Powers in products of terms of Pell's and Pell–Lucas Sequences

In this paper, we consider the usual Pell and Pell–Lucas sequences. The Pell sequence [Formula: see text] is given by the recurrence un= 2un-1+ un-2with initial condition u0= 0, u1= 1 and its associated Pell–Lucas sequence [Formula: see text] is given by the recurrence vn= 2vn-1+ vn-2with initial condition v0= 2, v1= 2. Let n, d, k, y, m be positive integers with m ≥ 2, y ≥ 2 and gcd (n, d) = 1. We prove that the only solutions of the Diophantine equation unun+d⋯un+(k-1)d= ymare given by u7= 132and u1u7= 132and the equation vnvn+d⋯vn+(k-1)d= ymhas no solution. In fact, we prove a more general result.