Artin's conjecture on primitive roots
In number theory, Artin's conjecture on primitive roots states that a given integer a which is neither a perfect square nor −1 is a primitive root modulo infinitely many primes p. The conjecture also ascribes an asymptotic density to these primes. This conjectural density equals Artin's constant or a rational multiple thereof.
The conjecture was made by Emil Artin to Helmut Hasse on September 27, 1927, according to the latter's diary. The conjecture is still unresolved as of May 2013. In fact, there is no single value of a for which Artin's conjecture is proved.
Let a be an integer which is not a perfect square and not −1. Write a = a0b2 with a0square-free. Denote by S(a) the set of prime numbers p such that a is a primitive root modulo p. Then
Similar conjectural product formulas exist for the density when a does not satisfy the above conditions. In these cases, the conjectural density is always a rational multiple of CArtin.
For example, take a = 2. The conjecture claims that the set of primes p for which 2 is a primitive root has the above density CArtin. The set of such primes is (sequence in the OEIS)
It has 38 elements smaller than 500 and there are 95 primes smaller than 500. The ratio (which conjecturally tends to CArtin) is 38/95 = 2/5 = 0.4.
In 1967, Hooley published a conditional proof for the conjecture, assuming certain cases of the Generalized Riemann hypothesis.
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