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Generalized Fibonacci and Lucas sequences (Un) and (Vn) are defined by the recurrence relations Un+1 = PUn+QUn−1 and Vn+1 = PVn +QVn−1, n ≥ 1, with initial conditions U0 = 0, U1 = 1 and V0 = 2, V1 = P. This paper deals with Fibonacci and Lucas numbers of the form Un(P,Q) and Vn(P,Q) with the special consideration that P ≥ 3 is odd and Q = −1. Under these consideration, we solve the equations Vn = 5kx, Vn = 7kx, Vn = 5kx±1, and Vn = 7kx±1 when k | P with k > 1. Moreover, we solve the equations Vn = 5x ± 1 and Vn = 7x ± 1.

GENERALIZED LUCAS NUMBERS OF THE FORM 5kx2AND 7kx2

GENERALIZED LUCAS NUMBERS OF THE FORM 5kx²AND 7kx²