lattice point on a circle

Question

consider a circle with center (sqrt[2],1/3) and any arbitrary radius. how do I prove that there is atmost one lattice point on the circle?

also, does there exist an unique cirle with exactly 2004 lattice points inside it?

Answer 1

Suppose there are two lattice points on a circle. Then the center of that circle is on the bissecting line between those two points.

Since the equation of that line is rational, if it contains $(\sqrt 2,1/3)$ then it must also contain its rational conjugate $(-\sqrt 2,1/3)$. Which means that the line must be the horizontal line $(L) : y=1/3$.

Now since the two lattice points are symmetric around $(L)$, their $y$ coordinates have to sum to $2/3$, which is impossible since they are integers.

Now, since a circle can only contain one lattice point at a time, as you increase the radius of the circle, the number of points inside the circle can only jump by $1$ at a time. So for any integer $n \ge 0$, there is an nonempty open interval $I$ such that every circle of radius $r \in I$ contains exactly $n$ lattice points.