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] 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: [1] [2] 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. [2]
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] 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: [1] [2] 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. [2]