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

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 $\mathrm{.}Inthispaper,weshowthattherearenopositiveintegers$r\ge 3,~x\ne 2,~n\ge 1$suchthat$U_n^x+U_{n+1}^x$isamemberof${\bf U}$.