3. The number of positive integer solutions to the equation $x^{4004}+y^{400 x}=z^{2002}$ is $(\quad)$.
A. 0
B. 1
C. finite
D. infinite
3. A

Let $x_{0}=x^{1001}, y_{0}=y^{1001}, z_{0}=z^{1001}$, then the original equation can be transformed into $x_{0}^{4}+y_{0}^{4}=z_{0}^{2}, x_{0}>0$, $y_{0}>0, z_{0}>0$. Using the method of infinite descent, it can be proven that the equation $x_{0}^{4} + y_{0}^{4}=z_{0}^{2}$ has no positive integer solutions. Therefore, the answer is A.