On the equation Vn=wx2∓1

Abstract

Let P≥3 be an integer and (Vn) denote Lucas sequence of the second kind defined by V0=2,V1=P, and Vn+1=PVn−Vn−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 Vn=wx2∓1. We showed that only V1 can be of the form wx2+1 and only V1 or V2 can be of the form wx2−1.

Keywords

• Lucas sequence of the second kind
• Congruences
• Diophantine equations