Question 118
Let $\mathbb{Z}$ denote the set of all integers and $\mathbb{Z}_{\gt 0}$ denote the set of all positive integers.
a. A function $f: \mathbb{Z} \rightarrow \mathbb{Z}$ is called $\mathbb{Z}$-good if it satisfies $f\left(a^{2}+b\right)=f\left(b^{2}+a\right)$ 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 $f: \mathbb{Z}_{\gt 0} \rightarrow \mathbb{Z}_{\gt 0}$ is called $\mathbb{Z}_{\gt 0}$-good if it satisfies $f\left(a^{2}+b\right)=f\left(b^{2}+a\right)$ for all $a, b \in \mathbb{Z}_{\gt 0}$. Determine the largest possible number of distinct values that can occur among $f(1), f(2), \ldots, f(2023)$, where $f$ is a $\mathbb{Z}_{\gt 0}$-good function.
a. A function $f: \mathbb{Z} \rightarrow \mathbb{Z}$ is called $\mathbb{Z}$-good if it satisfies $f\left(a^{2}+b\right)=f\left(b^{2}+a\right)$ 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 $f: \mathbb{Z}_{\gt 0} \rightarrow \mathbb{Z}_{\gt 0}$ is called $\mathbb{Z}_{\gt 0}$-good if it satisfies $f\left(a^{2}+b\right)=f\left(b^{2}+a\right)$ for all $a, b \in \mathbb{Z}_{\gt 0}$. Determine the largest possible number of distinct values that can occur among $f(1), f(2), \ldots, f(2023)$, where $f$ is a $\mathbb{Z}_{\gt 0}$-good function.