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{{About|the conjecture of Emil Artin on primitive roots|the conjecture of Artin on L-functions|Artin L-function}}

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 n|primitive root]] modulo infinitely many [[prime number|primes]] ''p''. The [[conjecture]] also ascribes an [[asymptotic density]] to these primes. This conjectural density equals Artin's constant or a [[rational number|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 2024. 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 &minus;1. Write ''a''&nbsp;=&nbsp;''a''<sub>0</sub>''b''<sup>2</sup> with ''a''<sub>0</sub> [[Square-free integer|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 ''a''<sub>0</sub> is not [[modular arithmetic|congruent]] to 1 modulo 4 {{OEIS|id=A085397}}, this density is independent of ''a'' and equals '''Artin's constant''', which can be expressed as an infinite product
#: <math>C_{\mathrm{Artin}}=\prod_{p\ \mathrm{prime}} \left(1-\frac{1}{p(p-1)}\right) = 0.3739558136\ldots</math> {{OEIS|A005596}}.

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, … {{OEIS|A085397}}

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, … {{OEIS|A120629}}

Similar conjectural product formulas<ref>
{{cite web
|url=http://www.numericana.com/answer/constants.htm#artin
|title=Artin's Constant
|first1=Gerard P.|last1=Michon
|date=2006-06-15
|work=Numericana
}}
</ref> 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''<sub>Artin</sub>. If ''a'' is a [[square number]] or ''a'' = −1, then the density is 0, and if ''a'' is a perfect ''p''th power for prime ''p'', then the number needs to be multiplied by <math>\frac{p(p-2)}{p^2-p-1}</math> (if there are more than one such prime ''p'', then the number needs to be multiplied by <math>\frac{p(p-2)}{p^2-p-1}</math> for all such primes ''p''), and if ''a''<sub>0</sub> is congruent to 1 mod 4, then the number needs to be multiplied by <math>\frac{p(p-1)}{p^2-p-1}</math> for all prime factors ''p'' of ''a''<sub>0</sub>, e.g. for ''a'' = 8 = 2<sup>3</sup>, the density is <math>\frac{3}{5}C</math>, and for ''a'' = 5 (which is congruent to 1 mod 4), the density is <math>\frac{20}{19}C</math>.

==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 ''C''<sub>Artin</sub>. The set of such primes is {{OEIS|id=A001122}}
: ''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''<sub>Artin</sub>) is 38/95&nbsp;=&nbsp;2/5&nbsp;=&nbsp;0.4.

==Partial results==

In 1967, [[Christopher Hooley]] published a [[conditional proof]] for the conjecture, assuming certain cases of the [[generalized Riemann hypothesis]].<ref>{{cite journal |last1=Hooley|first1=Christopher |year=1967 |title=On Artin's conjecture |journal=J. Reine Angew. Math. |volume=1967 |issue=225 |pages=209–220|mr=0207630|doi=10.1515/crll.1967.225.209|s2cid=117943829 }}</ref>

Without the generalized Riemann hypothesis, there is no single value of ''a'' for which Artin's conjecture is proved. [[Roger Heath-Brown|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''.<ref>{{ cite journal |author=D. R. Heath-Brown |title=Artin's Conjecture for Primitive Roots |journal=The Quarterly Journal of Mathematics |volume=37 |issue=1 |date=March 1986 |pages=27–38 |doi=10.1093/qmath/37.1.27}}</ref> 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 <math>E</math> given by <math>y^2 = x^3+ax+b</math>, Lang and Trotter gave a conjecture for rational points on <math>E(\mathbb{Q})</math> analogous to Artin's primitive root conjecture.<ref name=":0">{{Cite web |last=Moree |first=Pieter |title=Artin's Primitive Root Conjecture – a survey |url=http://guests.mpim-bonn.mpg.de/moree/surva.pdf}}</ref>

Specifically, they said there exists a constant <math>C_E</math>  for a given point of infinite order <math>P</math> in the set of rational points <math>E(\mathbb{Q})</math>  such that the number <math>N(P)</math> of primes (<math>p\leq x</math>) for which the reduction of the point <math>P\pmod p</math> denoted by <math>\bar{P}</math> generates the whole set of points in <math>\mathbb{F_p}</math> in <math>E</math>, denoted by <math>\bar{E}(\mathbb{F_p})</math>, is given by <math>N(P)\sim C_E\left ( \frac{x}{\log x} \right )</math>.<ref>{{Cite journal |last=Lang and 2 Trotter |date=1977 |title=Primitive points on Elliptic Curves |journal=Bull. Amer. Math. Soc. |volume=83 |issue=2 |pages=289–292|doi=10.1090/S0002-9904-1977-14310-3 |doi-access=free |url=https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society/volume-83/issue-2/Primitive-points-on-elliptic-curves/bams/1183538693.pdf }}</ref> Here we exclude  the primes which divide the denominators of the coordinates of <math>P</math>.

Gupta and Murty proved the Lang and Trotter conjecture for <math>E/\mathbb{Q}</math> with complex multiplication under the Generalized Riemann Hypothesis, for primes splitting in the relevant imaginary quadratic field.<ref>{{Cite journal |last=Gupta and Murty |date=1987 |title=Primitive points on elliptic curves |url=https://eudml.org/doc/89763 |journal=Compositio Mathematica |volume=58 |pages=13–44}}</ref>

=== Even order ===
Krishnamurty proposed the question how often the period  of the decimal expansion <math>1/p</math> of a prime <math>p</math> is even.

The claim is that the period of the decimal expansion of a prime in base <math>g</math> is even if and only if   <math>g^\left ( \frac{p-1}{2^j} \right )\not\equiv1\bmod p</math> where <math>j\geq1</math> and <math>j</math> is unique and p is such that <math>p\equiv 1+2^j\mod{2^j}</math>.

The result was proven by Hasse in 1966.<ref name=":0" /><ref>{{Cite journal |last=Hasse |first=H |date=1966 |title=About the density of prime numbers p, for a given integral number a not equal to 0 of even or odd order mod p |journal=Mathematische Annalen |pages=19–23|doi=10.1007/BF01361432 |s2cid=121171472 }}</ref>

== 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)]]

==References==
{{Reflist}}{{Prime number conjectures}}

[[Category:Analytic number theory]]
[[Category:Algebraic number theory]]
[[Category:Conjectures about prime numbers]]
[[Category:Unsolved problems in number theory]]