Document Type

Article

Publication Date

2017

Publication Title

New York Journal of Mathematics

ISSN

1076-9803

Volume

23

Abstract

Let φ(x) = x d + c be an integral polynomial of degree at least 2, and consider the sequence (φ n (0))∞n=0, which is the orbit of 0 under iteration by φ. Let Dd,c denote the set of positive integers n for which n | φ n (0). We give a characterization of Dd,c in terms of a directed graph and describe a number of its properties, including its cardinality and the primes contained therein. In particular, we study the question of which primes p have the property that the orbit of 0 is a single p-cycle modulo p. We show that the set of such primes is finite when d is even, and conjecture that it is infinite when d is odd.

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