Primitive prime divisors in zero orbits of polynomials
The authors study primitive prime divisors in the orbit of zero under iterated application of a unicritical polynomial \(\varphi(z) = z^d+c\), with \(c,d \in \mathbb{Z}\). The main theorem states that, when \(d \geq 2\) and \(0\) is a wandering point of \(\varphi\), then \(\varphi^n(0)\) has a primitive prime divisor for all \(n \geq 1\), except of course for \(n = 1\) when \(c = \pm 1\). (Zero is not wandering exactly when \(c=0\), or \(c=-1\) and \(d\) is even, or \(c=-2\) and \(d=2\).) Here, a primitive prime divisor is a prime that divides \(\varphi^n(0)\) but not \(\varphi^k(0)\) for any \(k < n\). This improves a theorem of \textit{B. Rice} [Integers 7, No. 1, A26, 16 p. (2007; Zbl 1165.11028)]. The proof is based on Rice's observation that these sequences are rigid divisibility sequences.
scientific article; zbMATH DE number 6054420