6. Let the number of distinct positive integer solutions $(x, y, z)$ to the indeterminate equation $x^{2}+y^{2}+z^{2}-x y z+10=0$ where $x, y, z$ are all greater than 2008 be $k$. Then $k$ satisfies (  .
$(A) k=0 \quad(B) 1 \leq k \leq 2008 \quad(C) k>2008$, but $k$ is a finite number $(D) k$ is infinite
Answer: Choose $D$.
$\left(x_{0}, y_{0}, z_{0}\right)=(3,4,5)$ is a particular solution to the original indeterminate equation. For any positive integer solution $\left(x_{1}, y, z\right)$ of the original indeterminate equation, assuming $x_{1} \leq y \leq z$, and $x_{1}z^{2}$, thus $x_{2}$ is a positive integer, and greater than $z$, so $\left(x_{2}, y, z\right)$ is also a solution to the original indeterminate equation, and rearranged in ascending order as $\left(y, z, x_{2}\right)$. By repeatedly applying the above conclusion, an infinite number of solutions can be obtained from a particular solution, therefore there are infinitely many solutions satisfying $x, y, z$ all greater than 2008.