Primitive prime divisors in zero orbits of polynomials

2010/09/21 by Doerksen, Kevin, Haensch, Anna
#11B37 (Primary) #37P05 (Secondary) #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1009.3971

Abstract

Let (b_n) = (b₁, b₂, ...) 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 ≠ ± 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 = ± 1, then every iterate after the first contains a primitive prime.

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