Pardon my ignorance of this topic.



> 
> **Q1**.
>  In which dimensions $d$ is it the case that, for every natural number $n$,
>  there exists a sphere having exactly $n$ lattice points on it $(d{-}1)$-dimensional surface?
> 
> 
> 


[Schinzel's Theorem](http://mathworld.wolfram.com/SchinzelsTheorem.html) establishes this in $\mathbb{R}^2$, and [Kulikowski's Theorem](http://mathworld.wolfram.com/KulikowskisTheorem.html)
establishes this in $\mathbb{R}^3$. Is it known in higher dimensions?



> 
> **Q2**. Why is it that the smallest integral radius of a circle (in $\mathbb{R}^2$)
>  centered at $(0,0)$ that
>  has $n$ lattice points on its circumference, is always a multiple of $5$?
>  Or rather: Is there a simple, intuitive explanation for the multiple-of-$5$?
> 
> 
> 


OEIS A046122: $1, 5, 25, 125, 65, 3125, 15625, 325, \ldots$. [A046122 link](http://oeis.org/A046112)


And this likely follows from answers to the previous two questions:



> 
> **Q3**. Does the multiple-of-$5$ phenomenon occur in higher dimensions as well?
> 
> 
> 


I realize these questions may be answered in the literature I have not encountered, so pointers are welcomed—Thanks!



Three questions concerning lattice points on sphere surfaces
Collecting my comments into an answer: 


It appears that Kulikowski's paper proved the result for all dimensions. I haven't seen the Kulikowski paper, but a short proof is given at this [site](http://www.cut-the-knot.org/arithmetic/algebra/Kulikowski.shtml). 


Question 2 is well-answered by Lev Borisov elsewhere on this page. 


For question 3, the sequence is tabulated at the [OEIS](http://oeis.org/A071609) and a glance at the first few terms  

($1,2,5,9,14,26,41,74,89,101,146,194,269\dots$) does not suggest any divisibility properties (although once you get past $14$ there seem to be a lot of prime factors congruent to 1 modulo 4). 


