On the integral solutions of the Egyptian fraction equation $\frac ap=\frac 1x+\frac 1y+\frac 1z$

Abstract

It is an interesting question to investigate the integral solutions for the Egyptian fraction equation $\frac{a}{p}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$, which is known as Erd\H{o}s-Straus equation when $a=4$. Recently, Lazar proved that this equation has not integral solutions with $xy<\sqrt{z/2}$ and $\gcd(x,y)=1$ when $a=4$. But his method is difficult to get an analogous result for arbitrary $\frac{a}{p}$, especially when $p$ and $a$ are lager numbers. In this paper, we extend Lazar's result to arbitrary integer $a$ with $4\le a\leq\frac{1+\sqrt{1+6p^3}}{p}$, and release the condition $\gcd(x,y)=1$. We show that $\frac{a}{p}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$ has no integral solutions satisfying that $xy<\sqrt{lz}$, where $l\leq\frac{(3p+a)p}{a^2}$ when $p\nmid y$ and $l\leq\frac{3p^2+a}{pa^2}$ when $p\mid y$. Besides, we extend Monks and Velingker's result to the case $4\le a<p$.

Keywords

• egyptian fraction
• erdős-straus conjecture
• integral solution