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

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

It was originally proved by and named after Andrzej Schinzel.

Proof

Schinzel proved this theorem by the following construction. If ${\displaystyle n}$ is an even number, with ${\displaystyle n=2k}$, then the circle given by the following equation passes through exactly ${\displaystyle n}$ points: $${\displaystyle \left(x-{\frac {1}{2}}\right)^{2}+y^{2}={\frac {1}{4}}5^{k-1}.}$$ This circle has radius ${\displaystyle 5^{(k-1)/2}/2}$, and is centered at the point ${\displaystyle ({\tfrac {1}{2}},0)}$. For instance, the figure shows a circle with radius ${\displaystyle {\sqrt {5}}/2}$ through four integer points.

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

On the other hand, if ${\displaystyle n}$ is odd, with ${\displaystyle n=2k+1}$, then the circle given by the following equation passes through exactly ${\displaystyle n}$ points: $${\displaystyle \left(x-{\frac {1}{3}}\right)^{2}+y^{2}={\frac {1}{9}}5^{2k}.}$$ This circle has radius ${\displaystyle 5^{k}/3}$, and is centered at the point ${\displaystyle ({\tfrac {1}{3}},0)}$.

Properties

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

References

1. ^ Schinzel, André (1958), "Sur l'existence d'un cercle passant par un nombre donné de points aux coordonnées entières", L'Enseignement mathématique (in French), 4: 71–72, MR 0098059
2. ^ Honsberger, Ross (1973), "Schinzel's theorem", Mathematical Gems I, Dolciani Mathematical Expositions, vol. 1, Mathematical Association of America, pp. 118–121
3. Weisstein, Eric W., "Schinzel Circle", MathWorld

• Theorems about circles
• Geometry of numbers
• Lattice points