1991 USAMO 第 3 题

Show that, for any fixed integer $\,n \geq 1,\,$ the sequence

$$2, \; 2^2, \; 2^{2^2}, \; 2^{2^{2^2}}, \ldots \pmod{n}$$

is eventually constant.

[The tower of exponents is defined by $a_1 = 2, \; a_{i+1} = 2^{a_i}$. Also $a_i \pmod{n}$ means the remainder which results from dividing $\,a_i\,$ by $\,n$.]

证明，对于任何固定整数 $\,n \geq 1,\,$ 序列

$$2、\; 2^2，\； 2^{2^2}, \; 2^{2^{2^2}}, \ldots \pmod{n}$$

最终是恒定的。

[指数塔由 $a_1 = 2, \; 定义a_{i+1} = 2^{a_i}$。另外 $a_i \pmod{n}$ 表示 $\,a_i\,$ 除以 $\,n$ 所得的余数。]