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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:
This circle has radius , and is centered at the point . For instance, the figure shows a circle with radius through four integer points.
As an equation of integers, is writing 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:
This circle has radius , and is centered at the point .
geometry of numbers Geometry of numbers is the part of number theory which uses geometry for the study of algebraic numbers. Typically, a ring of algebraic integers is viewed as a lattice in \mathbb R^n, and the study of these lattices provides fundamental information ...
, Schinzel's theorem is the following statement:
It was originally proved by and named after Andrzej Schinzel
Andrzej Bobola Maria Schinzel (5 April 1937 – 21 August 2021) was a Polish mathematician studying mainly number theory.
Education
Schinzel received an MSc in 1958 at Warsaw University, Ph.D. in 1960 from Institute of Mathematics of the Pol ...
.
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:
This circle has radius , and is centered at the point . For instance, the figure shows a circle with radius through four integer points.
As an equation of integers, is writing 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:
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, but they have the advantage that they are described by an explicit equation.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 Theorems about circles Geometry of numbers