关键词: 不定方程, 正整数解, 互异, Erd\"{o}s-Straus猜想

Abstract:

In this paper, we study the (distinct) positive integer solution of the equation

\begin{equation*}\label{eq12}\frac{k}{n} = \frac{1}{x_1}+\frac{1}{x_2}+\cdots+\frac{1}{x_t}\end{equation*} with $n>k\geq 2$ and $ t\geq 2$.

We show that the above equation has at least one distinct positive integer solution if it has a positive integer solution.

When $k=5$, we show the above equation has at least one distinct positive integer solution for all $n\geq 3$

except possibly when $n\equiv 1, 5041, 6301, 8821, 13861, 15121(\mbox{mod } 16380)$ with $t=3$,

and for all $n\geq 3$ except possibly when $n\equiv 1, 81901(\mbox{mod } 163800)$ with $t=4$.

Furthermore, we point out that the above equation has at least one distinct positive integer solution for all $n(>k)$

when $t\geq k\geq 2$.

Key words: Diophantine equation, Positive integer solution, Distinct, Erd\"{o}s-Straus conjecture