2024 Putnam A4

Find all primes $p > 5$ for which there exists an integer $a$ and an integer $r$ satisfying $1 \leq r \leq p-1$ with the following property: the sequence $1,a,a^2,\dots,a^{p-5}$ can be rearranged to form a sequence $b_0,b_1,b_2,\dots,b_{p-5}$ such that $b_n-b_{n-1}-r$ is divisible by $p$ for $1 \leq n \leq p-5$.

查找所有素数 $p > 5$，其中存在整数 $a$ 和整数 $r$ 满足 $1 \leq r \leq p-1$ 且具有以下属性：序列 $1,a,a^2,\dots,a^{p-5}$ 可以重新排列以形成序列 $b_0,b_1,b_2,\dots,b_{p-5}$ 使得$b_n-b_{n-1}-r$ 可被 $p$ 整除，即 $1 \leq n \leq p-5$。