Generalized Lucas Numbers of the form x2

Abstract

Let P ≥ 3 be an integer and (V n ) denote generalized Lucas sequence defined by V 0 = 2, V 1 = P, and V n+1 = P V n − V n−1 for n ≥ 1. In this study, we solve the equation V n = 11x 2 ∓ 1. We show that the equation V n = 11x 2 + 1 has a solution only when n = 1 and P ≡ 1(mod 11). Moreover, we show that if the equation V n = 11x 2 − 1 has a solution, then P ≡ 2(mod 8) and P ≡ −1(mod 11).