  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.

Number Theory