Fixed Points for Discrete Logarithms

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 Polya–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.