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Mathematics of Computation

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Some heuristics and results for small cycles of the discrete logarithm

Authors: Joshua Holden and Pieter Moree
Journal: Math. Comp. 75 (2006), 419-449
MSC (2000): Primary 11A07; Secondary 11N37, 94A60, 11-04
Published electronically: June 28, 2005
MathSciNet review: 2176407
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Abstract: Brizolis asked the question: does every prime $p$ have a pair $(g,h)$ such that $h$ is a fixed point for the discrete logarithm with base $g$? The first author previously extended this question to ask about not only fixed points but also two-cycles, and gave heuristics (building on work of Zhang, Cobeli, Zaharescu, Campbell, and Pomerance) for estimating the number of such pairs given certain conditions on $g$ and $h$. In this paper we extend these heuristics and prove results for some of them, building again on the aforementioned work. We also make some new conjectures and prove some average versions of the results.

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Additional Information

Joshua Holden
Affiliation: Department of Mathematics, Rose-Hulman Institute of Technology, Terre Haute, Indiana, 47803-3999

Pieter Moree
Affiliation: Max-Planck-Institut für Mathematik, Vivatsgasse 7, D-53111 Bonn, Germany
MR Author ID: 290905

Received by editor(s): January 4, 2004
Received by editor(s) in revised form: August 30, 2004
Published electronically: June 28, 2005
Additional Notes: The first author would like to thank the Rose-Hulman Institute of Technology for the special stipend which supported this project during the summer of 2002
The research of the second author was carried out while he was a visiting assistant professor at the University of Amsterdam and supported by Prof. E. M. Opdam’s Pioneer Grant of the Netherlands Organization for Scientific Research (NWO)
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.