GENERALIZED FIBONACCI AND LUCAS NUMBERS OF THE FORM wx(2) AND wx(2) 1

Abstract

Let P >= 3 be an integer and let (U-n) and (V-n) denote generalized Fibonacci and Lucas sequences 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 solve the equations V-n = kx(2) and Vn = 2kx(2) with k I P and k > 1. Then, when k I P and k > 1, we solve some other equations such as U-n = kx(2),U-n = 2kx(2),U-n = 3kx(2), V-n = kx(2) 1(n) = 2kx(2) 1, and Un = kx(2) 1. Moreover, when P is odd, we solve the equations V-n = wx(2) + 1 and V-n = wx(2) - 1 for w = 2, 3, 6. After that, we solve some Diophantine equations.