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Topic: Law of quadratic reciprocity

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	Reciprocity Laws. Rule of Quadratic Reciprocity - Numericana
		Quadratic residues: Half of the nonzero residues modulo an odd prime p.
		A multitude of proofs of the quadratic reciprocity law.
		In particular, g itself can't be a quadratic residue (the order of g must be p-1, and it would be at most (p-1)/2 if g was congruent to the square of some x, since the order of x divides p-1, by Fermat's little theorem).
	home.att.net /~numericana/answer/reciprocity.htm (1523 words)

	[No title]
		Newsgroups: sci.math Subject: Re: "what is a reciprocity law?" Date: Sat, 26 Dec 1998 19:05:49 -0000 The law of quadratic reciprocity, proved by Gauss, is fundamental to number theory.
		The law of quadratic reciprocity states, for p and q distinct odd primes (p/q) = (q/p) unless p == q == 3 (mod 4) in which case (p/q) = -(q/p), or, put another way, (p/q)(q/p) = (-1)^((p-1)(q-1)/4).
		LQR has many, many proofs, using results from supposedly disparate parts of mathematics such as combinatorics, trigonometry, algebra and fluid dynamics.
	www.math.niu.edu /~rusin/known-math/98/quadrecip (467 words)

	Orðasafn: Q
		quadratic equation annars stigs jafna, ferningsjafna (?), = quadratic, = quadric 2.
		quadratic reciprocity law ferningsgagnkvæmnissetning, setning um ferningsgagnkvæmni, = law of quadratic reciprocity.
		2 annars stigs jafna, ferningsjafna, = quadratic, = quadratic equation.
	www.hi.is /~mmh/ord/safn/safnQ.html (249 words)

	Quadratic reciprocity - Wikipedia, the free encyclopedia
		The proof of Hilbert reciprocity reduces to checking a few special cases, and the non-trivial cases turn out to be equivalent to the main law and the two supplementary laws of quadratic reciprocity for the Legendre symbol.
		Therefore it is a more natural way of expressing quadratic reciprocity with a view towards generalization: the Hilbert reciprocity law extends with very few changes to all global fields and this extensions can rightly be considered a generalization of quadratic reciprocity to all global fields.
		There are cubic, quartic (biquadratic) and other higher reciprocity laws; but since two of the cube roots of 1 (root of unity) are not real, cubic reciprocity is outside the arithmetic of the rational numbers (and the same applies to higher laws).
	en.wikipedia.org /wiki/Quadratic_reciprocity (1166 words)

	Highbeam Encyclopedia - Search Results for Quadratic
		The general quadratic in one unknown has the form ax2 + bx + c, where a, b, and c are constants and x is the variable.
		He is one of the first known to have used algebra; his writings include rules of arithmetic and of plane and spherical trigonometry, and solutions of quadratic equations.
		Saddlepoint approximation for the distribution of a ratio of quadratic forms in normal variables.
	www.encyclopedia.com /SearchResults.aspx?Q=Quadratic (633 words)

	Quadratic Residue (Site not responding. Last check: 2007-11-06)
		The deepest of the results regarding quadratic residues was first proven rigorously by Gauss, and is known as the Quadratic Reciprocity Law.
		Mathworld's article, Quadratic Residue includes a table giving the primes which have a given number, d, as a quadratic residue (left).
		Mathpages: The Jewel of Arithmetic: Quadratic Reciprocity -- Euler's Criterion for quadratic residue is that
	mcraefamily.com /MathHelp/BasicNumberSquareQuadraticResidues.htm (1244 words)

	PlanetMath: proof of quadratic reciprocity rule
		For a bibliography of the more than 200 known proofs of the QRL, see Lemmermeyer.
		"proof of quadratic reciprocity rule" is owned by mathcam.
		This is version 9 of proof of quadratic reciprocity rule, born on 2002-12-14, modified 2003-09-26.
	planetmath.org /encyclopedia/ProofOfQuadraticReciprocityRule.html (123 words)

	Legendre biography
		This is fair since Legendre's proof of quadratic reciprocity was unsatisfactory, while he offered no proof of the theorem on primes in an arithmetic progression.
		Gauss was correct, but one could understand how hurtful Legendre must have found an attack on the rigour of his results by such a young man. Of course Gauss did not state that he was improving Legendre's result but rather claimed the result for himself since his was the first completely rigorous proof.
		Again Gauss would claim that he had obtained the law for the asymptotic distribution of primes before Legendre, but certainly it was Legendre who first brought these ideas to the attention of mathematicians.
	www-groups.dcs.st-and.ac.uk /~history/Biographies/Legendre.html (1850 words)

	[No title]
		There is a theory of higher power reciprocity, but it's more difficult than quadratic reciprocity, and it involves working with number fields containing the appropriate roots of unity.
		For instance 4-th power reciprocity involves working with symbols (u/v)_4 where u and v are numbers of the form a + bi where i^2 = -1.
		You might want to look at the URL http://www.wolfram.com/Q/QuadraticRecipricityTheorem.html It seems that a generalized reciprocity law, the local Langlands correspondence, conjectured by Langlands around 1970, has recently been proved by Michael Harris of the University of Paris VII and Richard Taylor of Harvard.
	www.math.niu.edu /~rusin/known-math/00_incoming/reciprocity (828 words)

	Highbeam Encyclopedia - Search Results for Reciprocity (Site not responding. Last check: 2007-11-06)
		Millstone, Somerset co., N.J. He studied law in the office of his uncle, Theodore Frelinghuysen, who had adopted him when he was three, and on admission to the bar in 1839
		Differential effects of reciprocity and attitude similarity across long- versus short-term mating contexts.
		Turnabout is fair play: why a reciprocity requirement should be included in the America Law Institute's proposed federal statute.
	www.encyclopedia.com /SearchResults.aspx?Q=Reciprocity (694 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		The elementary properties of quadratic congruences and a method for their solution were studied in a previous chapter.
		The first complete proof of this law was given by Gauss in 1796.
		Gauss gave eight different proofs of the law and we discuss a proof that Gauss gave in 1808.
	www.math.columbia.edu /~rama/chapters/chap17.html (149 words)

	"Quadratic Reciprocity in Isabelle" by David Emmanuel Gray (Site not responding. Last check: 2007-11-06)
		Quadratic reciprocity is a deep and important result in number theory.
		In this spirit, we have formalized quadratic reciprocity in the semi-automated theorem prover Isabelle.
		In our proof of quadratic reciprocity, we derive Gauss’ Lemma from Euler’s Criterion, which states that for any odd prime p consider the set B = {a, 2a, 3a, …, {(p – 1) / 2}a}, where a T 0 (mod p).
	pressurecooker.phil.cmu.edu /Academic/Papers/quadRes.htm (3513 words)

	Reciprocity
		Reciprocity one a day max a pair reciprocity or other authority.
		No reciprocity visitor will be reciprocity used reciprocity to verify reciprocity your identity, and as valid in the reciprocity at reciprocity wiley.
		Locate reciprocity an embassy what is your canvas bridges of chapters reciprocity as a tree.
	reciprocity.andresmarin.net (474 words)

	Legendre (print-only)
		The 1785 paper on number theory contains a number of important results such as the law of quadratic reciprocity for residues and the results that every arithmetic series with the first term coprime to the common difference contains an infinite number of primes.
		Of course today we attribute the law of quadratic reciprocity to Gauss and the theorem concerning primes in an arithmetic progression to Dirichlet.
		However, these two results are of great importance and credit should go to Legendre for his work on them, although he was not the first to state the law of quadratic reciprocity since it occurs in Euler's work of 1751 and also of 1783 (see [15]).
	www-groups.dcs.st-and.ac.uk /~history/Printonly/Legendre.html (1820 words)

	The On-Line Encyclopedia of Integer Sequences
		With 241, 5 is a quadratic residue but 7 is not.
		Among these, the term 3818929 is interesting because it is the first break in a pattern established by the previous terms, wherein the smallest prime p such that -1 and the primes up to the kth prime are quadratic residues modulo p has the (k+1)st prime as a quadratic nonresidue.
		2689 is the smallest odd prime to have -1, 2, 3, 5, 7 and 11 as quadratic residues.
	www.research.att.com /~njas/sequences/A102295 (416 words)

	Background on 2002 Fields and Nevanlinna Awardees
		The roots of the Langlands program are found in one of the deepest results in number theory, the Law of Quadratic Reciprocity, which goes back to the time of Fermat in the 17th century and was first proved by Carl Friedrich Gauss in 1801.
		Other reciprocity laws that apply in more general situations were discovered by Teiji Takagi and by Emil Artin in the 1920s.
		One of the original motivations behind the Langlands Program was to provide a complete understanding of reciprocity laws that apply in even more general situations.
	www.ams.org /ams/fields2002-background.html (1845 words)

	Proofs of quadratic reciprocity - Wikipedia, the free encyclopedia
		In the mathematical field of number theory, the law of quadratic reciprocity, like the Pythagorean theorem, has lent itself to an unusual number of proofs.
		Eisenstein's proof of quadratic reciprocity is a simplification of Gauss's third proof.
		The proof presented here is by no means the simplest known; however, it is quite a deep one, in the sense that it motivates some of the ideas of Artin reciprocity.
	en.wikipedia.org /wiki/Proofs_of_quadratic_reciprocity (1180 words)

	Amazon.com: Reciprocity Laws: From Euler to Eisenstein (Springer Monographs in Mathematics): Books: Franz Lemmermeyer (Site not responding. Last check: 2007-11-06)
		This book is about the development of reciprocity laws, starting from conjectures of Euler and discussing the contributions of Legendre, Gauss, Dirichlet, Jacobi, and Eisenstein.
		Consists of the development of reciprocity laws, starting from conjectures of Euler and discussing the contributions of Legendre, Gauss, Didichlet, Jacobi, and Eisenstein.
		Discussions of the reciprocity laws and extensive bibliographies throughout.
	www.amazon.com /Reciprocity-Laws-Eisenstein-Monographs-Mathematics/dp/3540669574 (710 words)

	Quadratic Reciprocity Theorem -- from Wolfram MathWorld
		The quadratic reciprocity theorem was Gauss's favorite theorem from number theory, and he devised no fewer than eight different proofs of it over his lifetime.
		Nagell, T. "The Quadratic Reciprocity Law." §41 in Introduction to Number Theory.
		Riesel, H. "The Law of Quadratic Reciprocity." Prime Numbers and Computer Methods for Factorization, 2nd ed.
	mathworld.wolfram.com /QuadraticReciprocityTheorem.html (227 words)

	Gauß, Eisenstein, and the ``third'' proof of the Quadratic Reciprocity Theorem: Ein kleines Schauspiel
		The Quadratic Reciprocity Theorem compares the quadratic character of two primes with respect to each other.
		`The reciprocity laws are the cornerstone of the latter theory'
		Hochgeehrtester Herr Hofrath, I too have devoted much effort to the study of the quadratic reciprocity law and have given four different proofs of it.
	www.math.nmsu.edu /~history/schauspiel/schauspiel.html (1888 words)

	Carl Friedrich Gouss
		In 1795, for instance, he discovered independently of Euler the law of quadratic reciprocity in number theory.
		Its core is the theory of quadratic congruencies, forms, and residues; it culminates in the law of quadratic residues, that "theorema aureum" of which Gauss gave the first complete proof.
		It was a continuation of his theory of quadratic residues in the "Disquisitiones arithmeticae," but a continuation with the aid of a new method, the theory of complex numbers.
	johan-gauss.org /index.html (2808 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		Description of the Course This course is the study of the divisibility properties of the integers.
		Describe the inherent beauty of the Law of Quadratic Reciprocity as a unifying theme in number theory.
		The Law of Quadratic Reciprocity (6 hours) Quadratic residues, the Legendre symbol, and the Law of Quadratic Reciprocity 5.
	www.lhup.edu /ucc/mathematics/MATH302_rev.doc (648 words)

	Math Forum Discussions
		[HM] Gauss and the law of quadratic reciprocity
		Re: [HM] Gauss and the law of quadratic reciprocity
		The Math Forum is a research and educational enterprise of the Drexel School of Education.
	mathforum.org /kb/thread.jspa?threadID=384343&messageID=1186381 (134 words)

	The Jewel of Arithmetic: Quadratic Reciprocity
		These are called the quadratic residues (mod 7), and the remaining numbers 3,5,6 are called the non-quadratic residues.
		It's often extremely important when dealing with problems in number theory to know whether a certain prime p is a square (i.e., a quadratic residue) modulo some other particular prime q.
		It's just an abstract symbol equal to either +1 or -1 depending on whether p is or isn't a square modulo q.
	www.mathpages.com /home/kmath075.htm (1007 words)

	Logical Methods in the Humanities Workshop Abstracts Winter 2003 (Site not responding. Last check: 2007-11-06)
		Gauss is credited with the first proof of this law, given in Disquisitiones Arithmeticae.
		This work is also where he introduces his new theory of congruences, which is used in the proof of the law.
		On the face of it, the theory of congruences looks like a particularly innocuous example of a new calculus, but we can see Gauss as crediting it with his proof of the law of quadratic reciprocity.
	www-logic.stanford.edu /Abstracts/Workshop/Spring04.html (750 words)

	Quadratic residue - Wikipedia, the free encyclopedia
		In mathematics, a number q is called a quadratic residue modulo n if there exists an integer x such that:
		They occur in a rather random pattern; this has been exploited in applications to acoustics.
		In effect, a quadratic residue modulo n is a number that has a square root in modular arithmetic when the modulus is n.
	en.wikipedia.org /wiki/Quadratic_residue (247 words)

	Information Bridge: DOE Scientific and Technical Information - - Document #782710
		With the assumption of perfect flatness ({Omega}total= 1.0), this approach leads to the derivation of a cosmic seesaw congruence which unifies the concepts of space and mass.
		The law of quadratic reciprocity profoundly constrains the subgroup structure of the multiplicative group of units F{sub P{sub{alpha}}}* defined by the field.
		It is generally concluded that the organizing principle legislated by the alliance of quadratic reciprocity with the cosmic seesaw creates a universal optimized structure that functions in the regulation of a broad range of complex phenomena.
	www.osti.gov /bridge/product.biblio.jsp?osti_id=782710 (195 words)

	PlanetMath: quadratic reciprocity rule
		Note that the Legendre symbol may also appear as
		See Also: Euler's criterion, cubic reciprocity law, quadratic reciprocity for polynomials
		This is version 10 of quadratic reciprocity rule, born on 2001-08-13, modified 2006-10-04.
	planetmath.org /encyclopedia/QuadraticReciprocityRule.html (60 words)

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