2008 CMO 第 6 题

Find all triples $(p,q,n)$ that satisfy

$$q^{n+2} \equiv 3^{n+2} (\mod p^n) ,\quad p^{n+2} \equiv 3^{n+2} (\mod q^n)$$

where $p,q$ are odd primes and $n$ is an positive integer.

找到所有满足$(p,q,n)$的三元组

$$q^{n+2} \equiv 3^{n+2} (\mod p^n) ,\quad p^{n+2} \equiv 3^{n+2} (\mod q^n)$$

其中 $p,q$ 是奇素数，$n$ 是正整数。