Artin's conjecture on primitive roots: Difference between revisions

Latest revision as of 04:24, 2 January 2012

Statement

Infinitude version

Suppose $a$ is an integer that is not equal to $-1$ and is not a perfect square, i.e., $a$ is not the square of an integer. Then, there exist infinitely many primes $p$ such that $a$ is a primitive root modulo $p$ .

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 $a$	(special cases of) generalized Riemann hypothesis
Gupta-Ram Murty theorem	Artin's conjecture holds for infinitely many $a$	Unconditional
Heath-Brown theorem on Artin's conjecture	Artin's conjecture holds for all but two exceptional values of $a$ . However, no explicit information about the explicit values of $a$	Unconditional

External links

A survey of Artin's conjecture and the developments related to it (PDF)

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 ===Infinitude version===
-Suppose <math>a</math> is an integer that is not equal to <math>-1</math> and is not a [[perfect square]], i.e., <math>a</math> is not the square of an integer. Then, there exist infinitely many primes <math>p</math> such that <math>a</math> is a [[primitive root]] modulo <math>p</math>.
+Suppose <math>a</math> is an integer that is not equal to <math>-1</math> and is not a [[perfect square]], i.e., <math>a</math> is not the square of an integer. Then, there exist infinitely many primes <math>p</math> such that <math>a</math> is a [[fact about::primitive root]] modulo <math>p</math>.
-==Density version===
+===Density version===
+{{fillin}}
+==Relation with other conjectures and known facts==
+<section begin="related"/>
+{| class="sortable" border="1"
+! Name of conjecture/fact !! Statement !! Conditional to ...
+|-
+| [[Hooley's theorem]] || Artin's conjecture holds for all <math>a</math> || (special cases of) [[weaker than::generalized Riemann hypothesis]]
+|-
+| [[Gupta-Ram Murty theorem]] || Artin's conjecture holds for infinitely many <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
+|}
+<section end="related"/>
+==External links==
+* [http://guests.mpim-bonn.mpg.de/moree/surva.pdf A survey of Artin's conjecture and the developments related to it (PDF)]