关于单位分数的 Lazar 问题

设 $n$ 为任意正整数. 著名 Erd\H{o}s-Straus 猜想是指当 $n\ge 2$ 时, Diophantine 方程 $\frac{4}{n}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$ 总有正整数解 $(x,y,z)$. 虽然有许多作者研究这个猜想, 但是至今它还未被解决. 设 $p\ge 5$ 为任意素数. 最近, Lazar 证明 Diophantine 方程 $ \frac{4}{p}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$ 在区域 $xy<\sqrt{z/2}$ 内没有 $x$ 与 $y$ 互素的正整数解 $(x,y,z)$. 同时, Lazar 提出问题: 在上述方程中以 $5/p$ 替换 $4/p$, 是否有类似结果? 这也是 Sierpinski 提出的一个猜想. 在本文中, 我们证明 Diophantine 方程 $\frac{a}{p}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$ 没有满足\ $x, y$ 互素且\ $xy<\sqrt{z/2}$ 的正整数解 $(x,y,z)$, 其中 $a$ 为满足\ $a<7\le p$ 的正整数. 这回答了上述 Lazar 问题, 并推广了 Lazar 的结果. 我们的证明方法和工具主要是利用有理数\ $\frac{a}{p}$ 的连分数表示.

On a problem of Lazar on unit fractions

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 $xy<\sqrt{z/2}$ and $(x,y)=1$. Meanwhile, Lazar pointed out that it would be interesting to find an analog of this result for $5/p$ instead of $4/p$, which is also a conjecture due to Sierpinski. In this paper, we answer Lazar''s question affirmatively and also extended Lazar''s result by showing that the Diophantine equation $ \frac{a}{p}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$ cannot have any integer solution $(x,y,z)$ such that $x$ and $y$ are coprime and $xy<\sqrt{z/2}$, where $a$ is a positive integer such that $a<7\le p$. Our proof uses the continued fraction expansion of $\frac{a}{p}$.