Question 53
Consider the set $\mathcal{F}$ of functions $f: \mathbb{N} \to \mathbb{N}$ having the property that
$$
f(a^2 - b^2) = f(a)^2 - f(b)^2, \text{ for all } a, b \in \mathbb{N}, a \ge b.
$$
a) Determine the set $\{f(1) \mid f \in \mathcal{F}\}$.
b) Prove that $\mathcal{F}$ has exactly two elements.
$$
f(a^2 - b^2) = f(a)^2 - f(b)^2, \text{ for all } a, b \in \mathbb{N}, a \ge b.
$$
a) Determine the set $\{f(1) \mid f \in \mathcal{F}\}$.
b) Prove that $\mathcal{F}$ has exactly two elements.