Artin's conjecture on primitive roots
Encyclopedia
In number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...

, Artin's conjecture on primitive roots states that a given integer a which is not a perfect square
Square number
In mathematics, a square number, sometimes also called a perfect square, is an integer that is the square of an integer; in other words, it is the product of some integer with itself...

 and not −1 is a primitive root modulo
Primitive root modulo n
In modular arithmetic, a branch of number theory, a primitive root modulo n is any number g with the property that any number coprime to n is congruent to a power of g modulo n. In other words, g is a generator of the multiplicative group of integers modulo n...

 infinitely many primes
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...

 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
Emil Artin
Emil Artin was an Austrian-American mathematician of Armenian descent.-Parents:Emil Artin was born in Vienna to parents Emma Maria, née Laura , a soubrette on the operetta stages of Austria and Germany, and Emil Hadochadus Maria Artin, Austrian-born of Armenian descent...

 to Helmut Hasse
Helmut Hasse
Helmut Hasse was a German mathematician working in algebraic number theory, known for fundamental contributions to class field theory, the application of p-adic numbers to local classfield theory and diophantine geometry , and to local zeta functions.-Life:He was born in Kassel, and died in...

 on September 27, 1927, according to the latter's diary. Although significant progress has been made, the conjecture is still unresolved. In fact, there is no a single value of a for which Artin's conjecture is proved.

Formulation

Let a be an integer which is not a perfect square and not −1. Write a = a0b2 with a0 square-free
Square-free integer
In mathematics, a square-free, or quadratfrei, integer is one divisible by no perfect square, except 1. For example, 10 is square-free but 18 is not, as it is divisible by 9 = 32...

. Denote by S(a) the set of prime numbers p such that a is a primitive root modulo p. Then
  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 that 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


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 CArtin.

Example

For example, take a = 2. 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
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.

Proof attempts

In 1967, Hooley
Christopher Hooley
Christopher Hooley FLSW FRS is a British mathematician, emeritus professor of mathematics at Cardiff University. He did his PhD under the supervision of Albert Ingham. He won the Adams Prize of Cambridge University in 1973. He was elected a Fellow of the Royal Society in 1983...

 published a conditional proof
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....

 for the conjecture, assuming certain cases of the Generalized Riemann hypothesis
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, which are formally similar to the Riemann zeta-function...

. In 1984, R. Gupta and M. Ram Murty
M. Ram Murty
Maruti Ram Pedaprolu Murty, FRSC is an Indo-Canadian mathematician, currently head of the Department of Mathematics and Statistics at Queen's University, where he holds a Queen's Research Chair in mathematics.-Career:...

 showed unconditionally that Artin's conjecture is true for infinitely many a using sieve methods
Sieve theory
Sieve theory is a set of general techniques in number theory, designed to count, or more realistically to estimate the size of, sifted sets of integers. The primordial example of a sifted set is the set of prime numbers up to some prescribed limit X. Correspondingly, the primordial example of a...

. Roger Heath-Brown
Roger Heath-Brown
David Rodney "Roger" Heath-Brown F.R.S. , is a British mathematician working in the field of analytic number theory. He was an undergraduate and graduate student of Trinity College, Cambridge; his research supervisor was Alan Baker...

 improved on their result and showed unconditionally that there are at most two exceptional prime numbers a for which Artin's conjecture fails. This result is not constructive
Constructive proof
In mathematics, a constructive proof is a method of proof that demonstrates the existence of a mathematical object with certain properties by creating or providing a method for creating such an object...

, as far as the exceptions go. For example, it follows from the theorem of Heath-Brown that one out of 3, 5, and 7 is a primitive root modulo p for infinitely many p. But the proof does not provide us with a way of computing which one.
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