2004 IMO Shortlist S20

Given an integer ${n>1}$ , denote by $P_{n}$ the product of all positive integers $x$ less than $n$ and such that $n$ divides ${x^2-1}$ . For each ${n>1}$ , find the remainder of $P_{n}$ on division by $n$ .

*Proposed by John Murray, Ireland*

给定一个整数 ${n>1}$ ，用 $P_{n}$ 表示所有小于 $n$ 的正整数 $x$ 的乘积，并且使得 $n$ 除以 ${x^2-1}$ 。对于每个 ${n>1}$ ，求 $P_{n}$ 除以 $n$ 的余数。

*由爱尔兰约翰·默里提议*