New Conjectures and Results for Small Cycles of the Discrete Logarithm

Details New Conjectures and Results for Small Cycles of the Discrete Logarithm New Conjectures and Results for Small Cycles of the Discrete Logarithm

2003-05-21

arxiv.org/abs/math/0305305v2

In High Primes and Misdemeanours: lectures in honour of the 60th birthday of Hugh Cowie Williams, AMS, 2004, pp. 245--254.

Mathematics

Number Theory

10 pages, to appear in the proceedings of the "Conference in Number Theory in Honour of Professor H.C. Williams", 2003. Versio

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 give a summary of conjectures and results which follow from these heuristics, building again on the aforementioned work. We also make some new conjectures and prove some average versions of the results.

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