The Square Terms in Generalized Lucas Sequences

Abstract

Let P and Q be non-zero integers. The generalized Fibonacci sequence {U-n} and Lucas sequence {V-n} are defined by U-0 = 0, U-1 = 1 and Un+1 = PUn + QU(n-1) for n >= 1 and V-0 = 2, V-1 = P and Vn+1 = PVn + QV(n-1) for n >= 1, respectively. In this paper, we assume that Q = 1. Firstly, we determine indices n such that V-n = kx(2) when k|P and P is odd. Then, when P is odd, we show that there are no solutions of the equation V-n = 3 square for n > 2. Moreover, we show that the equation V-n = 6 square has no solution when P is odd. Lastly, we consider the equations V-n = 3V(m)square and V-n = 6V(m)square. It has been shown that the equation V-n = 3V(m)square has a solution when n = 3, m = 1, and P is odd. It has also been shown that the equation V-n = 6V(m)square has a solution only when n = 6. We also solve the equations V-n = 3 square and V-n = 3V(m)square under some assumptions when P is even.