Primitive Prime Divisors in the Critical Orbit of zd+c

Abstract

We prove the finiteness of the Zsigmondy set associated to the critical orbit of f(z) = z + c for rational values of c by uniformly bounding the size of the Zsigmondy set for all c ∈ Q and all d ≥ 2. We prove further that there exists an effectively computable bound M(c) on the largest element of the Zsigmondy set, and that under mild additional hypotheses on c, we have M(c) ≤ 3.