On square classes in generalized Fibonacci sequences

Abstract

Let P and Q be nonzero integers. The generalized Fibonacci and Lucas sequences are defined respectively as follows: U0=0,U1=1, V0=2,V1=P and Un+1=PUn+QUn−1, Vn+1=PVn+QVn−1 for n≥1. In this paper, when w∈{1,2,3,6}, for all odd relatively prime values of P and Q such that P≥1 and P2+4Q>0, we determine all n and m satisfying the equation Un=wUmx2. In particular, when k|P and k>1, we solve the equations Un=kx2 and Un=2kx2. As a result, we determine all n such that Un=6x2.