New Conjectures and Results for Small Cycles of the Discrete Logarithm
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.
External Access URL
Holden, Joshua and Pieter Moree. New conjectures and results for small cycles of the discrete logarithm. In: High Primes and Misdemeanours: lectures in honour of the 60th birthday of Hugh Cowie Williams, no. 41 in Fields Institute Communications, AMS, 2004.