In the geometry of numbers, Schinzel's theorem is the following statement:
Schinzel's theorem — For any given positive integer , 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. If is an even number, with , then the circle given by the following equation passes through exactly points:[1][2]
Multiplying both sides of Schinzel's equation by four produces an equivalent equation in integers,
On the other hand, if is odd, with , then the circle given by the following equation passes through exactly points:[1][2]
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.[2]