On the equation V n = w x 2 ? 1

摘要

Let P ≥ 3 be an integer and ( V n ) denote Lucas sequence of the second kind defined by V 0 = 2 , V 1 = P , and V n + 1 = P V n ? V n ? 1 for n ≥ 1 . In this study, when P is odd and w ∈ { 10 , 14 , 15 , 21 , 30 , 35 , 42 , 70 , 210 } , we solved the equation V n = w x 2 ? 1 . We showed that only V 1 can be of the form w x 2 + 1 and only V 1 or V 2 can be of the form w x 2 ? 1 .