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

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