Mersenne numbers which are products of two Pell numbers

Abstract

Let (P-n)(n >= 0) be the Pell sequence defined by P-0 = 0, P-1 = 1 and Pn+2 2P(n+1) + P-n for n >= 0 and M-k + 2(k) - 1 be the k-th Mersenne number for k >= 1: We show that the Diophantine equation PnPm + M-k with m <= n has only the unique positive integer solution (n, m, k) + (1; 1; 1).