Fixed points for discrete logarithms

Abstract

We establish a conjecture of Brizolis that for every prime p>3 there is a primitive root g and an integer x in the interval [1,p-1] with log g x=x. Here, log g is the discrete logarithm function to the base g for the cyclic group (Z/pZ)×. Tools include a numerically explicit "smoothed" version of the Pólya-Vinogradov inequality for the sum of values of a Dirichlet character on an interval, a simple lower bound sieve, and an exhaustive search over small cases. © 2010 Springer-Verlag Berlin Heidelberg.