Vol. 8, No. 7, 2014

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ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Bounded gaps between primes with a given primitive root

Paul Pollack

Vol. 8 (2014), No. 7, 1769–1786
Abstract

Fix an integer g 1 that is not a perfect square. In 1927, Artin conjectured that there are infinitely many primes for which g is a primitive root. Forty years later, Hooley showed that Artin’s conjecture follows from the generalized Riemann hypothesis (GRH). We inject Hooley’s analysis into the Maynard–Tao work on bounded gaps between primes. This leads to the following GRH-conditional result: Fix an integer m 2. If q1 < q2 < q3 < is the sequence of primes possessing g as a primitive root, then liminf n(qn+(m1) qn) Cm, where Cm is a finite constant that depends on m but not on g. We also show that the primes qn,qn+1,,qn+m1 in this result may be taken to be consecutive.

Keywords
primitive root, Artin's conjecture, bounded gaps, Maynard–Tao theorem
Mathematical Subject Classification 2010
Primary: 11A07
Secondary: 11N05
Milestones
Received: 27 April 2014
Revised: 21 June 2014
Accepted: 19 July 2014
Published: 21 October 2014
Authors
Paul Pollack
Department of Mathematics
University of Georgia
Boyd Graduate Studies Building
Athens, GA 30602
United States