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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$\begingroup$ $k$ has to be even, since if $x$ is an integer point on the sphere, $-x$ is as well. $\endgroup$– Ryan BudneyCommented May 22, 2013 at 22:23
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5$\begingroup$ @Ryan, the sphere need not have its center in the origin.. $\endgroup$– J.C. OttemCommented May 22, 2013 at 22:29
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$\begingroup$ Ryan, not if the sphere is not centered at the origin. Gerhard "Ask Me About System Design" Paseman, 2013.05.21 $\endgroup$– Gerhard PasemanCommented May 22, 2013 at 22:29
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$\begingroup$ 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 $\endgroup$– Gerhard PasemanCommented May 22, 2013 at 22:42
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2$\begingroup$ Not directly related to the question, but is it true that the problem is open if the $k$ lattice points are inside the hypersphere? $\endgroup$– JRNCommented May 23, 2013 at 1:20
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2 Answers
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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).
answered May 22, 2013 at 22:32