On the exponential Diophantine equation related to powers of two consecutive terms of Lucas sequences

Abstract

Let $r\ge 1$ be an integer and $\mathbf{U}:=(U_n{)}_{n\ge 0}$ be the Lucas sequence given by $U_0=0$, $U_1=1,$ and $U_{n+2}=r{U}_{n+1}+U_n$, for all $n\ge 0$. In this paper, we show that there are no positive integers $r\ge 3,x\ne 2,n\ge 1$ such that $U_n^x+U_{n+1}^x$ is a member of $\mathbf{U}$.

Accepted manuscript