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Kulikowski's Theorem

For every positive integer n, there exists a sphere which has exactly n lattice points on its surface. The sphere is given by the equation

(x-a)^2+(y-b)^2+(z-sqrt(2))^2=c^2+2,
(1)

where a and b are the coordinates of the center of the so-called Schinzel circle

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

and c is its radius.

See also

Circle Lattice Points, Lattice Point, Schinzel's Theorem

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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

Kulikowski's Theorem

Cite this as:

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

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