Characterization of arithmetic functions that preserve the sum-of-squares operation

We characterize all functions f: . . . such that f(m 2 + n 2) = f(m)2 + f(n)2 for all m, n . .. It turns out that all such functions can be grouped into three families, namely f . 0, f(n) = ±n (subject to some restrictions on when the choice of the sign is possible) and f(n) = ±12 (again subject to some restrictions on when the choice of the sign is possible).