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Primitive prime divisors in the critical orbit of \(z^d+c\). (English) Zbl 1329.11020
Summary: We prove the finiteness of the Zsigmondy set associated to the critical orbit of \(z^d+c\) for rational values of \(c\) by uniformly bounding the size of the Zsigmondy set for all \(c\in\mathbb{Q}\) and all \(d\geq2\). We prove further that there exists an effectively computable bound \(M(c)\) on the largest element of the Zsigmondy set, and that, under mild additional hypotheses on \(c\), we have \(M(c)\leq3\).

MSC:
11B83 Special sequences and polynomials
11A41 Primes
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
37P05 Arithmetic and non-Archimedean dynamical systems involving polynomial and rational maps
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