3. The number of all different positive integer solutions to the equation $\frac{x}{y}-\frac{2 y}{x}=\frac{2015}{x y}-1$ is $(\quad$.
(A) 0
(B) 1
(C) 2
(D) 4
3. A.
$$
\begin{array}{l}
\text { Given } \frac{x}{y}-\frac{2 y}{x}=\frac{2015}{x y}-1 \\
\Rightarrow x^{2}-2 y^{2}=2015-x y \\
\Rightarrow(x-y)(x+2 y)=2015 .
\end{array}
$$

Obviously, $x>y>0$.
Therefore, there are only four possibilities:
$$
\begin{array}{l}
\left\{\begin{array}{l}
x-y=1, \\
x+2 y=2
\end{array} 015 ;\left\{\begin{array}{l}
x-y=5, \\
x+2 y=403 ;
\end{array}\right.\right. \\
\left\{\begin{array}{l}
x-y=13, \\
x+2 y=155
\end{array} ;\left\{\begin{array}{l}
x-y=31, \\
x+2 y=65 .
\end{array}\right.\right.
\end{array}
$$

Furthermore, it is required that
$$
\begin{array}{l}
3|(2015-1), 3|(403-5), \\
3|(155-13), 3|(65-31),
\end{array}
$$

None of these conditions hold, hence there is no solution.