A remark on a conjecture of Erd\H{o}s and Straus

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