Fixed Points for Discrete Logarithms

  • Mariana Levin
  • Carl Pomerance
  • K. Soundararajan
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6197)


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.


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© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Mariana Levin
    • 1
  • Carl Pomerance
    • 2
  • K. Soundararajan
    • 3
  1. 1.Graduate Group in Science and Mathematics EducationUniversity of CaliforniaBerkeleyUSA
  2. 2.Department of MathematicsDartmouth CollegeHanoverUSA
  3. 3.Department of MathematicsStanford UniversityStanfordUSA

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