Schinzel's theorem

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

It was originally proved by and named after Andrzej Schinzel.Template:R

Proof

Circle through exactly four points given by Schinzel's construction

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:Template:R $${(x-\frac{1}{2})}^2+y^2=\frac{1}{4}{5}^{k-1}.$$ This circle has radius $5^{(k-1)/2}/2$, and is centered at the point $(\frac{1}{2},0)$. 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, $${(2x-1)}^2+(2y{)}^2=5^{k-1}.$$ 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:Template:R $${(x-\frac{1}{3})}^2+y^2=\frac{1}{9}{5}^{2k}.$$ This circle has radius $5^k/3$, and is centered at the point $(\frac{1}{3},0)$.

Properties

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