Artin's Primitive Root Conjecture – A Survey

Abstract

Abstract. One of the first concepts one meets in elementary number theory is that of the multiplicative order. We give a survey of the literature on this topic emphasizing the Artin primitive root conjecture (1927). The first part of the survey is intended for a rather general audience and rather colloquial, whereas the second part is intended for number theorists and ends with several open problems. The contributions in the survey on 'elliptic Artin' are due to Alina Cojocaru. Wojciec Gajda wrote a section on 'Artin for K-theory of number fields', and Hester Graves (together with me) on 'Artin's conjecture and Euclidean domains'.

Irregular primes with respect to Genocchi numbers and Artin's primitive root conjecture

“…Artin in 1927 conjectured that this set, when g is not a square, is infinite and also conjectured a density for it. To this day, this conjecture is open; see [25] for a survey.…”

An analogue of Artin’s conjecture for multiplicative subgroups of the rationals

“…The work of Hooley was generalized by several authors (including Moree [7], Murata [10], Lenstra [4], Wagstaff [15] and others) who determine, assuming the GRH, an asymptotic formula for N a (x, m). In particular, Lenstra, Moree and Stevenhagen, in [5], propose a complete characterization, assuming the GRH, of the pairs (a, m) for which there are no primes p a with ind p a = m. For a complete and updated account, we refer to Moree's Survey [8].…”

On Self-Dual Four Circulant Codes

“…In number theory, Artin's conjecture on primitive roots states that a given integer q which is neither a perfect square nor −1 is a primitive root modulo infinitely many primes l [9]. It was proved conditionally under GRH by Hooley [4].…”