10. Let $k$, $m$, $n$ be integers. From a point $P\left(m^{3}-m, n^{3}-n\right)$ outside the circle $x^{2}+y^{2}=(3 k+1)^{2}$, two tangents are drawn to the circle, touching it at points $A$ and $B$. The number of integer points (points with both coordinates as integers) on the line $AB$ is ( ).
(A) 2
(B) 1
(C) 0
(D) infinitely many
10. C.

It is known that the equation of the line $AB$ on which the chord of contact lies is
$$
\left(m^{3}-m\right) x+\left(n^{3}-n\right) y=(3 k+1)^{2} \text {. }
$$

If the line $AB$ contains an integer point $\left(x_{0}, y_{0}\right)$, then
$$
\begin{array}{l}
(m-1) m(m+1) x_{0}+(n-1) n(n+1) y_{0} \\
=(3 k+1)^{2} .
\end{array}
$$

Since $(m-1) m(m+1)$ and $(n-1) n(n+1)$ are both multiples of 3, the left side of the above equation can be divided by 3. However, the right side leaves a remainder of 1 when divided by 3, which is a contradiction.
Therefore, the line $AB$ does not contain any integer points.