128. How many solutions in integers does the equation

$$
\sqrt{x}+\sqrt{y}=9000 ?
$$
Solution. Transferring $\sqrt{x}$ to the right side and squaring both sides of the equation, we get the equality $y=9000^{2}-18000 \sqrt{x}+x$, from which it follows that $\sqrt{x}$ is a rational number. However, if the square root of an integer is rational, then the number itself is a perfect square. Therefore, $x$ is a perfect square. Similarly, it can be shown that the number $y$ is also a perfect square.

Thus, the given equation has as many solutions in integers as there are ways to represent the number 9000 as the sum of two addends (taking into account the order). Since there are 9001 such ways, the given equation has 9001 solutions.

Answer: 9001.