On The Lucas Sequence Equations V-n=7 square and V-n=7V(m)square

Abstract

Let P be a nonzero integer and let (U-n) and (V-n) denote Lucas sequences of first and second kind defined by U-0 = 0, U-1 = 1; V-0 = 2, V-1 = P; and Un+1 = PUn + Un-1, Vn+1 = PVn + Vn-1 for n >= 1. In this study, when P is odd, we show that the equation U-n = 7 square has only the solution (n, P) = (2, 7 square) when 7 vertical bar P and the equation V-n = 7 square has only the solution (n, P) = (1, 7 square) when 7 vertical bar P or (n, P) = (4, 1) when P-2 equivalent to I(mod 7). In addition, we show that the equation V-n = 7V(m)square has a solution if and only if P-2 = -3 + 7 square and (n, m) = (3, 1). Moreover, we show that the equation U-n = 7U(m)square has only the solution (a, m, P, square) = (8, 4, 1, 1) when P is odd.