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 Erdős-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 $.
Citation: Wei Zhao, Jian Lu, Lin Wang. On the integral solutions of the Egyptian fraction equation $ \frac ap = \frac 1x+\frac 1y+\frac 1z $[J]. AIMS Mathematics, 2021, 6(5): 4930-4937. doi: 10.3934/math.2021289
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 Erdős-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 $.
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Wei Zhao, Jian Lu, Lin Wang. On the integral solutions of the Egyptian fraction equation $ \frac ap = \frac 1x+\frac 1y+\frac 1z $[J]. AIMS Mathematics, 2021, 6(5): 4930-4937. doi: 10.3934/math.2021289
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