Quadratic Recurrences with a Positive Density of Prime Divisors

For f(x) ∈ Z[x] and a ∈ Z, we let f(x) be the nth iterate of f(x), P (f, a) = {p prime : p|f(a) for some n}, and D(P (f, a)) denote the natural density of P (f, a) within the set of primes. A conjecture of Jones [5] indicates that D(P (f, a)) = 0 for most quadratic f . In this paper, we find an exceptional family of (f ,a) such that D(P (f, a)) > 0 by considering ft(x) = (x + t) − 2 − t and at = ft(0) for t ∈ Z. We prove that if t is not of the form ±M±2 or ±2M±2, then D(P (ft, at)) = 1 3 . We also determine D(P (ft, at)) in some cases when the density is not equal to 13 . Our main technique involves the computation of the Galois group of the nth iterate of (x + t) − 2− t. Our results suggest a connection between the arithmetic dynamics of the conjugates of x and the conjugates of x − 2.