Schinzel's theorem

In the geometry of numbers, Schinzel's theorem is the following statement: Schinzel's theorem — For any given positive integer

n

, there exists a circle in the Euclidean plane that passes through exactly

integer points.

It was originally proved by and named after Andrzej Schinzel.

[1][2] Schinzel proved this theorem by the following construction.

is an even number, with

, then the circle given by the following equation passes through exactly

points:[1][2]

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

This circle has radius

, and is centered at the point

For instance, the figure shows a circle with radius

through four integer points.

Multiplying both sides of Schinzel's equation by four produces an equivalent equation in integers,

This writes

as a sum of two squares, where the first is odd and the second is even.

ways to write

as a sum of two squares, and half are in the order (odd, even) by symmetry.

, which produces the four points pictured.

On the other hand, if

is odd, with

, then the circle given by the following equation passes through exactly

points:[1][2]

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

This circle has radius

, and is centered at the point

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