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Line 1: {{ 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 ==Formulation== Line 31: 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 === | |||