Question 113
Determine all functions $f: \mathbb{N}^{\star} \rightarrow \mathbb{N}^{\star}$ such that:
(i) The integers $f(1), f(2), \ldots$ are coprime as a whole (their overall gcd is 1).
(ii) There exists $N \geqslant 1$ such that for every $n \geqslant N$, $f(n) \neq 1$ and for all $a, b \in \mathbb{N}^{\star}$,
$$
f(a)^{n} \mid f(a+b)^{a^{n-1}}-f(b)^{a^{n-1}}
$$
(i) The integers $f(1), f(2), \ldots$ are coprime as a whole (their overall gcd is 1).
(ii) There exists $N \geqslant 1$ such that for every $n \geqslant N$, $f(n) \neq 1$ and for all $a, b \in \mathbb{N}^{\star}$,
$$
f(a)^{n} \mid f(a+b)^{a^{n-1}}-f(b)^{a^{n-1}}
$$