Question 148
Denote by $\mathbb{N}$ the set of all positive integers. Let $f: \mathbb{N} \to \mathbb{N}$ be a *multiplicative* function such that $f(4) = 4$ and
$$
f(m^2 + n^2) = f(m^2) + f(n^2)
$$
holds for all positive integers $m$ and $n$. Prove that $f(m^2) = m^2$ holds for all positive integers $m$.
(A function is called *multiplicative* if $f(mn) = f(m)f(n)$ holds for all relatively prime positive integers $m$ and $n$.) (Mathematica Slovaca)
$$
f(m^2 + n^2) = f(m^2) + f(n^2)
$$
holds for all positive integers $m$ and $n$. Prove that $f(m^2) = m^2$ holds for all positive integers $m$.
(A function is called *multiplicative* if $f(mn) = f(m)f(n)$ holds for all relatively prime positive integers $m$ and $n$.) (Mathematica Slovaca)