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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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