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Schinzel's Theorem


For every positive integer n, there exists a circle in the plane having exactly n lattice points on its circumference. The theorem is based on the number r(n) of integral solutions (x,y) to the equation

 x^2+y^2=n,
(1)

given by

 r(n)=4(d_1-d_3),
(2)

where d_1 is the number of divisors of n of the form 4k+1 and d_3 is the number of divisors of the form 4k+3. It explicitly identifies such circles (the Schinzel circles) as

 {(x-1/2)^2+y^2=1/45^(k-1)   for n=2k; (x-1/3)^2+y^2=1/95^(2k)   for n=2k+1.
(3)

Note, however, that these solutions do not necessarily have the smallest possible radius.


See also

Browkin's Theorem, Kulikowski's Theorem, Schinzel Circle

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References

Honsberger, R. "Circles, Squares, and Lattice Points." Ch. 11 in Mathematical Gems I. Washington, DC: Math. Assoc. Amer., pp. 117-127, 1973.Kulikowski, T. "Sur l'existence d'une sphère passant par un nombre donné aux coordonnées entières." L'Enseignement Math. Ser. 2 5, 89-90, 1959.Schinzel, A. "Sur l'existence d'un cercle passant par un nombre donné de points aux coordonnées entières." L'Enseignement Math. Ser. 2 4, 71-72, 1958.Sierpiński, W. "Sur quelques problèmes concernant les points aux coordonnées entières." L'Enseignement Math. Ser. 2 4, 25-31, 1958.Sierpiński, W. "Sur un problème de H. Steinhaus concernant les ensembles de points sur le plan." Fund. Math. 46, 191-194, 1959.Sierpiński, W. A Selection of Problems in the Theory of Numbers. New York: Pergamon Press, 1964.

Referenced on Wolfram|Alpha

Schinzel's Theorem

Cite this as:

Weisstein, Eric W. "Schinzel's Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SchinzelsTheorem.html

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