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-1for n ≥ 1 and V0= 2, V1= P, and Vn+1= PVn+ QVn-1for 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= wVmx2or VnVm= wx2with w = 1, 2, 3 or 6 under the assumptions P2+ 4Q > 0 and Vm≠ 1 for all positive integers m.