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

Abstract

Let r ≥ 1 be an integer and U := (Un) n≥0 be the Lucas sequence given by U 0 = 0, U 1 = 1, and U n+2 = rU n+1 + Un, for all n ≥ 0. In this paper, we show that there are no positive integers r ≥ 3, x = 2, n ≥ 1 such that U x n + U x n+1 is a member of U.

On the Exponential Diophantine Equation $F_{n+1}^x - F_{n-1}^x = F_m^y$

“…m has no solutions in positive integers with 2 ≤ k < l and n ≥ 2. Similar problems with different recurrent sequences have been considered in [4,6,[19][20][21].…”