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.
> 
> 
> 



Integer lattice points on a hypersphere
There are several related and very interesting problems and theorems: 


* [Schinzel's theorem](http://mathworld.wolfram.com/SchinzelsTheorem.html) - solves the problem in $\mathbb{R}^2$ using so-called [Schinzel circles](http://mathworld.wolfram.com/SchinzelCircle.html). 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](http://mathworld.wolfram.com/KulikowskisTheorem.html) - gives explicit construction in $\mathbb{R}^3$ and generalizes to all higher dimensions:


*W. Sierpiński, ["Elementary Theory of Numbers: 2nd English Edition"](http://goo.gl/kI6UZ), 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:


* [Steinhaus' theorem](http://mathworld.wolfram.com/CircleLatticePoints.html)
* [Browkin's theorem](http://mathworld.wolfram.com/BrowkinsTheorem.html)


