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 U-0 = 0, U-1 = 1 and Un+1 = PUn - QU(n-1) for n >= 1, and V-0 = 2, V-1 = P and Vn+1 = PVn - QV(n-1) for n >= 1, respectively. In this paper, we assume that P >= 1, Q is odd, (P, Q) = 1, V-m not equal 1, and V-r not equal 1. We show that there is no integer x such that V-n = V(r)V(m)x(2) when m >= 1 and r is an even integer. Also we completely solve the equation V-n = V(m)V(r)x(2) for m >= 1 and r >= 1 when Q equivalent to 7 (mod 8) and x is an even integer. Then we show that when P equivalent to 3 (mod 4) and Q equivalent to 1 (mod 4), the equation V-n = V(m)V(r)x(2) 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 V-n such that V-n = VmVr for m > 1 and r > 1. Lastly, we show that there is no generalized Fibonacci number U-n such that U-n = UmUr for Q = +/- 1 and 1 < r < m.