Lucas numbers of the form (2t/k)

Abstract

Let L_m denote the m^th Lucas number. We show that the solutions to the diophantine equation (2^t/k) = L_m, in non-negative integers t, k ≤ 2^t−1, and m, are (t, k, m) = (1, 1, 0), (2, 1, 3), and (a, 0, 1) with non-negative integers a.