On the Lucas sequence equations $$V_{n}(P,1)=wkx^{2},$$ V n ( P , 1 ) = w k x 2 , $$w\in \left\{ 5,7\right\} $$ w ∈ 5 , 7

Abstract

Let P be an odd integer and (V n ) denote the generalized Lucas sequence defined by V 0 = 2, V 1 = P, and V n+1 = PV n + V n−1 for n ≥ 1. In this study, we solve the equations V n = 5kx 2 , V n = 7kx 2 , V n = 5kx 2 ± 1, and V n = 7kx 2 ± 1 when k|P with k > 1. Moreover, applying some of the results, we obtain complete solutions to the equations V n = σ x 2 , σ ∈ {15, 21, 35}.