3. Satisfy
$$
(2 m+1)^{2}-(2 n+1)^{2}=2016
$$

the number of positive integer pairs $(m, n)$ is ( ).
(A) 3
(B) 4
(C) 5
(D) 6
3. D.

Notice,
$$
\begin{array}{l}
(2 m+1)^{2}-(2 n+1)^{2} \\
=4(m+n+1)(m-n) .
\end{array}
$$

Since $m+n+1$ and $m-n$ have opposite parity, we have
$$
\begin{array}{l}
(m-n, m+n+1) \\
=(1,504),(3,168),(7,72), \\
(8,63),(9,56),(21,24) \\
\Rightarrow(m, n) \\
\quad=(252,251),(85,82),(39,32), \\
(35,27),(32,23),(22,1) .
\end{array}
$$

Therefore, there are 6 pairs of positive integers $(m, n)$ that satisfy the condition.