2. The positive integer pairs $(x, y)$ that satisfy $y=\sqrt{x+3}+\sqrt{x+2007}$ (   ).
(A) There is only one pair
(B) There are exactly two pairs
(C) There are at least three pairs
(D) None exist
2.B.

Let $a^{2}=x+3, b^{2}=x+2007$, where $a, b$ are natural numbers. Then
$$
\begin{array}{l}
y=a+b, \\
b^{2}-a^{2}=(b-a)(b+a) \\
=2004=2^{2} \times 3 \times 167 .
\end{array}
$$

Since $b+a$ and $b-a$ have the same parity, and $b+a > b-a$, we have
$$
\left\{\begin{array} { l } 
{ b + a = 1 0 0 2 , } \\
{ b - a = 2 }
\end{array} \text { or } \left\{\begin{array}{l}
b+a=334, \\
b-a=6 .
\end{array}\right.\right.
$$

Solving these, we get $\left\{\begin{array}{l}a=500, \\ b=502\end{array}\right.$ or $\left\{\begin{array}{l}a=164, \\ b=170 .\end{array}\right.$