New Conjectures and Results for Small Cycles of the Discrete Logarithm

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 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.