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Brizolis asked for which primes p greater than 3 does there exist a pair (g,h) such that h is a fixed point of the discrete exponential map with base g, orequivalently h is a fixed point of the discrete logarithm with base g. Zhang(1995) and Cobeli and Zaharescu (1999) answered with a "yes" for sufficientlylarge primes and gave estimates for the number of such pairs when g and h areprimitive roots modulo p. In 2000, Campbell showed that the answer to Brizoliswas "yes" for all primes. The first author has extended this question toquestions about counting fixed points, two-cycles, and collisions of thediscrete exponential map. In this paper, we use p-adic methods, primarilyHensel's lemma and p-adic interpolation, to count fixed points, two cycles,collisions, and solutions to related equations modulo powers of a prime p.

Counting Fixed Points, Two-Cycles, and Collisions of the Discrete Exponential Function using p-adic Methods