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Published in:
2010 | OriginalPaper | Chapter
Fixed Points for Discrete Logarithms
Authors : Mariana Levin, Carl Pomerance, K. Soundararajan
Published in: Algorithmic Number Theory
Publisher: Springer Berlin Heidelberg
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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/
p
ℤ)
×
. 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.