In the geometry of numbers, Schinzel's theorem is the following statement:
It was originally proved by and named after Andrzej Schinzel.
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:
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. There are exactly ways to write as a sum of two squares, and half are in the order (odd, even) by symmetry. For example, , so we have or , and or , 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:
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, but they have the advantage that they are described by an explicit equation.