Middle European Mathematical Olympiad 2023 Problem T-1

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

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

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