2011 IMO Shortlist N8

Let $k$ be a positive integer and set $n=2^{k}+1$. Prove that $n$ is a prime number if and only if the following holds: there is a permutation $a_{1}, \ldots, a_{n-1}$ of the numbers $1,2, \ldots, n-1$ and a sequence of integers $g_{1}, g_{2}, \ldots, g_{n-1}$ such that $n$ divides $g_{i}^{a_{i}}-a_{i+1}$ for every $i \in\{1,2, \ldots, n-1\}$, where we set $a_{n}=a_{1}$.

令$k$为正整数并设置$n=2^{k}+1$。证明 $n$ 是素数当且仅当以下成立：存在数字 $1,2, \ldots, n-1$ 的排列 $a_{1}, \ldots, a_{n-1}$ 和整数序列 $g_{1}, g_{2}, \ldots, g_{n-1}$ 使得 $n$ 整除$g_{i}^{a_{i}}-a_{i+1}$ 对于每个 $i \in\{1,2, \ldots, n-1\}$，其中我们设置 $a_{n}=a_{1}$。