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.
Contents
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.
Formulation
Let a be an integer which is not a perfect square and not −1. Write a = a0b2 with a0 square-free. Denote by S(a) the set of prime numbers p such that a is a primitive root modulo p. Then
- S(a) has a positive asymptotic density inside the set of primes. In particular, S(a) is infinite.
- Under the conditions that a is not a perfect power and that a0 is not congruent to 1 modulo 4 (sequence A085397 in the OEIS), this density is independent of a and equals Artin's constant which can be expressed as an infinite product
C A r t i n = ∏ p p r i m e ( 1 − 1 p ( p − 1 ) ) = 0.3739558136 … (sequence A005596 in the OEIS).
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.
Example
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 A001122 in the OEIS)
S(2) = {3, 5, 11, 13, 19, 29, 37, 53, 59, 61, 67, 83, 101, 107, 131, 139, 149, 163, 173, 179, 181, 197, 211, 227, 269, 293, 317, 347, 349, 373, 379, 389, 419, 421, 443, 461, 467, 491, ...}.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.
Proof attempts
In 1967, Hooley published a conditional proof for the conjecture, assuming certain cases of the Generalized Riemann hypothesis.