On Gaps Between Primitive Roots in the Hamming Metric

Rainer Dietmann, Christian Elsholtz*, Igor Shparlinski

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


We consider a modification of the classical number theoretic question about the gaps between consecutive primitive roots modulo a prime p, which by the well-known result of Burgess are known to be at most p1/4+o(1). Here we measure the distance in the Hamming metric and show that if p is a sufficiently large r-bit prime, then, for any integer n∈[1, p], one can obtain a primitive root modulo p by changing at most 0.11002786… r binary digits of n. This is stronger than what can be deduced from the Burgess result. Experimentally, the number of necessary bit changes is very small. We also show that each Hilbert cube contained in the complement of the primitive roots modulo p has dimension at most O(p1/5+ϵ), improving on previous results of this kind.
Original languageEnglish
Pages (from-to)1043-1055
JournalThe Quarterly Journal of Mathematics
Issue number4
Publication statusPublished - 2013

Fields of Expertise

  • Information, Communication & Computing

Treatment code (Nähere Zuordnung)

  • Basic - Fundamental (Grundlagenforschung)
  • Theoretical
  • Experimental

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