In number theory, Artin's conjecture on primitive roots states that a given integer a that is neither a square number 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.
In fact, there is no single value of a for which Artin's conjecture is proved.
Let a be an integer that is not a square number and not −1.
Denote by S(a) the set of prime numbers p such that a is a primitive root modulo p. Then the conjecture states The positive integers satisfying these conditions are: The negative integers satisfying these conditions are: Similar conjectural product formulas[1] 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.
If a is a square number or a = −1, then the density is 0, and if a is a perfect pth power for prime p, then the number needs to be multiplied by
for all such primes p), and if a0 is congruent to 1 mod 4, then the number needs to be multiplied by
for all prime factors p of a0, e.g. for a = 8 = 23, the density is
, and for a = 5 (which is congruent to 1 mod 4), the density is
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) 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, Christopher Hooley published a conditional proof for the conjecture, assuming certain cases of the generalized Riemann hypothesis.
[2] Without the generalized Riemann hypothesis, there is no single value of a for which Artin's conjecture is proved.
D. R. Heath-Brown proved in 1986 (Corollary 1) that at least one of 2, 3, or 5 is a primitive root modulo infinitely many primes p.[3] He also proved (Corollary 2) that there are at most two primes for which Artin's conjecture fails.
, Lang and Trotter gave a conjecture for rational points on
analogous to Artin's primitive root conjecture.
[4] Specifically, they said there exists a constant
for a given point of infinite order
in the set of rational points
generates the whole set of points in
[5] Here we exclude the primes which divide the denominators of the coordinates of
Gupta and Murty proved the Lang and Trotter conjecture for
with complex multiplication under the Generalized Riemann Hypothesis, for primes splitting in the relevant imaginary quadratic field.
[6] Krishnamurty proposed the question how often the period of the decimal expansion
The claim is that the period of the decimal expansion of a prime in base
The result was proven by Hasse in 1966.