Find all functions $f:\mathbb N_0\to \mathbb N_0$ such that $f(a^2+b^2)=f(a)^2+f(b)^2$ https://ift.tt/eA8V8J

I think the answer is that there are only 2 such functions: the zero function and the identity function, but I'm not able to prove it.

A few findings:

1. $f(0)=0$ and thus $f(a^2)=f(a)^2$.

2. If $f(1)=0$, then $f(2^n)=0$ for all $n\in \mathbb N_0$; if $f(1)=1$, then $f(2^n)=2^n$ for all $n\in \mathcal N_0$ (can be proven by PMI).

3. For any functions satisfying the condition, say $f,g$, $f\circ g$ also satisfies the condition.

Source of this problem: https://www2.math.binghamton.edu/p/pow/problem2f20

from Hot Weekly Questions - Mathematics Stack Exchange
Yuqiao Huang