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I am interested in the following question:

Is it known that $2$ is a primitive root modulo $p$ for infinitely many primes $p$?

There is some information about Artin's conjecture in Wikipedia. I need to know if it is up-to-date and if one can say something about the case $n=2$.

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    $\begingroup$ No. $\!\!\!\!\!$ $\endgroup$
    – David Hansen
    Commented Dec 2, 2010 at 1:25
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    $\begingroup$ @David: there were two questions. @Kate: Pieter Moree at Bonn will know the most recent advances if there were any. $\endgroup$
    – Franz Lemmermeyer
    Commented Dec 2, 2010 at 9:00
  • $\begingroup$ 11 years later: still no (to the best of my knowledge). $\endgroup$
    – Gerry Myerson
    Commented Mar 6, 2022 at 23:23

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It follows from GRH. Not known on its own.

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I'm not an expert, but the content of the article Artin's primitive root conjecture -a survey - (modified December 2004) by Pieter Moree suggests the Wikipedia article is reasonably up-to-date.

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