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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 http://en.wikipedia.org/wiki/Artin%27s_conjecture_on_primitive_roots

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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No. $\!\!\!\!\!$ –  David Hansen Dec 2 '10 at 1:25
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@David: there were two questions. @Kate: Pieter Moree at Bonn will know the most recent advances if there were any. –  Franz Lemmermeyer Dec 2 '10 at 9:00
    
@Franz: Thanks for the information –  Kate Juschenko Dec 2 '10 at 19:28
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2 Answers

I'm not an expert, but the content of this article (last modified December 2004) suggests the Wikipedia article is reasonably up-to-date.

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It follows from GRH. Not known on its own.

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yes, the citations to this result are included here: en.wikipedia.org/wiki/Artin%27s_conjecture_on_primitive_roots –  Kate Juschenko Dec 2 '10 at 19:24
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