One of the consequences of Fermat's last theorem is the existence of a countable infinite number of rational points on the unit circle, which allows in turn, to find the rational points on the unit sphere via the inverse stereographic projection of the homothecies of the rational points on the unit circle. We proceed to iterate this process and obtain the rational points on the unit $S^3$ via the inverse stereographic projection of the homothecies of the rational points on the previous unit $S^2$. One may continue this iteration/recursion process ad infinitum in order to find the rational points on unit hyper-spheres of arbitrary dimension $S^4, S^5, cdots, S^N$. As an example, it is shown how to obtain the rational points of the unit $ S^{24}$ that is associated with the Leech lattice. The physical applications of our construction follow and one finds a direct relation among the $N+1$ quantum states of a spin-N/2 particle and the rational points of a unit $S^N$ hyper-sphere embedded in a flat Euclidean $R^{N+1}$ space.