Primitive Divisors in Arithmetic Dynamics

Let φ(z) ∈ Q(z) be a rational function of degree d ≥ 2 with φ(0) = 0 and such that φ does not vanish to order d at 0. Let α ∈ Q have infinite orbit under iteration of φ and write φ(α) = An/Bn as a fraction in lowest terms. We prove that for all but finitely many n ≥ 0, the numerator An has a primitive divisor, i.e., there is a prime p such that p | An and p ∤ Ai for all i < n. More generally, we prove an analogous result when φ is defined over a number field and 0 is a periodic point for φ. Introduction Let A = (An)n≥1 be a sequence of integers. A prime p is called a primitive divisor of An if p | An and p ∤ Ai for all 1 ≤ i < n. The Zsigmondy set of A is the set Z(A) = { n ≥ 1 : An does not have a primitive divisor } . A classical theorem of Bang [2] (for b = 1) and Zsigmondy [30] in general says that if a, b ∈ Z are integers with a > b > 0, then Z ( (a − b)n≥1 ) is a finite set. Indeed assuming that gcd(a, b) = 1, they prove that Z (