Characterization of arithmetic functions that preserve the sum-of-squares operation (Q2452416)

The author finds all functions \(f\colon \mathbb{N} \to \mathbb{C}\) satisfying \(f(m^2+n^2)=f(m)^2+f(n)^2\) for all \(m,n \in \mathbb{N}\). The result is too long to be stated here: except for the trivial function \(f \equiv 0\), there are two groups of functions (one group with values \(f(n) = \pm n\) and the other group with values \(f(n)=\pm \tfrac 12\)). The proof starts with a (quite nontrivial) lemma asserting that \(f(1) \in \{0,1,-1,\tfrac 12,-\tfrac 12\}\). Then the values of \(f(n)\) are determined in the range \(2 \leq n \leq 10\) and, finally, the conclusion for all \(n \in \mathbb{N}\) is attained applying the identity \((ad+cd)^2+(ad-bc)^2=(ab-cd)^2+(ad+bc)^2\).