For which parameters a sphere has at least 3 rational points? Let $R>0$, $x_0,y_0,z_0\in\mathbb{R}$ and 
$S_{x_0,y_0,z_0,R}:=\{(x,y,z)\in\mathbb{R}^3:(x-x_0)^2+(y-y_0)^2+(z-z_0)^2=R^2\}$.
Question: For which $R,x_0,y_0,z_0$ a sphere $S_{x_0,y_0,z_0,R}$ has at least 3
rational points?
Thanks.
UPD:
Nevertheless, the question reamins. I am interesting in cardinality of such spheres.
 A: I will not directly address your question but give a hint for finding as many points you want with rational coordinates on the unit sphere.
Two facts: A) and B)


*

*A) There exists a bijective correspondence between the unit sphere and the unit disk, called stereographic projection (https://en.wikipedia.org/wiki/Stereographic_projection) through the following direct and inverse formulas :


$$\tag{1}(a)\begin{cases}x=\frac {X}{1-Z}\\y=\frac{Y}{1-Z}\end{cases} \ \ \iff \ \ \ \ (b) \begin{cases}X=\frac{2x}{x^{2}+y^{2}+1}\\Y=\frac{2y}{x^{2}+y^{2}+1}\\ Z=\frac{x^{2}+y^{2}-1}{x^{2}+y^{2}+1}\end{cases}$$
(where $(X,Y,Z)$ are the 3D coordinates, and $(x,y)$ are the 2D coordinates).
This projection is a central projection from north pole to equatorial plane. 


*

*B) Rational points $(x,y)$ on the circle centered in $(x_0,y_0)$ are well known ; they are all obtained by taking any $a,b \in \mathbb{Z}$ into the formulas 


$$\tag{2}\begin{cases}x=x_0+R \frac{a^2-b^2}{a^2+b^2}\\y=y_0+R\frac{2ab}{a^2+b^2}\end{cases}$$
($x_0,y_0,R \in \mathbb{Q}$). It suffices then to take as many of these points as you want , plug formulas (2) into formulas (1)(b) ; the result will clearly be rational numbers.
A reference: (https://mathoverflow.net/q/125224)
