Let $n$ be a positive integer. The well-known Erd\H{o}s-Straus conjecture asserts that the positive integral solution of the Diophantine equation $\frac{4}{n}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$ always exists when $n\ge 2$. This problem remains unresolved and produced numerous related problems. Recently, Lazar investigated some properties of the solutions to above Diophantine equation in the special case that $n$ is a prime number. Let $p\ge 5$ be a prime number. Lazar showed that there are no triple of positive integers $(x,y,z)$ which is solution of the Diophantine equation $ \frac{4}{p}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$ in the range $xy
