A circle having a given number of lattice points on its circumference. The Schinzel circle
having
lattice points is given by the equation
(1)
Note that these solutions do not necessarily have the smallest possible radius. For example, while the Schinzel circle centered at (1/3, 0) and with radius 625/3
has nine lattice points on its circumference, so
does the circle centered at (1/3, 0) with radius 65/3.
A table of minimal circles to is given by Pegg (2008).
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. 25,
89-90, 1959.Pegg, E. "Lattice Circles." http://demonstrations.wolfram.com/LatticeCircles/.Schinzel,
A. "Sur l'existence d'un cercle passant par un nombre donné de points
aux coordonnées entières." L'Enseignement Math. Ser. 24,
71-72, 1958.Sierpiński, W. "Sur quelques problèmes
concernant les points aux coordonnées entières." L'Enseignement
Math. Ser. 24, 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.
lattice points is given by the equation
is given by Pegg (2008).
