Primitive prime divisors in zero orbits of polynomials

TL;DR

This paper investigates primitive prime divisors in zero orbits of specific polynomials, establishing conditions under which all iterates contain primitive primes, thus extending understanding of prime divisors in polynomial dynamics.

Contribution

It proves that for polynomials of the form $f(z) = z^d + c$, all iterates have primitive prime divisors under certain conditions, advancing knowledge of prime divisors in polynomial orbits.

Findings

Every iterate of $f(z) = z^d + c$ has a primitive prime if the zero orbit is infinite and $c eq \u00b1$.

If $c = \u00b1$, then all iterates after the first have a primitive prime.

The results extend primitive prime divisor theory to polynomial zero orbits in number theory.

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 $c \neq \pm 1$, 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.