  In this paper, we mainly discuss the characterization of a class of
arithmetic functions $f: N \rightarrow C$ such that
$f(u^{2}+kv^2)=f^{2}(u)+kf^{2}(v)$ $(k, u, v \in N)$. We obtain a
characterization with given condition, propose a conjecture and show the result
holds for $k \in \{2, 3, 4, 5 \}$.
