Proof Attempts
In 1967, Hooley published a conditional proof for the conjecture, assuming certain cases of the Generalized Riemann hypothesis. In 1984, R. Gupta and M. Ram Murty showed unconditionally that Artin's conjecture is true for infinitely many a using sieve methods. Roger Heath-Brown improved on their result and showed unconditionally that there are at most two exceptional prime numbers a for which Artin's conjecture fails. This result is not constructive, as far as the exceptions go. For example, it follows from the theorem of Heath-Brown that one out of 3, 5, and 7 is a primitive root modulo p for infinitely many p. But the proof does not provide us with a way of computing which one.
Read more about this topic: Artin's Conjecture On Primitive Roots
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