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.
Is the following statement true?



There are several related and very interesting problems and theorems:
W. Sierpiński, "Elementary Theory of Numbers: 2nd English Edition", page 386, the last paragraph:
And similar problems related to interior points: 


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). 

