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

TL;DR

This paper employs p-adic methods to analyze fixed points, cycles, and collisions of the discrete exponential function, extending previous results and providing new counts for these phenomena modulo prime powers.

Contribution

It introduces p-adic techniques to count fixed points, two-cycles, and collisions of the discrete exponential map, advancing understanding beyond prime moduli.

Findings

Counted fixed points, two-cycles, and collisions modulo prime powers.

Extended previous results to all primes using p-adic methods.

Provided estimates for the number of solutions to related equations.

Abstract

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, or equivalently h is a fixed point of the discrete logarithm with base g. Zhang (1995) and Cobeli and Zaharescu (1999) answered with a "yes" for sufficiently large primes and gave estimates for the number of such pairs when g and h are primitive roots modulo p. In 2000, Campbell showed that the answer to Brizolis was "yes" for all primes. The first author has extended this question to questions about counting fixed points, two-cycles, and collisions of the discrete exponential map. In this paper, we use p-adic methods, primarily Hensel's lemma and p-adic interpolation, to count fixed points, two cycles, collisions, and solutions to related equations modulo powers of a prime p.