Prime factors of dynamical sequences

Let phi(t) is an element of Q(t) have degree d >= 2. For a given rational number x(0), define x(n+1) = phi(x(n)) for each n >= 0. If this sequence is not eventually periodic, and if phi does not lie in one of two explicitly determined affine conjugacy classes of rational functions, then x(n+1) - x(n) has a primitive prime factor in its numerator for all sufficiently large n. The same result holds for the exceptional maps provided that one looks for primitive prime factors in the denominator of x(n+1) - x(n) Hence the result for each rational function f of degree at least 2 implies (a new proof) that there are infinitely many primes. The question of primitive prime factors of x(n+Delta) - x(n) is also discussed for D uniformly bounded.