Weird Functional Equation

Algebra Level 4

ff(a)(b)ff(b)(a)=(f(a+b))2f^{f(a)}(b) f^{f(b)}(a) = \big(f(a+b)\big)^2

Let f:NNf:\mathbb{N} \to \mathbb{N} be an injective function such that the above holds true for all a,bNa,b \in \mathbb{N}. Let SS be the sum of all possible values of f(2017)f(2017). Find Smod1000S \bmod{1000}.


Note: fk(n)f^{k} (n) means f(f(f(f(n))))number of f’s = k.\underbrace{f\big(f(f(\ldots f(n)\ldots))\big)}_{\text{number of }f \text{'s}\ = \ k}.

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