On square classes in generalized Lucas sequences

Abstract

Let P and Q be nonzero integers. Generalized Fibonacci and Lucas sequences are defined as follows: U0 = 0, U1 = 1, and Un+1 = PUn + QUn-1 for n ≥ 1 and V0 = 2, V1 = P, and Vn+1 = PVn + QVn-1 for n ≥ 1, respectively. For all odd relatively prime values of P and Q such that P ≥ 1, we determine all indices n and m such that Vn = wVmx2 or VnVm = wx2 with w = 1, 2, 3 or 6 under the assumptions P2 + 4Q > 0 and Vm ≠ 1 for all positive integers m. © 2015 World Scientific Publishing Company.