Artin's conjecture on primitive roots states that a given integer a that is neither a square number nor -1 is a primitive root modulo infinitely many primes p, and ascribes an asymptotic density to those primes equal to Artin's constant or a rational multiple of it. The conjecture was made by Emil Artin to Helmut Hasse on September 27, 1927, according to Hasse's diary, and remains unresolved. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
Facts
StatementA given integer a that is neither a perfect square nor -1 is a primitive root modulo infinitely many primes p, and the set of such primes has an asymptotic density equal to Artin's constant or a rational multiple of it. 1 Proposed Year Progress Toward ResolutionUnresolved unconditionally. Hooley proved it in 1967 assuming certain cases of the generalized Riemann hypothesis. Heath-Brown proved in 1986 that at least one of 2, 3 or 5 is a primitive root modulo infinitely many primes, and that there are at most two primes for which the conjecture fails. 1 Classification
Resolution Status Prize Status
Prize Status (category) Connections
In Branch
Source Artin's Conjecture on Primitive Roots (Wikipedia)
Posed By
Source Artin's Conjecture on Primitive Roots (Wikipedia)
Sources
1. Artin's Conjecture on Primitive Roots (Wikipedia)
Wikimedia FoundationLead section
made by Emil Artin to Helmut Hasse on September 27, 1927
Lead section, statement sentence
is a primitive root modulo infinitely many primes p
Partial results section, Heath-Brown Corollary 1
D. R. Heath-Brown proved in 1986 (Corollary 1) that at least one of 2, 3, or 5 is a primitive root modulo infinitely many primes p.
Partial results
In 1967, Christopher Hooley published a conditional proof for the conjecture, assuming certain cases of the extended Riemann hypothesis. Without the generalized Riemann hypothesis, there is no single value of a for which Artin's conjecture is proved.
Partial results section, Heath-Brown Corollary 2
He also proved (Corollary 2) that there are at most two primes for which Artin's conjecture fails.
In Branch: Number Theory, Lead paragraph
In number theory, Artin's conjecture on primitive roots states that a given integer a that is neither a square number nor −1 is a primitive root modulo infinitely many primes p.
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