1986 IMO 第 2 题

(SWE 2) Let $f(x)=x^{n}$ where $n$ is a fixed positive integer and $x=$ $1,2, \ldots$. Is the decimal expansion $a=0 . f(1) f(2) f(3) \ldots$ rational for any value of $n$ ? The decimal expansion of $a$ is defined as follows: If $f(x)=d_{1}(x) d_{2}(x) \ldots$ $\ldots d_{r(x)}(x)$ is the decimal expansion of $f(x)$, then $a=0.1 d_{1}(2) d_{2}(2) \ldots$ $\ldots d_{r(2)}(2) d_{1}(3) \ldots d_{r(3)}(3) d_{1}(4) \ldots$.

(SWE 2) 设$f(x)=x^{n}$，其中$n$是固定正整数，$x=$$1,2,\ldots$。是十进制扩展$a=0 。 f(1) f(2) f(3) \ldots$对于$n$的任意值有理数？$a$的十进制扩展定义如下： 如果$f(x)=d_{1}(x) d_{2}(x) \ldots$$\ldots d_{r(x)}(x)$是$f(x)$的十进制扩展，则$a=0.1 d_{1}(2) d_{2}(2) \ldots$$\ldots d_{r(2)}(2) d_{1}(3) \ldots d_{r(3)}(3) d_{1}(4) \ldots$。