Problem

Source: Turkey Team Selection Test 2018 P8

Tags: number theory



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