# Difference between revisions of "Artin's conjecture on primitive roots"

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==Relation with other conjectures and known facts== | ==Relation with other conjectures and known facts== | ||

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| [[Heath-Brown theorem on Artin's conjecture]] || Artin's conjecture holds for all but two exceptional values of <math>a</math>. However, no explicit information about the explicit values of <math>a</math> || Unconditional | | [[Heath-Brown theorem on Artin's conjecture]] || Artin's conjecture holds for all but two exceptional values of <math>a</math>. However, no explicit information about the explicit values of <math>a</math> || Unconditional | ||

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==External links== | ==External links== | ||

* [http://guests.mpim-bonn.mpg.de/moree/surva.pdf A survey of Artin's conjecture and the developments related to it (PDF)] | * [http://guests.mpim-bonn.mpg.de/moree/surva.pdf A survey of Artin's conjecture and the developments related to it (PDF)] |

## Latest revision as of 04:24, 2 January 2012

## Contents

## Statement

### Infinitude version

Suppose is an integer that is not equal to and is not a perfect square, i.e., is not the square of an integer. Then, there exist infinitely many primes such that is a primitive root modulo .

### Density version

*Fill this in later*

## Relation with other conjectures and known facts

Name of conjecture/fact | Statement | Conditional to ... |
---|---|---|

Hooley's theorem | Artin's conjecture holds for all | (special cases of) generalized Riemann hypothesis |

Gupta-Ram Murty theorem | Artin's conjecture holds for infinitely many | Unconditional |

Heath-Brown theorem on Artin's conjecture | Artin's conjecture holds for all but two exceptional values of . However, no explicit information about the explicit values of | Unconditional |