TOPICS

# Schinzel's Theorem

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

 (1)

given by

 (2)

where is the number of divisors of of the form and is the number of divisors of the form . It explicitly identifies such circles (the Schinzel circles) as

 (3)

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

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

## Explore with Wolfram|Alpha

More things to try:

## 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