Question 66
Let $p$ be a prime and let $\mathbb{F}_p$ be the set of integers modulo $p$. Call a function $f: \mathbb{F}_p^2 \rightarrow \mathbb{F}_p$ quasiperiodic if there exist $a, b \in \mathbb{F}_p$, not both zero, so that $f(x+a, y+b)=f(x, y)$ for all $x, y \in \mathbb{F}_p$. Find, with proof, the number of functions $\mathbb{F}_p^2 \rightarrow \mathbb{F}_p$ that can be written as the sum of some number of quasiperiodic functions.