Artin prime producing polynomials
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- by Amir Akbary and Keilan Scholten PDF
- Math. Comp. 84 (2015), 1861-1882 Request permission
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 $g$.References
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Additional Information
- Amir Akbary
- Affiliation: Department of Mathematics and Computer Science, University of Lethbridge, Lethbridge, AB T1K 3M4, Canada
- MR Author ID: 650700
- 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
- 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.
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp. 84 (2015), 1861-1882
- MSC (2010): Primary 11A07, 11N32
- DOI: https://doi.org/10.1090/S0025-5718-2014-02902-5
- MathSciNet review: 3335895