Mersenne numbers as a difference of two Lucas numbers

Abstract

Let (L-n)(n >= 0) be the Lucas sequence. We show that the Diophantine equation L-n - L-m = M-k has only the nonnegative integer solutions (n, m, k) = (2, 0, 1), (3, 1, 2), (3, 2, 1), (4, 3, 2), (5, 3, 3), (6, 2, 4), (6, 5, 3) where M-k = 2(k) - 1 is the kth Mersenne number and n > m.