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

Let $r\ge 1$ be an integer and ${\bf U}:=(U_{n})_{n\ge 0}$ be the Lucas sequence given by $U_0=0$, $U_1=1,$ and $U_{n+2}=rU_{n+1}+U_n$, 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 ${\bf U}$.

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