New York Journal of Mathematics
Volume 23 (2017) 1045-1063


Annie S. Chen, T. Alden Gassert, and Katherine E. Stange

Index divisibility in dynamical sequences and cyclic orbits modulo p

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Published: August 18, 2017
Keywords: arithmetic dynamics, dynamical portrait, index divisibility, cycle, orbit, functional digraph, dynamical sequence, polynomial map, iteration, quadratic map, divisibility sequence, integer sequence, post-critical orbit
Subject: Primary: 37P05, 37P25, 11Y55, Secondary: 11B37, 11B39, 11B50, 11G99

Let ϕ(x) = xd + c be an integral polynomial of degree at least 2, and consider the sequence (ϕn(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 | ϕn(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.


The third author's work was supported by the National Security Agency grant H98230-16-1-0040 and National Science Foundation grant DMS-1643552.

Author information

Annie S. Chen:
Stanford University, 450 Serra Mall, Stanford, CA 94305

T. Alden Gassert:
Hobart and William Smith Colleges, 300 Pulteney Drive, Geneva, NY 14456

Katherine E. Stange:
Department of Mathematics, University of Colorado, Campus Box 395, Boulder, Colorado 80309-0395