Let $ (P_m)_{m\ge 0} $ be the sequence of Pell numbers given by $ P_0=0, ~ P_1=1 $, and $ P_{m+2}=2P_{m+1}+P_m $ for all $ m\ge 0 $. In this paper, for an integer $d\ge 2$ which is square free, we show that there is at most one value of the positive integer $x$ participating in the Pell equation $x^{2}-dy^{2} =\pm 1$, which is a product of two Pell numbers.

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.

Let r ≥ 1 be an integer and U := {Un} n≥0 be the Lucas sequence given by U0 = 0, U1 = 1, and Un+2 = rUn+1 + Un for n ≥ 0. In this paper, we explain how to find all the solutions of the Diophantine equation,where A, B, C, D are given integers with A = 0, B = 0, m, n, m1, n1 are nonnegative integer unknowns and r is also unknown.