2. Find the largest positive integer $n$, such that $n^{2}+2000 n$ is a perfect square.
2. Let $n^{2}+2000 n=(n+k)^{2}, k \in \mathbf{Z}^{+}$. Rearranging gives: $n=\frac{k^{2}}{2000-2 k}$, it is easy to see that $k$ is even.

Let $k=2 m$, then $n=\frac{m^{2}}{500-m}$, here $m$ should be the largest and make $500-m$ divide $m^{2}$. As long as $m=499$, at this time $k=2 \times$ $499=998 . n=\frac{998^{2}}{4}=249001$