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

Let r ≥ 1 be an integer and U : = ( U n ) n ≥ 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 ≥ 0 . In this paper, we show that there are no positive integers r ≥ 3 , x ≠ 2 , n ≥ 1 such that U n x + U n + 1 x is a member of U .