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Circle Lattice Points

For every positive integer n, there exists a circle which contains exactly n lattice points in its interior. H. Steinhaus proved that for every positive integer n, there exists a circle of area n which contains exactly n lattice points in its interior.

SchinzelCircles

Schinzel's theorem shows that for every positive integer n, there exists a circle in the plane having exactly n 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 R be the smallest integer radius of a circle centered at the origin (0, 0) with L(r) 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 0<=y<=|_R/sqrt(2)_|, where |_z_| is the floor function. Calling this N(R), then for r>=1, L(R)=8N(R)-4, so L(R)=4 (mod 8). The multiplication by eight counts all octants, and the subtraction by four eliminates points on the axes which the multiplication counts twice. (Since sqrt(2) 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 R

N(R)=1+4|_R_|+4sum_(i=1)^(|_R_|)|_sqrt(R^2-i^2)_|.
(2)

Gauss showed that

N(R)=piR^2+E(R),
(3)

where

|E(R)|<=2sqrt(2)piR.
(4)
CircleLatticePoints000

The number of lattice points on the circumference of circles centered at (0, 0) with radius R is N(R)=r_2(R^2), where r_k(n) 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 R<=330010000 for a circle centered at (0, 0) having a given number of lattice points N(R) (OEIS A006339). Note that 8[N(n)-4] is also the least hypotenuse of n 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).

CircleLatticePoints050
CircleLatticePoints033

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. a_n be the radius of the circle centered at (0, 0) having 8n+4 lattice points on its circumference,

2. b_n/2 be the radius of the circle centered at (1/2, 0) having 4n+2 lattice points on its circumference,

3. c_n/3 be the radius of circle centered at (1/3, 0) having 2n+1 lattice points on its circumference.

Then the sequences {a_n}, {b_n}, and {c_n} are equal, with the exception that b_n=0 if 2|n and c_n=0 if 3|n. 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 n, there exists a three-dimensional sphere which has exactly n lattice points on its surface. The sphere is given by the equation

(x-a)^2+(y-b)^2+(z-sqrt(2))^2=c^2+2,
(5)

where a and b are the coordinates of the center of the so-called Schinzel circle and c is its radius (Honsberger 1973).

See also

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

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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.Sloane, N. J. A. Sequences A006339, A062875, and A062876 in "The On-Line Encyclopedia of Integer Sequences."

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Circle Lattice Points

Cite this as:

Weisstein, Eric W. "Circle Lattice Points." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CircleLatticePoints.html

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