MEMO 2023 T1

Let $Z$ denote the set of all integers and $Z_{>0}$ denote the set of all positive integers.

(a) A function $f: Z \rightarrow Z$ is called $Z$-good if it satisfies $f(a^2 + b) = f(b^2 + a)$ for all $a, b \in Z$. Determine the largest possible number of distinct values that can occur among $f(1), f(2), \ldots, f(2023)$, where $f$ is a $Z$-good function.

(b) A function $f: Z_{>0} \rightarrow Z_{>0}$ is called $Z_{>0}$-good if it satisfies $f(a^2 + b) = f(b^2 + a)$ for all $a, b \in Z_{>0}$. Determine the largest possible number of distinct values that can occur among $f(1), f(2), \ldots, f(2023)$, where $f$ is a $Z_{>0}$-good function.