Question 12
Let $p=2017$ be a prime and $\mathbb{F}_p$ be the integers modulo $p$. A function $f: \mathbb{Z} \rightarrow \mathbb{F}_p$ is called good if there is $\alpha \in \mathbb{F}_p$ with $\alpha \not \equiv 0 (\bmod p)$ such that
$$
f(x) f(y) = f(x+y) + \alpha^{y} f(x-y) \quad (\bmod p)
$$
for all $x, y \in \mathbb{Z}$. How many good functions are there that are periodic with minimal period $2016$?
$$
f(x) f(y) = f(x+y) + \alpha^{y} f(x-y) \quad (\bmod p)
$$
for all $x, y \in \mathbb{Z}$. How many good functions are there that are periodic with minimal period $2016$?