4. $C$ is a circle with radius $r$, centered at the point $(\sqrt{2}, \sqrt{3})$, where $r$ is a positive real number. Then the maximum number of integer points on $C$ is ( ) points.
A. 0
B. 1
C. 4
D. Infinite
4. The equation of circle $C$ is: $(x-\sqrt{2})^{2}+(y-\sqrt{3})^{2}=r^{2}$.

If circle $C$ has two lattice points $A\left(x_{1}, y_{1}\right)$ and $B\left(x_{2}, y_{2}\right)$, then $\left(x_{1}-\sqrt{2}\right)^{2}+\left(y_{1}-\sqrt{3}\right)^{2}=r^{2},\left(x_{2}\right.$ $-\sqrt{2})^{2}+\left(y_{2}-\sqrt{3}\right)^{2}=r^{2}$.
Subtracting the two equations gives $2 \sqrt{2}\left(x_{2}-x_{1}\right)+2 \sqrt{3}\left(y_{2}-y_{1}\right)+\left(x_{1}^{2}+y_{1}^{2}\right)-\left(x_{2}^{2}+y_{2}^{2}\right)=0$.
However, $x_{1}=x_{2}, y_{1}=y_{2}$ cannot hold simultaneously. Therefore, $2 \sqrt{2}\left(x_{2}-x_{1}\right)+2 \sqrt{3}\left(y_{2}-y_{1}\right) \notin \mathbf{Q}$ (where $\mathbf{Q}$ is the set of rational numbers), which contradicts the above equation. Hence, circle $C$ can pass through at most one lattice point.
Furthermore, when $r=\sqrt{5}$, circle $C$ passes through the origin, so circle $C$ can pass through at most one lattice point.