Simple continued fractions and special relativity

Abstract:

Let ${E_0},{E_1}, \ldots ,{E_n}$ be inertial frames of reference in a one dimensional relativistic universe where the speed of light is $c = \sqrt k$, k some natural number. For $n \geqslant 1$ let ${E_n}$ have velocity 1 with respect to ${E_{n - 1}}$. Let ${x_n}$ denote the velocity of ${E_n}$ with respect to ${E_0}$. Then only if $k = 2,3$ or 5 will ${x_n}$ be a simple continued fraction convergent of $\sqrt k$ infinitely often.