Primitive Prime Divisors in Zero Orbits of Polynomials

Let (bn) = (b1, b2, . . . ) be a sequence of integers. A primitive prime divisor of a term bk is a prime which divides bk but does not divide any of the previous terms of the sequence. A zero orbit of a polynomial φ(z) is a sequence of integers (cn) where the n-th term is the n-th iterate of φ at 0. We consider primitive prime divisors of zero orbits of polynomials. In this note, we show that for c, d in Z, where d ≥ 2 and c "= ±1, every iterate in the zero orbit of φ(z) = zd + c contains a primitive prime divisor whenever zero has an infinite orbit. If c = ±1, then every iterate after the first contains a primitive prime divisor.