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	Artin conjecture -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-08-19)
		In (A science (or group of related sciences) dealing with the logic of quantity and shape and arrangement) mathematics, there are two notable Artin conjectures, the legacy of (Click link for more info and facts about Emil Artin) Emil Artin.
		Hooley proved that the second conjecture is a consequence of the first (a (Click link for more info and facts about conditional proof) conditional proof).
		He assumed the regularity of L-functions for certain extensions built by (Click link for more info and facts about Kummer theory) Kummer theory, by adjoining k-th (Click link for more info and facts about roots of unity) roots of unity and the k-th root of a to the (An integer or a fraction) rational numbers.
	www.absoluteastronomy.com /encyclopedia/a/ar/artin_conjecture.htm (549 words)

	Artin (Site not responding. Last check: 2007-08-19)
		Artin's childhood was not a particularly happy one and he recounted later in his life how he had felt lonely.
		Artin himself proved that when O is the field of algebraic numbers, the subfield K of real algebraic numbers solves the problem and, moreover, it is the unique solution up to automorphisms of the field O.
		Artin and Schreier published in their famous 1926 paper their studies of all formally real fields and real closed fields, showing that a specific ordering could be defined on them.
	www-history.mcs.st-andrews.ac.uk /history/Mathematicians/Artin.html (2579 words)

	Least primitive root of prime numbers
		The Artin conjecture has been generalized in the following way [4]: let N(x;a_1,...,a_n) be the number of primes p <= x such that a_1,...,a_n are simultaneously primitive roots mod p.
		The generalized Artin conjecture states that N(x;a_1,...,a_n) is given asymptotically by A(a_1,...,a_n) pi(x) for some non-negative constant A(a_1,...,a_n).
		Using the inclusion-exclusion principle applied to the Matthews' generalized Artin conjecture, it is expected that the ratios N_g(x;r) / pi(x), N_G(x;r) / pi(x), and N_B(x;r) / pi(x) approach constants when x goes to infinity.
	www.ieeta.pt /~tos/p-roots.html (1612 words)

	Shalen Abstract UGA Math (Site not responding. Last check: 2007-08-19)
		Artin further conjectured that these L-functions should be holomorphic away from 1.
		One important special case of Artin's conjecture can be reformulated as describing a correspondence between weight one cuspidal modular forms which are eigenforms for the Hecke operators, and two-dimensional odd irreducible complex representations of the absolute Galois group of the rationals.
		In this form the conjecture, often known as the `Strong Artin conjecture', stands as one of two notable classical cases of the Langlands conjectures for number fields; the modularity of elliptic curves over the rationals is the other.
	www.math.uga.edu /~wag/colloquium/abst-dickinson.html (159 words)

	Ram Murty - Publications by Year
		Selberg conjectures and Artin L-functions, Bulletin of the American Mathematical Society, 31 (1) (1994) 1-14.
		Artin's conjecture and elliptic analogues, in Sieve Methods, Exponential Sums, and their Applications in Number Theory, (eds.
		Artin's conjecture for polynomials over finite fields, (with Erik Jensen), in Number Theory (edited by R.P. Bambah, V. Dumir and R.J. Hans Gill), Birkhauser-Verlag, 2000, pp.
	www.mast.queensu.ca /~murty/index2.html (1878 words)

	U of R Number Theory Seminar (Site not responding. Last check: 2007-08-19)
		Mike Knapp- "Artin's conjecture for forms of degree 7 and 11".
		A conjecture commonly attributed to Artin states that a homogeneous polynomial of degree d should have a nontrivial zero in each p-adic field Q_p provided only that the number of variables involved in the polynomial is at least d^2 + 1.
		While this conjecture is false in general, it turns out to be true if the fields are restricted to Q_p with p sufficiently large.
	www.math.rochester.edu /research/algebra_and_number_theory/9.17.02.html (115 words)

	Artin's Primitive Root Conjecture For Quadratic Fields (ResearchIndex) (Site not responding. Last check: 2007-08-19)
		1.3: A Quadratic Analogue Of Artin's Conjecture On Primitive Roots.
		Prime Divisors Of Linear Recurrences And Artin's Primitive Root..
		A Quadratic Analogue Of Artin's Conjecture On Primitive Roots.
	citeseer.ist.psu.edu /489894.html (282 words)

	[No title]
		A refined version of the conjecture gives a formula for the relative density of such primes (always positive for the allowed integers).
		Artin's conjecture deals with the _multiplicative_ structure of Z/pZ for integral primes p, with particular emphasis on modular exponentiation.
		Artin's conjecture is the heuristic dual: for every integer g, will there always exist a prime p for which g is a primitive root?
	www.math.niu.edu /~rusin/known-math/98/artins_conj (798 words)

	UIUC Dept. of Mathematics Seminar Calendar (Site not responding. Last check: 2007-08-19)
		This conjecture was quickly disproved (assuming Continuum Hypothesis), but also it was shown by Richard Laver in 1976, to be consistently true.
		Similarly, given a family of "small" sets of reals, the corresponding Borel Conjecture is the statement that being a member of this family is equivalent to being countable.
		Abstract: The Birch and Swinnerton-Dyer conjecture provides a way for methods of analytic number theory to produce results on ranks of families of elliptic curves by studying properties of their L-functions.
	torus.math.uiuc.edu /cal/math/cal?...&month=04&day=21&interval=day (1985 words)

	On Arithmetic Of Hyperelliptic Curves (ResearchIndex) (Site not responding. Last check: 2007-08-19)
		Abstract: In this expos'e, Pell's equation is put in a geometric perspective, and a version of Artin's primitive roots conjecture is formulated for hyperelliptic jacobians.
		Introduction It is well known that there are close connections between the arithmetic behavior of algebraic number fields and that of the algebraic function fields...
		The Abc Conjecture Implies Roth's Theorem And Mordell's..
	citeseer.ist.psu.edu /366325.html (384 words)

	Some number-theoretical constants
		Density of the set of primes q, relative to the set of all primes, such that a given positive integer (not a proper power and with squarefree part incongruent 1 mod 4) is a primitive root modulo q.
		Matthews, A generalisation of Artin's conjecture for primitive roots, Acta Arith.
		This is part of a conjectural density formula for the number of twin primes not exceeding a given bound.
	www.gn-50uma.de /alula/essays/Moree/Moree.en.shtml (1108 words)

	Abstract of: Computer verification of the Ankeny-Artin-Chowla conjecture for all primes less than 100 000 000 000 (Site not responding. Last check: 2007-08-19)
		Let p be a prime congruent to 1 modulo 4 and let t, u be rational integers such that (t+u √p)/2 is the fundamental unit of the real quadratic field Q(√p).
		Even testing the conjecture can be quite challenging because of the size of the numbers t, u; for example, when p = 40 094 470 441, then both t and u exceed 10
		In this paper we describe a new technique for testing the AAC conjecture and we provide some results of a computer run of the method for all primes p up to 10
	db.cwi.nl /rapporten/abstract.php?abstractnr=804 (220 words)

	Math JS Milne Preprints
		Prove the full conjecture of Birch and Swinnerton-Dyer in the case of a constant abelian variety over a global field of prime characteristic; in particular, give the first examples of nonzero abelian varieties whose Tate-Shafarevich groups are known to be finite.
		Proves the conjecture of Artin and Tate (relating the order of the Brauer group of a surface over a finite field to its zeta function) for rational surfaces.
		Examines the conjecture of Langlands and Rapoport and its consequences; introduces the notion of a canonical integral model.
	www.jmilne.org /math/Preprints (2322 words)

	Professor Laurent Lafforgue - CIRS
		In rank 1, this conjecture is nothing other than the now traditional "class field theory" of Emil Artin.
		For that purpose, he built varieties similar to modular curves and showed certain cases of the conjecture of Langlands in rank 2.
		This turned out to make the general case accessible, after formidable technical difficulties were surmounted.The crucial contribution by Laurent Lafforgue to solve this question is the construction of compactifications of certain varieties of modules.
	www.cirs.net /investigadores/mathematics/LAFFORGUE.htm (347 words)

	LC '98 abstract: Hans Schoutens (Site not responding. Last check: 2007-08-19)
		This lead Artin to the following general conjecture: any Henselian excellent local ring has the Artin Approximation Property, i.e., is existentially closed inside its completion (with respect to the maximal ideal topology).
		Hence a new proof to Artin's Conjecture would be provided by showing that a Henselian local Gorenstein ring $R$ with algebraically closed residue field is existentially closed in $\Cal C_{d,e}$.
		Of course, as part of the latter one needs to show that $R$ is existentially closed inside its completion, but perhaps there exist other means to prove the result without going through this particular exemplification of it.
	www.math.cas.cz /~lc98/abstracts/Schoutens.html (508 words)

	ipedia.com: Robert Langlands Article
		Langlands is the author of the Langlands program, a deep web of conjectures connecting number theory and representation theory.
		Taken to its logical conclusion, this leads to his famous functoriality conjecture, which altered our understanding of what the key issues are in number theory.
		The functoriality conjecture is far from proved, but a special case (the octahedral Artin conjecture, proved by Langlands and Tunnell) was the starting point of Andrew Wiles' attack on the Taniyama-Shimura conjecture and the proof of Fermat's last theorem.
	www.ipedia.com /robert_langlands.html (387 words)

	Joint Mathematics Colloquium (Site not responding. Last check: 2007-08-19)
		Emil Artin reformulated these results as saying that the L-functions associated to 1-dimensional complex Galois representations had analytic continuation apart from well-understood poles and conjectured that the same should be true for n-dimensional complex Galois representations.
		This conjecture could now be regarded as a precursor to the Langlands programme.
		This talk will be for non-experts; I will spend about half the talk defining complex Galois representations and their L-functions, and the other half giving statements of results and sometimes indications of proofs.
	www.math.neu.edu /bhmn/buzzard.html (153 words)

	[No title] (Site not responding. Last check: 2007-08-19)
		From: nikl+sm000315@pchelwig1.mathematik.tu-muenchen.de (Gerhard Niklasch) Newsgroups: sci.math Subject: Re: Artin Conjecture Date: 20 Aug 1998 13:09:07 GMT In article
		For a very readable introduction to the Artin Conjecture about primitive roots, see H.W.Lenstra Jr., `Euclidean Fields' (3 parts), in The Mathe- matical Intelligencer 2 #1 (1979), 6--15, 2 #2 (1980), 73--77 and 99--103.
		This Artin Conjecture can be stated as saying that for a given positive integer g, not a square, infinitely many primes p have the property that g^k for 0
	www.math.niu.edu /~rusin/known-math/98/artin_conj (156 words)

	Icosahedral Galois representations (Site not responding. Last check: 2007-08-19)
		Artin conjectured that the L-series associated to any continuous irreducible representation
		Exciting work of Taylor and others suggests that a complete proof of Artin's conjecture, in the case when n=2 and
		These ongoing computations are laying a part of the foundation necessary for a full proof of the Artin conjecture for odd two-dimensional
	modular.fas.harvard.edu /job/Short/node7.html (118 words)

	Kimball Martin--the math
		A Symplectic Case of Artin's Conjecture, revised 13 June 2003.
		This gives a new case of Artin's conjecture in GSp(4,C) by establishing the more general Langlands' reciprocity law in this case.
		Langlands' Conjecture for the Tetrahedral and Octahedral Cases, a short introduction to Langlands' reciprocity conjecture with an exposition of the proof in the tetrahedral and octahedral cases, i.e., the Langlands-Tunnell Theorem.
	www.math.columbia.edu /~kimball/math.html (316 words)

	Welcome to Mathsoft
		Wrench, Evaluation of Artin's constant and the twin prime constant, Math.
		Lenstra, Jr., On Artin's conjecture and Euclid's algorithm in global fields, Inventiones Math.
		Roskam, A quadratic analogue of Artin's conjecture on primitive roots, J.
	www.mathsoft.com /mathsoft_resources/mathsoft_constants/ref/2308.asp (382 words)

	Index to On-Line Encyclopedia of Integer Sequences (Site not responding. Last check: 2007-08-19)
		Artin's conjecture or constant, sequences related to (start):
		Artin's conjecture, Artin's constants: A005596* A048296* A065414 A065417 A066517
		Artin's conjecture: see also primes by primitive root
	www.research.att.com /~njas/sequences/Sindx_Ar.html (353 words)

	A two variable Artin conjecture (Site not responding. Last check: 2007-08-19)
		A two variable Artin conjecture, by Pieter Moree and Peter Stevenhagen
		Let a and b be non-zero rational numbers that are multiplicatively independent.
		Our result, in combination with earlier work of the second author, allows us to deduce that any second order linear recurrence with reducible characteristic polynomial having integer elements, has a positive density of prime divisors (under GRH).
	www.math.uiuc.edu /Algebraic-Number-Theory/0216 (145 words)

	Queen's Math. & Stats - Ram Murty's Home Page
		On Artin L-Functions, in Class Field Theory, Its Centenary and Prospect, (ed.
		Francesco Pappalardi, On Artin's conjecture for primitive roots, Ph.D. thesis, McGill University, 1993.
		Satya Mohit, The Wieferich criterion, the ABC conjecture and Shimura's correspondence, M.Sc.
	www.mast.queensu.ca /~murty (494 words)

	Number Theory Seminars in JHU (Site not responding. Last check: 2007-08-19)
		Some previous lower rank cases and the relation to the Gross-Prasad conjecture will be discussed along the way.
		Artin conjectured that the L-functions of Galois representations are entire (excluding the zeta functions).
		This would follow from Langlands' conjecture that Galois representations can associated to automorphic forms.
	www.math.jhu.edu /~qzhang/Seminar/Seminar-2004Fall.htm (312 words)

	82a (Site not responding. Last check: 2007-08-19)
		Selberg has made two conjectures concerning the Dirichlet series in the Selberg class
		Conjecture B can be interpreted as saying that the primitive functions form an orthonormal system.
		It implies, among other things, Artin's conjecture on the holomorphy of non-abelian
	www.aimath.org /WWN/rh/articles/html/82a (82 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-08-19)
		An empirical deduction of mine is that prime powers can never be primitive roots; a bolder assertion still would be that perfect powers can never be primitive roots.
		Which numbers really can never be primitive roots and how do we prove that?
		There is, however, a Conjecture by Artin, which states the following: Artin's Conjecture: Let a be any integer not a perfect square and not equal to -1.
	mathforum.org /library/drmath/view/61084.html (250 words)

	DESCRIPTIONS OF AREAS/COURSES IN NUMBER THEORY, LECTURE NOTES
		A proof of the full Shimura-Taniyama-Weil conjecture is announced, Henri Darmon, Notices of the AMS, December 1999
		Catalan's Conjecture: Another old diophantine problem solved, Tauno Metsänkylä, Bull.
		An exposition by Yuri Bilu of Preda Mihailescu's recent proof of Catalan's conjecture: dvi, ps
	www.numbertheory.org /ntw/N4.html (1977 words)

	Buzzard-Stein: Artin's Conjecture (Site not responding. Last check: 2007-08-19)
		A mod five approach to modularity of icosahedral Galois representations
		We give eight new examples of icosahedral Galois representations that satisfy Artin's conjecture on holomorphicity of their L-function.
		We give in detail one example of an icosahedral representation of conductor 1376=2
	modular.fas.harvard.edu /Tables/artin.html (54 words)

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