4. Let the function $f: \mathbf{Z}_{+} \rightarrow \mathbf{Z}_{+}$, and satisfy:
(1) $\operatorname{gcd}(f(1), f(2), \cdots, f(n), \cdots)=1$;
(2) For sufficiently large $n, f(n) \neq 1$, for any positive integers $a, b$ and sufficiently large $n$ we always have
$$
(f(a))^{n} \mid\left((f(a+b))^{a^{n-1}}-(f(b))^{a^{n-1}}\right) \text {. }
$$

Find $f$.
First, we have $f(1)=1$. Using the Lifting Theorem, we get $a \mid f(a)$, i.e., $f(p)=p$.
Thus, $p \mid (f(k p + b) - f(b)) \left(b \in \mathbf{Z}_{+}\right)$.
Therefore, $f(x)=x$.