Question 93
Let $p$ be a prime, $n$ be a positive integer, and let $Z_{p^n}$ denote the set of congruence classes modulo $p^n$. Determine the number of functions $f : Z_{p^n} \to Z_{p^n}$ satisfying the condition
$$
f(a) + f(b) \equiv f(a + b + pab) \pmod{p^n}
$$
for all $a, b \in Z_{p^n}$.
$$
f(a) + f(b) \equiv f(a + b + pab) \pmod{p^n}
$$
for all $a, b \in Z_{p^n}$.