Primitive prime divisors in backward orbits

Abstract

Let f be a polynomial, a,b be two algebraic numbers, and (an)n≥1 be a sequence satisfying f(a1)=a and f(an)=an−1 for all n≥2. We generalize the concept of primitive prime divisors to the shifted sequence (an−b)n≥1 and study the generalized Zsigmondy set, that is, the set of n such that all prime ideal divisors of an−b divide at least one prior term. The finiteness of Zsigmondy sets are determined in various cases.