PRIMEFREE SHIFTED BINARY LINEAR RECURRENCE SEQUENCES

We say a sequence X = (x(n))(n >= 0) is primefree if |x(n)| is not prime for all n >= 0 and, to rule out trivial situations, we require that no single prime divides all terms of chi. For a, b, w(0), w(1) is an element of Z, we let W (w(0), w(1), a, b) = (w(n))(n >= 0) denote the general linear binary recurrence that is defined by w(n) = aw(n-1) + bw(n-2) for n >= 2. It has been shown recently for any sequence chi is an element of {W (0, 1, a, 1), W (2, a, a, 1)}, that there exist in fi nitely positive integers k such that both of the shifted sequences chi +/- k are simultaneously primefree, and moreover, each term has at least two distinct prime divisors. In this article, we extend these theorems by establishing analogous results for all but finitely many sequences chi is an element of {W(0, 1, a, -1), W (2, a, a, -1), W (1, 1, a,-1), W (- 1, 1, a, -1)}, which provides additional evidence to support a conjecture of Ismailescu and Shim.