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Annie S. Chen, T. Alden Gassert, and Katherine E. Stange
Index divisibility in dynamical sequences and cyclic orbits modulo p view print
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| Published: |
August 18, 2017
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| 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 |
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Abstract
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.
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| Acknowledgements
The third author's work was supported by the National Security Agency grant H98230-16-1-0040 and National Science Foundation grant DMS-1643552.
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| Author information
Annie S. Chen:
Stanford University, 450 Serra Mall, Stanford, CA 94305
asc8@stanford.edu
T. Alden Gassert:
Hobart and William Smith Colleges, 300 Pulteney Drive, Geneva, NY 14456
gassert@hws.edu
Katherine E. Stange:
Department of Mathematics, University of Colorado, Campus Box 395, Boulder, Colorado 80309-0395
kstange@math.colorado.edu
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