Simple continued fractions and special relativity Academic Article

Let E 0 , E 1 , , E n {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 = k c = sqrt k , k some natural number. For n 1 n geqslant 1 let E n {E_n} have velocity 1 with respect to E n 1 {E_{n - 1}} . Let x n {x_n} denote the velocity of E n {E_n} with respect to E 0 {E_0} . Then only if k = 2 , 3 k = 2,3 or 5 will x n {x_n} be a simple continued fraction convergent of k sqrt k infinitely often.