Artin's conjecture on primitive roots

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 2025. In fact, there is no single value of a for which Artin's conjecture is proved.

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

   $C_{\mathrm{A}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{n}}=\prod _{p\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{m}\mathrm{e}}(1-\frac{1}{p(p-1)})=0.3739558136\dots$ (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 a square number or a = −1, then the density is 0; more generally, if a is a perfect pth power for prime p, then the number needs to be multiplied by $\frac{p(p-2)}{p^2-p-1};$ if there is more than one such prime p, then the number needs to be multiplied by $\frac{p(p-2)}{p^2-p-1}$ for all such primes p). Similarly, if a₀ is congruent to 1 mod 4, then the number needs to be multiplied by $\frac{p(p-1)}{p^2-p-1}$ for all prime factors p of a₀.

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 C_Artin. 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 C_Artin) is 38/95 = 2/5 = 0.4.

For a = 8 = 2³, which is a power of 2, the conjectured density is $\frac{3}{5}C$, and for a = 5, which is congruent to 1 mod 4, the density is $\frac{20}{19}C$.

Partial results

In 1967, Christopher Hooley published a conditional proof for the conjecture, assuming certain cases of the generalized 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.

Some variations of Artin's problem

Elliptic curve

An elliptic curve $E$ given by $y^2=x^3+ax+b$, Lang and Trotter gave a conjecture for rational points on $E(\mathbb{\mathbb{Q}})$ analogous to Artin's primitive root conjecture.

Specifically, they said there exists a constant $C_E$ for a given point of infinite order $P$ in the set of rational points $E(\mathbb{\mathbb{Q}})$ such that the number $N(P)$ of primes ($p\le x$) for which the reduction of the point $P(\mathrm{mod}p)$ denoted by $\overline{P}$ generates the whole set of points in ${\mathbb{F}}_{\mathbb{𝕡}}$ in $E$, denoted by $\overline{E}({\mathbb{F}}_{\mathbb{𝕡}})$, is given by $N(P)\sim C_E\left(\frac{x}{\log x}\right)$. Here we exclude the primes which divide the denominators of the coordinates of $P$.

Gupta and Murty proved the Lang and Trotter conjecture for $E/\mathbb{\mathbb{Q}}$ with complex multiplication under the Generalized Riemann Hypothesis, for primes splitting in the relevant imaginary quadratic field.

Even order

Krishnamurty proposed the question how often the period of the decimal expansion $1/p$ of a prime $p$ is even.

The claim is that the period of the decimal expansion of a prime in base $g$ is even if and only if $g^{\left(\frac{p-1}{2^j}\right)}\equiv ̸1modp$ where $j\ge 1$ and $j$ is unique and p is such that $p\equiv 1+2^{j}mod{2}^j$.

The result was proven by Hasse in 1966.

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 (group theory)