Three. (50 points) Let the infinite sequence of positive integers $\left\{a_{n}\right\}$ $\left(n \in \mathbf{N}_{+}\right)$ satisfy
$$
a_{4}=4, a_{n}^{2}-a_{n-1} a_{n+1}=1(n \geqslant 2) \text {. }
$$

Find the general term formula for $\left\{a_{n}\right\}$.
Three, from the known we get $\frac{a_{n}}{a_{n+1}}>\frac{a_{n-1}}{a_{n}}$.
If there is some $n$ such that $\frac{a_{n-1}}{a_{n}} \geqslant 1$, then $a_{n}>a_{n+1}$.
Thus, $a_{n-1} \geqslant a_{n}>a_{n+1}>a_{n+2}>\cdots$, which is clearly impossible (since $\left\{a_{n}\right\}\left(n \in \mathbf{N}_{+}\right)$ is an infinite sequence of positive integers).
Therefore, the terms in the sequence $\left\{a_{n}\right\}$ are strictly increasing.
From $a_{4}=4$, we know that $a_{1}=1, a_{2}=2, a_{3}=3$.
Thus, by the recursive formula of $\left\{a_{n}\right\}$ and mathematical induction, we have $a_{n}=n\left(n \in \mathbf{N}_{+}\right)$.