Schinzel's theorem

In the geometry of numbers, it is a theorem of Andrzej Schinzel that, for any given natural number ${\displaystyle n}$, there exists a circle in the Euclidean plane that passes through exactly ${\displaystyle n}$ integer points.[1][2]

Circle through exactly four points given by Schinzel's construction

Schinzel proved this theorem by the following construction. If ${\displaystyle n}$ is an even number, with ${\displaystyle n=2k}$, then the circle given by the following equation passes through exactly ${\displaystyle n}$ points:[1][2]

$${\displaystyle \left(x-{\frac {1}{2}}\right)^{2}+y^{2}={\frac {1}{4}}5^{k-1}.}$$

This circle has radius ${\displaystyle 5^{(k-1)/2}/2}$, and is centered at the point ${\displaystyle ({\tfrac {1}{2}},0)}$. For instance, the figure shows a circle with radius ${\displaystyle {\sqrt {5}}}$ through four integer points.

On the other hand, if ${\displaystyle n}$ is odd, with ${\displaystyle n=2k+1}$, then the circle given by the following equation passes through exactly ${\displaystyle n}$ points:[1][2]

$${\displaystyle \left(x-{\frac {1}{3}}\right)^{2}+y^{2}={\frac {1}{9}}5^{2k}.}$$

This circle has radius ${\displaystyle 5^{k}/3}$, and is centered at the point ${\displaystyle ({\tfrac {1}{3}},0)}$.

The circles generated by this construction are not the smallest possible circles through the given number of integer points,[3] but they have the advantage that they are described by an explicit equation.[2]