$$
\begin{array}{r}
\text { Three. (25 points) In }\left[\frac{1^{2}}{2008}\right],\left[\frac{2^{2}}{2008}\right], \cdots, \\
{\left[\frac{2008^{2}}{2008}\right] \text {, how many different integers are there (where }[x]}
\end{array}
$$
indicates the greatest integer not greater than $x$)?
Three, let $f(n)=\frac{n^{2}}{2008}$.
When $n=2,3, \cdots, 1004$, we have
$$
\begin{array}{l}
f(n)-f(n-1) \\
=\frac{n^{2}}{2008}-\frac{(n-1)^{2}}{2008}=\frac{2 n-1}{2008}1 .
\end{array}
$$

And $f(1005)=\frac{1005^{2}}{2008}=\frac{(1004+1)^{2}}{2008}$
$$
=502+1+\frac{1}{2008}>503,
$$

Therefore, $\left[\frac{1005^{2}}{2008}\right],\left[\frac{1006^{2}}{2008}\right], \cdots,\left[\frac{2008^{2}}{2008}\right]$ are distinct integers.

Thus, in $\left[\frac{1^{2}}{2008}\right],\left[\frac{2^{2}}{2008}\right], \cdots,\left[\frac{2008^{2}}{2008}\right]$, there are a total of $503+1004=1507$ different integers.