Schinzel's theorem

{{Short description|Theorem about circles through lattice points}}

In the geometry of numbers, Schinzel's theorem is the following statement:

{{math theorem|name=Schinzel's theorem|For any given positive integer $n$ , there exists a circle in the Euclidean plane that passes through exactly $n$ integer points.}}

It was originally proved by and named after Andrzej Schinzel.{{r|schinzel|honsberger}}

Proof

File:Schinzel Circle.png

Schinzel proved this theorem by the following construction. If $n$ is an even number, with $n=2k$ , then the circle given by the following equation passes through exactly $n$ points:{{r|schinzel|honsberger}}

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

This circle has radius $5^{(k-1)/2}/2$ , and is centered at the point $(\tfrac12,0)$ . For instance, the figure shows a circle with radius $\sqrt 5/2$ through four integer points.

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

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

This writes $5^{k-1}$ as a sum of two squares, where the first is odd and the second is even. There are exactly $4k$ ways to write $5^{k-1}$ as a sum of two squares, and half are in the order (odd, even) by symmetry. For example, $5^1=(\pm 1)^2 + (\pm 2)^2$ , so we have $2x-1=1$ or $2x-1=-1$ , and $2y=2$ or $2y=-2$ , which produces the four points pictured.

On the other hand, if $n$ is odd, with $n=2k+1$ , then the circle given by the following equation passes through exactly $n$ points:{{r|schinzel|honsberger}}

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

This circle has radius $5^k/3$ , and is centered at the point $(\tfrac13,0)$ .

Properties

The circles generated by Schinzel's construction are not the smallest possible circles passing through the given number of integer points,{{r|mathworld}} but they have the advantage that they are described by an explicit equation.{{r|honsberger}}