Primitive prime divisors in zero orbits of polynomials

Abstract

Let $(b_n)=(b_1,b_2,...)$ be a sequence of integers. A primitive prime divisor of a term $b_k$ is a prime which divides $b_k$ but does not divide any of the previous terms of the sequence. A zero orbit of a polynomial $f(z)$ is a sequence of integers $(c_n)$ where the $n$-th term is the $n$-th iterate of $f$ at 0. We consider primitive prime divisors of zero orbits of polynomials. In this note, we show that for integers $c$ and $d$, where $d>1$ and $c\ne \pm 1$, every iterate in the zero orbit of $f(z)=z^d+c$ contains a primitive prime whenever zero has an infinite orbit. If $c=\pm 1$, then every iterate after the first contains a primitive prime.

6 pages