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.
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.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.
, 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
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
