THE SQUARE TERMS IN GENERALIZED LUCAS SEQUENCE WITH PARAMETERS P AND Q

Abstract

Let P and Q be nonzero integers. Generalized Lucas sequence is defined as follows: V-0 = 2, V-1 = P and Vn+1 = PVn QV(n-1) for n >= 1. We assume that P and Q are odd relatively prime integers. Firstly, we determine all indices n such that V-n = kx(2) and V-n = 2kx(2) when k vertical bar P. Then, as an application of our these results, we find all solutions of the equations V-n = 3x(2) and V-n = 6x(2). Moreover, we find integer solutions of some Diophantine equations.