Strongly primitive elements

  • Authors:

  • Daniel Goldstein;Alfred W. Hales

  • Affiliations:

  • Center for Communications Research, San Diego, CA;Center for Communications Research, San Diego, CA

  • Venue:
  • SSC'07 Proceedings of the 2007 international conference on Sequences, subsequences, and consequences
  • Year:
  • 2007

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Abstract

Let n be a positive integer. A nonzero element γ of the finite field F of order q = 2n is said to be "strongly primitive" if every element (aγ+b)/(cγ+d), with a, b, c, d in {0, 1} and ad-bc not zero, is primitive in the usual sense. We show that the number N of such strongly primitive elements is asymptotic to θθ′ ċ q where θ is the product of (1-1/p) over all primes p dividing (q - 1) and θ′ is the product of (1 - 2/p) over the same set. Using this result and the accompanying error estimates, with some computer assistance for small n, we deduce the existence of such strongly primitive elements for all n except n = 1, 4, 6. This extends earlier work on Golomb's conjecture concerning the simultaneous primitivity of γ and γ + 1. We also discuss analogous questions concerning strong primitivity for other finite fields.