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. The conjecture is still unresolved as of 2026. In fact, there is no single value of a for which Artin's conjecture is proved. However, it is proven that there cannot be more than two primes a for which Artin's conjecture fails.
Formulation
Let a be an integer that is not a square number 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 the conjecture states 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 a0 is not congruent to 1 modulo 4 , this density is independent of a and equals Artin's constant, which can be expressed as an infinite product .
The positive integers satisfying these conditions are:
2, 3, 6, 7, 10, 11, 12, 14, 15, 18, 19, 22, 23, 24, 26, 28, 30, 31, 34, 35, 38, 39, 40, 42, 43, 44, 46, 47, 48, 50, 51, 54, 55, 56, 58, 59, 60, 62, 63, …
The negative integers satisfying these conditions are:
2, 4, 5, 6, 9, 10, 13, 14, 16, 17, 18, 20, 21, 22, 24, 25, 26, 29, 30, 33, 34, 36, 37, 38, 40, 41, 42, 45, 46, 49, 50, 52, 53, 54, 56, 57, 58, 61, 62, …
Similar conjectural product formulasArtin's ConstantHeath-Brown: Artin Primitive Roots Conjecture 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 either square number or −1, then this density is 0.
If a is not a perfect power but a0 is congruent to 1 modulo 4, then this density is .
If a is a perfect power, let r be the largest number such that a is an rth power, then this density is:
if a0 is not congruent to 1 modulo 4;
if a0 is congruent to 1 modulo 4.
For , the conjectural densities given by the formula are CArtin times
0, 1, 1, 0, , 1, 1, , 0, 1, 1, 1, , 1, 1, 0, , 1, 1, , , 1, 1, 1, 0, 1, , 1, , 1, 1, , , 1, 1, 0, , 1, 1, 1, , 1, 1, 1, , 1, 1, 1, 0, 1, ... ( for the numerators, for the denominators)
For , the conjectural densities given by the formula are CArtin times
0, 1, , 1, 1, 1, , , 1, 1, , , 1, 1, , 1, 1, 1, , 1, 1, 1, , 1, 1, 1, , , 1, 1, , , 1, 1, , 1, 1, 1, , 1, 1, 1, , , 1, 1, , , 1, 1, ... ( for the numerators, for the denominators)
Examples For example, take a = 2. The conjecture is that the set of primes p for which 2 is a primitive root has the density CArtin. The set of such primes is 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.
For a = 4 = 22, which is a square number, the density is 0, and for a = 8 = 23, which is a power of 2, the conjectured density is , and for a = 5, which is congruent to 1 mod 4, the conjectured density is .
Partial results
In 1967, Christopher Hooley published a conditional proof for the conjecture, assuming certain cases of the extended Riemann hypothesis.
Without the generalized Riemann hypothesis, there is no single value of a for which Artin's conjecture is proved. However, 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. He also proved (Corollary 2) that there are at most two primes for which Artin's conjecture fails, (Corollary 3) that there are at most three square-free integers for which Artin's conjecture fails, and (Corollary 4) that there is a constant such that among all integers of magnitude up to x at most fail Artin's conjecture.
If p is a prime of the form with q itself a prime, then is known to be a primitive root modulo p. Proving that there are infinitely many p of this type would establish as an explicit a for which Artin's conjecture is known to hold.
Abelian varieties
Unconditionally, the infinite part of Artin's primitive root conjecture for general abelian varieties was proved in:
C. Virdol, Artin's conjecture for abelian varieties, Kyoto Journal of Mathematics, 2016.
Moreover, unconditionally, the strong form of Artin's primitive root conjecture for general CM elliptic curves was proved in:
C. Virdol, On Artin's conjecture for CM elliptic curves, Kyoto Journal of Mathematics, 2020.
Conditionally, under the Generalized Riemann Hypothesis, Gupta and Murty proved Artin's primitive root conjecture for elliptic curves with complex multiplication, for primes splitting in the relevant imaginary quadratic field.
See also Stephens' constant, a number that plays the same role in a generalization of Artin's conjecture as Artin's constant plays here Brown–Zassenhaus conjecture Full reptend prime Cyclic number
References