Circle lattice points are lattice points in the interior of a circle or on its circumference.
For every positive integer 
, there is a circle of area 
containing exactly 
lattice points in its interior,
a result of H. Steinhaus (Honsberger 1973, p. 118).
Schinzel's theorem shows that for every positive integer 
,
there exists a circle in the plane
having exactly 
lattice points on its circumference
(Schinzel 1958, pp. 71-72). The theorem also explicitly identifies such "Schinzel circles" as
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(1)
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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.
For a nonnegative integer radius 
, let 
be the number of lattice points
on the circumference of the circle
centered at the origin (0, 0), with 
. For 
, it is enough to count those in the first octant, where
and 
.
Calling this count 
gives 
, so 
. Multiplication by eight counts all octants,
and subtraction by four removes the extra copies of points on the axes. Since 
is irrational,
no lattice point on this circle
lies on an octant boundary away from the axes.
Gauss's circle problem asks for the number of lattice points in the closed
disk of radius 
centered at the origin
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(2)
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where 
is the floor function. An elementary estimate gives
 |
(3)
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where, for 
,
 |
(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 smallest positive integer radii of circles centered at the origin
with 
lattice points on their circumferences,
for 
,
1, ..., are 1, 5, 25, 125, 65, 3125, 15625, 325, ... (OEIS A006339).
For 
,
each is also the least common hypotenuse of exactly
distinct Pythagorean triangles, since their
number for a given integer hypotenuse 
is ![[L(R)-4]/8](/images/equations/CircleLatticePoints/Inline27.svg)
. The record values of these minimum radii
are 1, 5, 25, 125, 3125, ... (OEIS A062875),
and their corresponding numbers of lattice points
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).
More generally, for a positive integer 
, let 
count the lattice points
on the circle with center 
and radius 
, and let 
count those on the circle
with center 
and radius 
. Then
 |
(5)
|
and
 |
(6)
|
For 
,
1, ..., let
1. 
be the smallest positive integer radius
for which the circle centered at 
has 
lattice points on its
circumference,
2. 
be the smallest radius of the form 
, with 
a positive integer, for
which the circle centered at 
has 
lattice points on its
circumference,
3. 
be the smallest radius of the form 
, with 
a positive integer, for
which the circle centered at 
has 
lattice points on its
circumference.
Then 
,
so the actual minimum radii are 
, 
, and 
.
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
 |
(7)
|
where 
and 
are the coordinates of the center of the so-called Schinzel
circle and 
is its radius (Kulikowski 1959, pp. 89-90; Honsberger
1973).
