Circle Lattice Points -- from Wolfram MathWorld

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

${(x-1/2)^2+y^2=1/45^(k-1) for n=2k; (x-1/3)^2+y^2=1/95^(2k) for n=2k+1.$

(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, a mid-arc point 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 radius is , where is the sum of squares function. The numbers of lattice points falling on the circumference of circles centered at the origin of radii 0, 1, 2, ... are therefore 1, 4, 4, 4, 4, 12, 4, 4, 4, 4, 12, 4, 4, ... (OEIS A046109).

The following table gives the smallest radius for a circle centered at (0, 0) having a given number of lattice points (OEIS A006339). Note that is also the least hypotenuse of distinct Pythagorean triples. The high-water numbers of lattice points are 1, 5, 25, 125, 3125, ... (OEIS A062875), and the corresponding radii are 4, 12, 20, 28, 44, ... (OEIS A062876).

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, ... (OEIS 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, ... (OEIS 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 ${a_n}$ , ${b_n}$ , and ${c_n}$ 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, ... (OEIS A006339).

Kulikowski's theorem states that for every positive integer , there exists a three-dimensional sphere which has exactly lattice points on its surface. The sphere is given by the equation