## Circle Lattice Points

For every Positive Integer , there exists a Circle which contains exactly lattice points in its interior. H. Steinhaus proved that for every Positive Integer , there exists a Circle of Area which contains exactly lattice points in its interior.

Schinzel's Theorem shows that for every Positive Integer , there exists a Circle in the Plane having exactly Lattice Points on its Circumference. The theorem also explicitly identifies such Schinzel Circles'' as

 (1)

Note, however, 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.

Let be the smallest Integer Radius of a Circle centered at the Origin (0, 0) with Lattice Points. In order to find the number of lattice points of the Circle, it is only necessary to find the number in the first octant, i.e., those with , where is the Floor Function. Calling this , then for , , so . The multiplication by eight counts all octants, and the subtraction by four eliminates points on the axes which the multiplication counts twice. (Since is Irrational, the Midpoint of a are is never a Lattice Point.)

Gauss's Circle Problem asks for the number of lattice points within a Circle of Radius

 (2)

Gauß showed that
 (3)

where
 (4)

The number of lattice points on the Circumference of circles centered at (0, 0) with radii 0, 1, 2, ... are 1, 4, 4, 4, 4, 12, 4, 4, 4, 4, 12, 4, 4, ... (Sloane's A046109). The following table gives the smallest Radius for a circle centered at (0, 0) having a given number of Lattice Points . Note that the high water mark radii are always multiples of five.

 1 0 108 1,105 4 1 132 40,625 12 5 140 21,125 20 25 156 203,125 28 125 180 5,525 36 65 196 274,625 44 3,125 252 27,625 52 15,625 300 71,825 60 325 324 32,045 68 390,625 420 359,125 76 540 160,225 84 1,625 92 100 4,225

If the Circle is instead centered at (1/2, 0), then the Circles of Radii 1/2, 3/2, 5/2, ... have 2, 2, 6, 2, 2, 2, 6, 6, 6, 2, 2, 2, 10, 2, ... (Sloane's A046110) on their Circumferences. If the Circle is instead centered at (1/3, 0), then the number of lattice points on the Circumference of the Circles of Radius 1/3, 2/3, 4/3, 5/3, 7/3, 8/3, ... are 1, 1, 1, 3, 1, 1, 3, 1, 3, 1, 1, 3, 1, 3, 1, 1, 5, 3, ... (Sloane's A046111).

Let

1. be the Radius of the Circle centered at (0, 0) having lattice points on its Circumference,

2. be the Radius of the Circle centered at (1/2, 0) having lattice points on its Circumference,

3. be the Radius of Circle centered at (1/3, 0) having lattice points on its Circumference.
Then the sequences , , and are equal, with the exception that if and if . However, the sequences of smallest radii having the above numbers of lattice points are equal in the three cases and given by 1, 5, 25, 125, 65, 3125, 15625, 325, ... (Sloane's A046112).

Kulikowski's Theorem states that for every Positive Integer , there exists a 3-D Sphere which has exactly Lattice Points on its surface. The Sphere is given by the equation

where and are the coordinates of the center of the so-called Schinzel Circle and is its Radius (Honsberger 1973).

See also Circle, Circumference, Gauss's Circle Problem, Kulikowski's Theorem, Lattice Point, Schinzel Circle, Schinzel's Theorem

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.

Weisstein, E. W. Circle Lattice Points.'' Mathematica notebook CircleLatticePoints.m.

© 1996-9 Eric W. Weisstein
1999-05-26