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Is the following statement true?

For every integer $n\ge2$ and every integer $k\ge0$ there exists a hypersphere in $\mathbb{R}^n$ (circle, sphere etc) containing exactly $k$ integer lattice points on its surface.

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$k$ has to be even, since if $x$ is an integer point on the sphere, $-x$ is as well. – Ryan Budney May 22 '13 at 22:23
@Ryan, the sphere need not have its center in the origin.. – J.C. Ottem May 22 '13 at 22:29
Ryan, not if the sphere is not centered at the origin. Gerhard "Ask Me About System Design" Paseman, 2013.05.21 – Gerhard Paseman May 22 '13 at 22:29
Note that if it is true for n=2, then it is true for all n, by using a hypersphere of well chosen irrational radius with all the lattice points sitting in a two dimensional subspace. Gerhard "Irrational Solutions To Rational Problems" Paseman, 2013.05.21 – Gerhard Paseman May 22 '13 at 22:42
Not directly related to the question, but is it true that the problem is open if the $k$ lattice points are inside the hypersphere? – Joel Reyes Noche May 23 '13 at 1:20
up vote 17 down vote accepted

There are several related and very interesting problems and theorems:

  • Schinzel's theorem - solves the problem in $\mathbb{R}^2$ using so-called Schinzel circles. It seems intuitively clear that it generalizes to higher dimensions by slightly adjusting radius of a hypersphere so that it contains exactly the same lattice points as its section in lower dimension, but of course, a rigorous proof is needed. Indeed, there is:
  • Kulikowski's theorem - gives explicit construction in $\mathbb{R}^3$ and generalizes to all higher dimensions:

W. Sierpiński, "Elementary Theory of Numbers: 2nd English Edition", page 386, the last paragraph:

T. Kulikowski [1] has proved that for any natural number n there exists a sphere (in the three-dimensional space), on the boundary of which there are precisely n points whose coordinates are integers. He generalized this theorem for spheres in spaces of an arbitrary $\ge 3$ dimension.

And similar problems related to interior points:

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Yes, this seems to be Kulikowski's theorem, see:

T. Kulikowski Sur l'existence d'une sphère passant par un nombre donné de points aux coordonnées entières. Enseignement Math. (2) 5 1959 89–90.

(the Mathworld link seems to mention the case $n=3$ only, but according to the MathScinet review, the theorem is proved in all dimensions).

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