Schinzel's theorem

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

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.[1][2]

Proof

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Circle through exactly four points given by Schinzel's construction

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.

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:[1][2]   This circle has radius  , and is centered at the point  .

Properties

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The circles generated by Schinzel's construction are not the smallest possible circles passing through the given number of integer points,[3] but they have the advantage that they are described by an explicit equation.[2]

References

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  1. ^ 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
  2. ^ 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
  3. ^ Weisstein, Eric W., "Schinzel Circle", MathWorld