Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Index Divisibility in Dynamical Sequences and Cyclic Orbits Modulo $p$   Let $\phi(x) = x^d + c$ be an integral polynomial of degree at least 2, and
consider the sequence $(\phi^n(0))_{n=0}^\infty$, which is the orbit of $0$
under iteration by $\phi$. Let $D_{d,c}$ denote the set of positive integers
$n$ for which $n \mid \phi^n(0)$. We give a characterization of $D_{d,c}$ 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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The categories of this paper are: hep-th
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The categories of this paper are: acc-phys
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The categories of this paper are: hep-ph
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The categories of this paper are: comp-gas
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The categories of this paper are: dg-ga
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The categories of this paper are: nucl-th
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The categories of this paper are: solv-int
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The categories of this paper are: cs
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The categories of this paper are: patt-sol
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The categories of this paper are: ao-sci
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The categories of this paper are: gr-qc
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The categories of this paper are: bayes-an
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The categories of this paper are: chem-ph
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The categories of this paper are: plasm-ph
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The categories of this paper are: adap-org
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The categories of this paper are: chao-dyn