Question 66
For integers $m \ge 3$, $n$ and $x_1, x_2, \dots, x_m$, if $x_{i+1} - x_i \equiv x_i - x_{i-1} \pmod n$ for every $2 \le i \le m-1$, we say that the $m$-tuple $(x_1, \dots, x_m)$ is an *arithmetic sequence in (mod n)*. Let $p \ge 5$ be a prime number and $1 \lt a \lt p-1$ be an integer. Let $\{a_1, a_2, \dots, a_k\}$ be the set of all possible remainders when positive powers of $a$ are divided by $p$. Show that if a permutation of $\{a_1, a_2, \dots, a_k\}$ is an arithmetic sequence in (mod $p$), then $k = p-1$.