On the Lucas Sequence Equations V-n = kV(m) and U-n = kU(m)

Abstract

Let P and Q be nonzero integers. The sequences of generalized Fibonacci and Lucas numbers are defined by 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. In this paper, we assume that P≥1, Q is odd, (P,Q)=1, Vm≠1, and Vr≠1. We show that there is no integer xsuch that Vn=VrVmx2 when m≥1 and ris an even integer. Also we completely solve the equation Vn=VmVrx2 for m≥1 and r≥1 when Q≡7(mod8) and x is an even integer. Then we show that when P≡3(mod4) and Q≡1(mod4), the equation Vn=VmVrx2 has no solutions for m≥1 and r≥1. Moreover, we show that when P>1 and Q=±1, there is no generalized Lucas number Vn such that Vn=VmVr for m>1 and r>1. Lastly, we show that there is no generalized Fibonacci number Un such that Un=UmUrfor Q=±1 and 1<r<m.