Circle with 3 lattice points
Azmi Tamid
3 lattice points ?
Thanks
Jan Kristian Haugland
On 3 May 2002, Azmi Tamid wrote:
> Is there a circle in the plane that has on it exactly
> 3 lattice points ?
I was going to ask about the general case the other day,
but found what seemed to be a complete solution.
The circle
(x + 1/2)^2 + y^2 = 5^(n/2 - 1) / 4
goes through exactly n lattice points if n is even, and
(x + 1/4)^2 + y^2 = 5^(n-1) / 16
does it when n is odd.
Not entirely sure this is correct, but I remember that
something similar seemed to work. For n = 3 you can just
take the circle through (0, 0), (2, 0) and (1, 2) anyway.
--
J K Haugland
http://hjem.sol.no/neutreeko
Azmi Tamid
Thanks Jan,
Let me ask also what happeneds if you replace the word "circle"
with "Square" in the plane (the boundary of the square as before) .
Azmi
Jan Kristian Haugland
> Let me ask also what happeneds if you replace the word "circle"
> with "Square" in the plane (the boundary of the square as before) .
Easy. The square with vertices
(2/3, 0) (n + 1/3, 0) (2/3, n - 1/3) (n + 1/3, n - 1/3)
goes through exactly n lattice points.
James Buddenhagen
New question: if a circle is centered at (0,0) and passes
through n lattice points, what values can n have?
--Jim Buddenhagen
Jan Kristian Haugland
> New question: if a circle is centered at (0,0) and passes
> through n lattice points, what values can n have?
Any multiple of 4: x^2 + y^2 = 5^(k-1) passes
through 4k lattice points. Because of symmetry,
n must be divisible by 4 (unless it is a circle
with radius 0).
James Buddenhagen
> James Buddenhagen wrote:
>
> > New question: if a circle is centered at (0,0) and passes
> > through n lattice points, what values can n have?
>
> Any multiple of 4: x^2 + y^2 = 5^(k-1) passes
> through 4k lattice points. Because of symmetry,
> n must be divisible by 4 (unless it is a circle
> with radius 0).
Well that turned out to have a slick and simple answer!
Here are more circle/lattice point thoughts:
Let C0, C1, C2 ... be all circles centered at (0,0) which
pass through lattice points. These are ordered by increaing
radius: 0=r0<r1<r2... . (The radii need not be integers).
Let n0, n1, n2 ... be the corresponding sequence that counts
lattice points, i.e. n_i=#(C_i). What does this sequence
{n_i} look like? Is it in Sloane?
Which circles have the property that they pass through more
lattice points than any smaller circle? If I calculate
correctly these start: C1, C13, C30, C121, C362, C1232,
C1584, ... Here is a small table:
(r_n)^2 n_i/4 i
1 1 1
25 3 13
65 4 30
325 6 121
1105 8 362
4225 9 1232
5525 12 1584
These sequences don't appear to be in Sloane, but some
related ones are. Can any explict formulas be given for
the n_th terms. What about growth rates, asymtotics etc.?
--Jim Buddenhagen
Gerry Myerson
azmi_...@my-deja.com (Azmi Tamid) wrote:
> Is there a circle in the plane that has on it exactly
> 3 lattice points ?
Ross Honsberger discusses this & related problems in Chapter 11
of Mathematical Gems. E.g.,
Steinhaus has proved that for every natural number n there exists
a circle of area n which contains in its interior exactly n lattice
points.
Schinzel proved that for every natural number n there exists a circle
in the plane which has exactly n lattice points on its circumference.
Schinzel and Kulikowski proved that for every nonempty plane bounded
convex figure C and for every natural number n there is a figure in
the plane with the shape of C which contains exactly n lattice points
in its interior.
Kulikowski proved that for every natural number n there is a sphere
which has exactly n latticce points on its surface. Honsberger notes
that the proof generalizes immediately to any number of dimensions.
Browkin proved that for any natural number n there is a cube in 3-space
which contains exactly n lattice points in its interior.
It's given as an exercise to prove that for every natural number n
there is a square in the plane with exactly n lattice points on its
boundary.
--
Gerry Myerson (ge...@mpce.mq.edi.ai) (i -> u for email)
Gerry Myerson
"James Buddenhagen" <jbud...@texas.net> wrote:
=> Let C0, C1, C2 ... be all circles centered at (0,0) which
=> pass through lattice points. These are ordered by increaing
=> radius: 0=r0<r1<r2... . (The radii need not be integers).
=>
=> Which circles have the property that they pass through more
=> lattice points than any smaller circle? If I calculate
=> correctly these start: C1, C13, C30, C121, C362, C1232,
=> C1584, ... Here is a small table:
=>
=> (r_n)^2 n_i/4 i
=> 1 1 1
=> 25 3 13
=> 65 4 30
=> 325 6 121
=> 1105 8 362
=> 4225 9 1232
=> 5525 12 1584
=>
=> These sequences don't appear to be in Sloane, but some
=> related ones are. Can any explict formulas be given for
=> the n_th terms. What about growth rates, asymtotics etc.?
The number of lattice points on the circle of radius r centered at
(0, 0) is the number of representations of r^2 as a sum of two squares.
There's a formula for this in terms of the prime factorization of n,
in particular it's the primes of the form 4k + 1 that matter. E.g.,
25 = 5^2, 65 = 5 x 13, 325 = 5^2 x 13, 1105 = 5 x 13 x 17, etc.
That tells you where to look for your champions, and it says you'll
have to know something about the distribution of primes to say much
about growth rates & asymptotics.
Jan Kristian Haugland
James Buddenhagen wrote:
> New question: if a circle is centered at (0,0) and passes
> through n lattice points, what values can n have?
Any multiple of 4: x^2 + y^2 = 5^(n-1) passes through 4n lattice points.
Jan Kristian Haugland
Azmi Tamid wrote:
> Let me ask also what happeneds if you replace the word "circle"
> with "Square" in the plane (the boundary of the square as before) .
Easy. The square with vertices
(2/3, 0) (n + 1/3, 0) (2/3, n - 1/3) (n + 1/3, n - 1/3)
goes through exactly n lattice points.
Jan Kristian Haugland
Azmi Tamid wrote:
> Let me ask also what happeneds if you replace the word "circle"
> with "Square" in the plane (the boundary of the square as before) .
Easy. The square with vertices
(2/3, 0) (n + 1/3, 0) (2/3, n - 1/3) (n + 1/3, n - 1/3)
goes through exactly n lattice points.
Jan Kristian Haugland
Azmi Tamid wrote:
> Let me ask also what happeneds if you replace the word "circle"
> with "Square" in the plane (the boundary of the square as before) .
Easy. The square with vertices
(2/3, 0) (n + 1/3, 0) (2/3, n - 1/3) (n + 1/3, n - 1/3)
goes through exactly n lattice points.
Jan Kristian Haugland
Azmi Tamid wrote:
> Let me ask also what happeneds if you replace the word "circle"
> with "Square" in the plane (the boundary of the square as before) .
Easy. The square with vertices
(2/3, 0) (n + 1/3, 0) (2/3, n - 1/3) (n + 1/3, n - 1/3)
goes through exactly n lattice points.
Ahmed
In the sequence of (r_n)^2 didn't you forget to include
(r_n)^2 = 5 ?
A.F.
James Buddenhagen
news:8a9347ac.02051...@posting.google.com...
> "James Buddenhagen" <jbud...@texas.net> wrote in message
news:<_PcB8.69088$Ii2.6...@bin2.nnrp.aus1.giganews.com>...
[...]
Good catch! You are correct. The second line of the table
should be:
5 2 4
I believe the other lines are correct (but independent
verification would be nice). The 'lattice point count
jump' circles are now C1, C4, C13, C30, C121, C362, C1232,
C1584 ...
--Jim Buddenhagen