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Let $p$ be a prime number. Is there a prime number $q$ such that $p$ is a primitive root of $q$?

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Have a look at Artin's conjecture. –  Felipe Voloch Jun 29 '13 at 12:55
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This question appears to be off-topic because it is about a well-known open problem. –  Felipe Voloch Jun 29 '13 at 12:55
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Ali asks if there is one (as opposed to infinitely many) such prime number $q$. This may be easier. –  Péter Komjáth Jun 29 '13 at 13:08

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Gupta and R. Murta showed that there were infinitely many primes $p$ for which there are infinitely many $q$. Heath-Brown generalized this to show that there are at most two prime numbers $p$ for which there are not infinitely many $q$ that they are primitive roots modulo $q$.

Heath-Brown, D. R.(4-OXM) Artin's conjecture for primitive roots. Quart. J. Math. Oxford Ser. (2) 37 (1986), no. 145, 27–38. 11A07 (11N13 11N35)

Gupta, Rajiv(1-IASP); Murty, M. Ram(3-MGL) A remark on Artin's conjecture. Invent. Math. 78 (1984), no. 1, 127–130. 11A07

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Thank you very very much. –  Ali Jun 29 '13 at 17:10

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