Artin's conjecture on primitive roots

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

1. S(a) has a positive asymptotic density inside the set of primes. In particular, S(a) is infinite.
2. Under the conditions that a is not a perfect power and 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

   ${\displaystyle C_{\mathrm{A}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{n}}=\prod _{p\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{m}\mathrm{e}}\left(1-\frac{1}{p^2-p}\right)=0.3739558136\dots }$ (sequence A005596 in the OEIS).

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, … (sequence A085397 in the OEIS)

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, … (sequence A120629 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 C_Artin.

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 ${\displaystyle C_{\text{Artin}}\left(1-\prod _{p|a_0,p\text{prime}}\frac{1}{-p^2+p+1}\right)}$.

If a is a perfect power, let r be the largest number such that a is an rth power, then this density is:

${\displaystyle C_{\text{Artin}}\prod _{p|r,p\text{prime}}\frac{p^2-2p}{p^2-p-1}}$ if a₀ is not congruent to 1 modulo 4;

${\displaystyle C_{\text{Artin}}\prod _{p|r,p\text{prime}}\frac{p^2-2p}{p^2-p-1}\left(1-\prod _{p|a_0,p\text{prime}}\frac{1}{-p^2+p+1}\prod _{p|gcd(a_0,r),p\text{prime}}\frac{p^2-p-1}{p-2}\right)}$ if a₀ is congruent to 1 modulo 4.

For ${\displaystyle n=1,2,3,\cdots }$, the conjectural densities given by the formula are C_Artin times

0, 1, 1, 0, ⁠20/19⁠, 1, 1, ⁠3/5⁠, 0, 1, 1, 1, ⁠156/155⁠, 1, 1, 0, ⁠272/271⁠, 1, 1, ⁠20/19⁠, ⁠204/205⁠, 1, 1, 1, 0, 1, ⁠3/5⁠, 1, ⁠812/811⁠, 1, 1, ⁠15/19⁠, ⁠544/545⁠, 1, 1, 0, ⁠1332/1331⁠, 1, 1, 1, ⁠1640/1639⁠, 1, 1, 1, ⁠20/19⁠, ... (OEIS: A399880 for the numerators, OEIS: A399881 for the denominators)

For ${\displaystyle n=-1,-2,-3,\cdots }$, the conjectural densities given by the formula are C_Artin times

0, 1, ⁠6/5⁠, 1, 1, 1, ⁠42/41⁠, ⁠3/5⁠, 1, 1, ⁠110/109⁠, ⁠6/5⁠, 1, 1, ⁠94/95⁠, 1, 1, 1, ⁠342/341⁠, 1, 1, 1, ⁠506/505⁠, 1, 1, 1, ⁠6/5⁠, ⁠42/41⁠, 1, 1, ⁠930/929⁠, ⁠15/19⁠, 1, 1, ⁠778/779⁠, 1, 1, 1, ⁠774/775⁠, 1, 1, 1, ⁠1806/1805⁠, ⁠110/109⁠, 1, ... (OEIS: A399882 for the numerators, OEIS: A399883 for the denominators)