GENERALIZED LUCAS NUMBERS OF THE FORM 5kx²AND 7kx²

Abstract

Abstract. Generalized Fibonacci and Lucas sequences (Un) and (Vn) are defined by the recurrence relations U n+1 = P Un+QU n−1 and V n+1 = P Vn + QV n−1 , n ≥ 1, with initial conditions U 0 = 0, U 1 = 1 and V 0 = 2, V 1 = 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 2 , Vn = 7kx 2 , Vn = 5kx 2 ±1, and Vn = 7kx 2 ±1 when k | P with k > 1. Moreover, we solve the equations Vn = 5x 2 ± 1 and Vn = 7x 2 ± 1.

Generalized Lucas Numbers of the form x2

“…Öğüt and Keskin [22] showed that only U 1 (P, −1) and U 2 (P, −1) may be of the form 11x 2 + 1 if P is odd. When P is odd, Karaatlı and Keskin [13] solved the equations V n (P, −1) = 5x 2 ∓ 1 and V n (P, −1) = 7x 2 ∓ 1.…”