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 |
