Question 141
Let $N$ denote the set of all positive integers. Find all injective functions $f: N \to N$ such that
$$
f^{f(a)}(b) f^{f(b)}(a) = (f(a+b))^2
$$
holds for all positive integers $a, b$. Here $f^k(n)$ denotes $\underbrace{f(f(\dots f(n)\dots))}_{k}$.
$$
f^{f(a)}(b) f^{f(b)}(a) = (f(a+b))^2
$$
holds for all positive integers $a, b$. Here $f^k(n)$ denotes $\underbrace{f(f(\dots f(n)\dots))}_{k}$.