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 , there exists a circle in the Euclidean plane that passes through exactly integer points.}}
It was originally proved by and named after Andrzej Schinzel.{{r|schinzel|honsberger}}
Proof
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:{{r|schinzel|honsberger}}
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:{{r|schinzel|honsberger}}
This circle has radius , and is centered at the point .
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=
| 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}}
| 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}}
}}