We explore some questions related to one of Brizolis: does every prime p
have a pair (g,h) such that h is a fixed point for the discrete logarithm
with base g? We extend this question to ask about not only fixed points
but also two-cycles. Campbell and Pomerance have not only answered the
fixed point question for sufficiently large p but have also rigorously
estimated the number of such pairs given certain conditions on g and h. We
attempt to give heuristics for similar estimates given other conditions on
g and h and also in the case of two-cycles. These heuristics are
well-supported by the data we have collected, and seem suitable for
conversion into rigorous estimates in the future.