Schinzel's theorem
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In the geometry of numbers, it is a theorem of Andrzej Schinzel that, for any given natural number , there exists a circle in the Euclidean plane that passes through exactly integer points.[1][2]
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:[1][2]
This circle has radius , and is centered at the point . For instance, the figure shows a circle with radius through four integer points.
On the other hand, if is odd, with , then the circle given by the following equation passes through exactly points:[1][2]
This circle has radius , and is centered at the point .
The circles generated by this construction are not the smallest possible circles through the given number of integer points,[3] but they have the advantage that they are described by an explicit equation.[2]
References[]
- ^ a b c 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
- ^ a b c d Honsberger, Ross (1973), "Schinzel's theorem", Mathematical Gems I, Dolciani Mathematical Expositions, vol. 1, Mathematical Association of America, pp. 118–121
- ^ Weisstein, Eric W., "Schinzel Circle", MathWorld
Categories:
- Theorems about circles
- Geometry of numbers