solution such that $x$ and $y$ are coprime with $xy<\sqrt{z/2}$. The proof uses the continued fraction expansion of $4/n$.