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

$$\cases{ (x-{\textstyle{1\over 2}})^2+y^2={\textstyle{1\over... ...r 3}})^2+y^2={\textstyle{1\over 9}} 5^{2k} & for $n=2k+1$.\cr}$$ (3)

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

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

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.

Sierpinski, W. ``Sur quelques problèmes concernant les points aux coordonnées entières.'' L'Enseignement Math. Ser. 2 4, 25-31, 1958.

Sierpinski, W. ``Sur un problème de H. Steinhaus concernant les ensembles de points sur le plan.'' Fund. Math. 46, 191-194, 1959.

Sierpinski, W. A Selection of Problems in the Theory of Numbers. New York: Pergamon Press, 1964.