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ff(a)(b)ff(b)(a)=(f(a+b))2f^{f(a)}(b) f^{f(b)}(a) = \big(f(a+b)\big)^2ff(a)(b)ff(b)(a)=(f(a+b))2
Let f:N→Nf:\mathbb{N} \to \mathbb{N}f:N→N be an injective function such that the above holds true for all a,b∈Na,b \in \mathbb{N}a,b∈N. Let SSS be the sum of all possible values of f(2017)f(2017)f(2017). Find S mod 1000S \bmod{1000}Smod1000.
Note: fk(n)f^{k} (n)fk(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}.number of f’s = kf(f(f(…f(n)…))).
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