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In
number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
, Artin's conjecture on primitive roots states that a given
integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
''a'' that is neither a
square number In mathematics, a square number or perfect square is an integer that is the square (algebra), square of an integer; in other words, it is the multiplication, product of some integer with itself. For example, 9 is a square number, since it equals ...
nor −1 is a primitive root modulo infinitely many primes ''p''. The
conjecture In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis or Fermat's conjecture (now a theorem, proven in 1995 by Andrew Wiles), ha ...
also ascribes an asymptotic density to these primes. This conjectural density equals Artin's constant or a
rational Rationality is the quality of being guided by or based on reason. In this regard, a person acts rationally if they have a good reason for what they do, or a belief is rational if it is based on strong evidence. This quality can apply to an ...
multiple thereof. The conjecture was made by
Emil Artin Emil Artin (; March 3, 1898 – December 20, 1962) was an Austrians, Austrian mathematician of Armenians, Armenian descent. Artin was one of the leading mathematicians of the twentieth century. He is best known for his work on algebraic number t ...
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'' = ''a''0''b''2 with ''a''0
square-free {{no footnotes, date=December 2015 In mathematics, a square-free element is an element ''r'' of a unique factorization domain ''R'' that is not divisible by a non-trivial square. This means that every ''s'' such that s^2\mid r is a unit of ''R''. ...
. 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 ''a''0 is not
congruent Congruence may refer to: Mathematics * Congruence (geometry), being the same size and shape * Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure * In modu ...
to 1 modulo 4 , this density is independent of ''a'' and equals Artin's constant, which can be expressed as an infinite product #: C_=\prod_ \left(1-\frac\right) = 0.3739558136\ldots . 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 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 In mathematics, a square number or perfect square is an integer that is the square (algebra), square of an integer; in other words, it is the multiplication, product of some integer with itself. For example, 9 is a square number, since it equals ...
or ''a'' = −1, then the density is 0; more generally, if ''a'' is a perfect ''p''th power for prime ''p'', then the number needs to be multiplied by \frac; if there is more than one such prime ''p'', then the number needs to be multiplied by \frac for all such primes ''p''). Similarly, if ''a''0 is congruent to 1 mod 4, then the number needs to be multiplied by \frac for all prime factors ''p'' of ''a''0.


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 : ''S''(2) = . 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 \fracC, and for ''a'' = 5, which is congruent to 1 mod 4, the density is \fracC.


Partial results

In 1967,
Christopher Hooley Christopher Hooley (7 August 1928 – 13 December 2018) was a British mathematician and professor of mathematics at Cardiff University. He did his PhD under the supervision of Albert Ingham. He won the Adams Prize of Cambridge University ...
published a
conditional proof A conditional proof is a proof that takes the form of asserting a conditional, and proving that the antecedent of the conditional necessarily leads to the consequent. Overview The assumed antecedent of a conditional proof is called the condi ...
for the conjecture, assuming certain cases of the
generalized Riemann hypothesis The Riemann hypothesis is one of the most important conjectures in mathematics. It is a statement about the zeros of the Riemann zeta function. Various geometrical and arithmetical objects can be described by so-called global ''L''-functions, whi ...
. 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) 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) such that the number N(P) of primes (p\leq x) for which the reduction of the point P\pmod p denoted by \bar generates the whole set of points in \mathbb in E, denoted by \bar(\mathbb), is given by N(P)\sim C_E\left ( \frac \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 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 \right )\not\equiv1\bmod p where j\geq1 and j is unique and p is such that p\equiv 1+2^j\mod. 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 In number theory, a full reptend prime, full repetend prime, proper primeDickson, Leonard E., 1952, ''History of the Theory of Numbers, Volume 1'', Chelsea Public. Co. or long prime in base ''b'' is an odd prime number ''p'' such that the Fermat ...
* Cyclic number (group theory)


References

{{Prime number conjectures Analytic number theory Algebraic number theory Conjectures about prime numbers Unsolved problems in number theory