Some Heuristics and Results for Small Cycles of the Discrete Logarithm

Details Some Heuristics and Results for Small Cycles of the Discrete Logarithm Some Heuristics and Results for Small Cycles of the Discrete Logarithm

2004-01-03

arxiv.org/abs/math/0401013v2

Math. Comp. 75 (2006), 419-449

Mathematics

Number Theory

32 pages; to appear in Mathematics of Computation; revisions as suggested by referees, including notation and minor improvemen

Scientific paper

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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