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Artin prime producing polynomials


Authors: Amir Akbary and Keilan Scholten
Journal: Math. Comp. 84 (2015), 1861-1882
MSC (2010): Primary 11A07, 11N32
DOI: https://doi.org/10.1090/S0025-5718-2014-02902-5
Published electronically: December 2, 2014
MathSciNet review: 3335895
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Abstract: We define an Artin prime for an integer $ g$ to be a prime such that $ g$ is a primitive root modulo that prime. Let $ g\in \mathbb{Z}\setminus \{-1\}$ and not a perfect square. A conjecture of Artin states that the set of Artin primes for $ g$ has a positive density. In this paper we study a generalization of this conjecture for the primes produced by a polynomial and explore its connection with the problem of finding a fixed integer $ g$ and a prime producing polynomial $ f(x)$ with the property that a long string of consecutive primes produced by $ f(x)$ are Artin primes for $ g$. By employing some results of Moree, we propose a general method for finding such polynomials $ f(x)$ and integers $ g$. We then apply this general procedure for linear, quadratic, and cubic polynomials to generate many examples of polynomials with very large Artin prime production length. More specifically, among many other examples, we exhibit linear, quadratic, and cubic (respectively) polynomials with $ 6355$, $ 37951$, and $ 10011$ (respectively) consecutive Artin primes for certain integers $ g$.


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Additional Information

Amir Akbary
Affiliation: Department of Mathematics and Computer Science, University of Lethbridge, Lethbridge, AB T1K 3M4, Canada
Email: amir.akbary@uleth.ca

Keilan Scholten
Affiliation: Department of Mathematics and Computer Science, University of Lethbridge, Lethbridge, AB T1K 3M4, Canada
Email: Keilan.Scholten@outlook.com

DOI: https://doi.org/10.1090/S0025-5718-2014-02902-5
Keywords: Artin's primitive root conjecture, prime producing polynomials
Received by editor(s): May 27, 2013
Received by editor(s) in revised form: September 8, 2013, and October 8, 2013
Published electronically: December 2, 2014
Additional Notes: Research of the first author was supported by NSERC. Research of the second author was supported by an NSERC USRA award.
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.