1997 USAMO 第 1 题

Let $p_1,p_2,p_3,...$ be the prime numbers listed in increasing order, and let $x_0$ be a real number between $0$ and $1$. For positive integer $k$, define

$$x_{k}=\begin{cases}0&\text{ if }x_{k-1}=0\\ \left\{\frac{p_{k}}{x_{k-1}}\right\}&\text{ if }x_{k-1}\ne0\end{cases}$$

where $\{x\}$ denotes the fractional part of $x$. (The fractional part of $x$ is given by $x-\lfloor{x}\rfloor$ where $\lfloor{x}\rfloor$ is the greatest integer less than or equal to $x$.) Find, with proof, all $x_0$ satisfying $0<x_0<1$ for which the sequence $x_0,x_1,x_2,...$ eventually becomes $0$.

令 $p_1,p_2,p_3,...$ 为按升序排列的素数，令 $x_0$ 为 $0$ 到 $1$ 之间的实数。对于正整数 $k$，定义

$$

x_{k}=\begin{cases}0&\text{ 如果 }x_{k-1}=0\\ \left\{\frac{p_{k}}{x_{k-1}}\right\}&\text{ 如果 }x_{k-1}\ne0\end{cases}

$$

其中$\{x\}$表示$x$的小数部分。 ($x$ 的小数部分由 $x-\lfloor{x}\rfloor$ 给出，其中 $\lfloor{x}\rfloor$ 是小于或等于 $x$ 的最大整数。)通过证明，找到所有满足 $0<x_0<1$ 的 $x_0$，其中序列 $x_0,x_1,x_2,...$ 最终变为 $0$。