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}}

References

{{reflist|refs=

{{citation

| last = Honsberger | first = Ross | author-link = Ross Honsberger

| contribution = Schinzel's theorem

| pages = 118–121

| publisher = Mathematical Association of America

| series = Dolciani Mathematical Expositions

| title = Mathematical Gems I

| volume = 1

| year = 1973}}

{{mathworld|urlname=SchinzelCircle|title=Schinzel Circle|mode=cs2}}

{{citation

| last = Schinzel | first = André | author-link = Andrzej Schinzel

| journal = L'Enseignement mathématique

| language = fr

| mr = 98059

| pages = 71–72

| title = Sur l'existence d'un cercle passant par un nombre donné de points aux coordonnées entières

| volume = 4

| year = 1958}}

}}

Category:Theorems about circles

Category:Geometry of numbers

Category:Lattice points