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Maximal number of distinct values

Source: MEMO 2023 T1

August 25, 2023
algebra

Problem Statement

(a) A function f:Z→Zf:\mathbb{Z} \rightarrow \mathbb{Z} is called Z\mathbb{Z}-good if f(a2+b)=f(b2+a)f(a^2+b)=f(b^2+a) for all a,b∈Za, b \in \mathbb{Z}. What is the largest possible number of distinct values that can occur among f(1),…,f(2023)f(1), \ldots, f(2023), where ff is a Z\mathbb{Z}-good function?
(b) A function f:N→Nf:\mathbb{N} \rightarrow \mathbb{N} is called N\mathbb{N}-good if f(a2+b)=f(b2+a)f(a^2+b)=f(b^2+a) for all a,b∈Na, b \in \mathbb{N}. What is the largest possible number of distinct values that can occur among f(1),…,f(2023)f(1), \ldots, f(2023), where ff is a N\mathbb{N}-good function?
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