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	PlanetMath: Jacobi symbol
		The Jacobi symbol is a generalization of the Legendre symbol to all odd positive integers.
		A further generalization of the Legendre symbol, due to Kronecker, is the Kronecker symbol.
		This is version 6 of Jacobi symbol, born on 2002-04-22, modified 2004-08-24.
	planetmath.org /encyclopedia/JacobiSymbol.html (89 words)

	PlanetMath: Legendre symbol
		The Legendre symbol can be computed by means of Euler's criterion or Gauss' lemma.
		A generalization of this symbol is the Jacobi Symbol.
		This is version 7 of Legendre symbol, born on 2001-10-08, modified 2006-11-01.
	planetmath.org /encyclopedia/LegendreSymbol.html (100 words)

	Legendre symbol - Wikipedia, the free encyclopedia
		The Legendre symbol is used by mathematicians in the area of number theory, particularly in the fields of factorization and quadratic residues.
		The Legendre symbol is a special case of the Jacobi symbol.
		The Jacobi symbol is a generalization of the Legendre symbol that allows composite bottom numbers.
	www.wikipedia.org /wiki/Legendre_symbol (288 words)

	Carl Gustav Jakob Jacobi -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-11-03)
		Jacobi wrote the classic treatise (1829) on (Click link for more info and facts about elliptic function) elliptic functions, of great importance in mathematical physics, because of the need to "integrate second order kinetic energy equations".
		Jacobi was also the first mathematician to apply elliptic functions to (Click link for more info and facts about number theory) number theory, for example, proving the polygonal number theorem of (French mathematician who founded number theory; contributed (with Pascal) to the theory of probability (1601-1665)) Pierre de Fermat.
		It was in analytical development that Jacobi’s peculiar power mainly lay, and he made many important contributions of this kind to other departments of mathematics, as a glance at the long list of papers that were published by him in Crelle’s Journal and elsewhere from 1826 onwards will sufficiently indicate.
	www.absoluteastronomy.com /encyclopedia/C/Ca/Carl_Gustav_Jakob_Jacobi.htm (715 words)

	[No title]
		Symbol, a maker of bar code scanners and radio frequency identification devices, hired Nuti from Cisco as president and chief operating officer in 2002 to help lead the company's turnaround and end its five-year string of losses.
		The major difference between the Nazi swastika and the ancient symbol of many different cultures, is that the Nazi swastika is at a slant, while the ancient swastika is rested flat.
		Jacobi generalized the Legendre symbol to allow lower entries that are odd (but not necessarily prime) as follows: Let the factorization of n be.
	www.lycos.com /info/symbol--miscellaneous.html?page=2 (345 words)

	Legendre symbol Summary
		The Legendre symbol is a notation used for stating a central theorem of elementary number theory, the quadratic reciprocity law.
		The Legendre symbol is a special case of the Jacobi symbol.
		The Jacobi symbol is a generalization of the Legendre symbol that allows composite bottom numbers.
	www.bookrags.com /Legendre_symbol (863 words)

	Legendre symbol
		The Legendre symbol is used by mathematicians in the theory of numbers, particularly in the fields of factorization and quadratic residues.
		The Legendre symbol can be used to compactly formulate the law of quadratic reciprocity.
		The Jacobi symbol can be 1 without a being a quadratic residue of b.
	www.ebroadcast.com.au /lookup/encyclopedia/ja/Jacobi_symbol.html (214 words)

	The Jacobi Symbol (Site not responding. Last check: 2007-11-03)
		If n is odd, the jacobi symbol [a\n], which looks exactly like the legendre symbol, is the product [a\q] for all prime factors q in n, including multiplicities.
		Unlike the legendre symbol, the value of the jacobi symbol does not tell you whether a is a square mod n.
		If the jacobi symbol is split apart, as above, assume there are k primes in a that are 3 mod 4, and l primes in n that are 3 mod 4.
	www.mathreference.com /num-mod,jac.html (276 words)

	Jacobi's Symbol
		Jacobi (in 1846) published a paper in which he defined his own symbol by extending the Legendre symbol (a/p).
		It turns out that Jacobi's symbol (a/b) is very easy to evaluate, using only the eight rules listed below (in which b is assumed to be ≥ 3).
		Rule 8 allows you to "flip" the symbol (a/b) whenever a is odd and 3 ≤ a < b.
	wwwmaths.anu.edu.au /DoM/thirdyear/MATH3301/jacobi.html (443 words)

	Jacobi symbol -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-11-03)
		The Jacobi symbol is used by (A person skilled in mathematics) mathematicians in the area of (Click link for more info and facts about number theory) number theory.
		The Jacobi symbol is a generalization of the (Click link for more info and facts about Legendre symbol) Legendre symbol using the (Click link for more info and facts about prime factorization) prime factorization of the bottom number.
		In the case where we are unable to say that a is a quadratic residual of n.
	www.absoluteastronomy.com /encyclopedia/J/Ja/Jacobi_symbol.htm (285 words)

	Reciprocity Laws. Rule of Quadratic Reciprocity - Numericana
		The Legendre symbol (ap) can be extended to values of p besides odd primes.
		The Legendre symbol was introduced specifically to stress the nice symmetrical relationship between (mn) and (nm) when m and n are both odd primes.
		The Jacobi symbol and Kronecker symbol are just to the restricted Legendre symbol what real and complex exponents are to integral exponents [ for a positive base, of course ].
	home.att.net /~numericana/answer/reciprocity.htm (1523 words)

	Arithmetic Functions
		The Legendre symbol ((n/m)): for prime m this checks whether or not n is a quadratic residue modulo m.
		Quadratic reciprocity is used to calculate this symbol, which has the values -1, 0 or 1.
		This is the extension of the Jacobi symbol to all integers m, by multiplicativity, and by defining ((n/2))=(- 1)^((n^2 - 1)/8) for odd n (and 0 for even n) and ((n/- 1)) equals plus or minus 1 according to the sign of n for n != 0 (and 1 for n = 0).
	www.umich.edu /~gpcc/scs/magma/text535.htm (452 words)

	APPLIED CRYPTOGRAPHY, SECOND EDITION: Protocols, Algorithms, and Source Code in C:Mathematical Background
		The Jacobi symbol, written J(a,n), is a generalization of the Legendre symbol to composite moduli; it is defined for any integer a and any odd integer n.
		The Jacobi symbol is a function on the set of reduced residues of the divisors of n and can be calculated by several formulas [1412].
		The Jacobi symbol cannot be used to determine whether a is a quadratic residue mod n (unless n is prime, of course).
	friedo.szm.sk /krypto/AC/ch11/11-09.html (586 words)

	[No title]
		Jacobi blithely goes ahead to formally compute (a/p), even when p isn't prime.
		Maple's problem is in not clearly distinguishing between their function L(a,b) and Legendre's symbol (a/b), which is defined _only_ for prime b.
		Jacobi's symbol is the generalization of (a/b) to all odd b, but it loses some of the usefulness for non-prime b, since it no longer correctly identifies non-residues.
	www.math.niu.edu /~rusin/known-math/97/jacobi (1550 words)

	[No title] (Site not responding. Last check: 2007-11-03)
		% % Parameters: % % Input, integer Q, an integer whose Jacobi symbol with % respect to P is desired.
		% % Input, integer P, the number with respect to which the Jacobi % symbol of Q is desired.
		% % Output, integer J, the Jacobi symbol (Q/P).
	www.csit.fsu.edu /~burkardt/m_src/polpak/jacobi_symbol.m (258 words)

	The Jacobi Symbol
		Jacobi's Symbol is crutial to the understanding of several encryption ciphers including the Rabin cipher.
		Since, in class we only talked about Jacobi's symbol enough to use it, I felt that a more thorough understanding of the symbol was needed.
		Also, there was some new and different notation that I found useful in describing Jacobi's Symbol.
	www.vonhagel.com /papers/theory/node13.html (139 words)

	Number theory - Wikipedia, the free encyclopedia
		Independently of Euler and Legendre, Gauss discovered the law about 1795, and was the first to give a general proof.
		To the subject have also contributed: Cauchy; Dirichlet whose Vorlesungen über Zahlentheorie is a classic; Jacobi, who introduced the Jacobi symbol; Liouville, Zeller(?), Eisenstein, Kummer, and Kronecker.
		The theory extends to include cubic and biquadratic reciprocity, (Gauss, Jacobi who first proved the law of cubic reciprocity, and Kummer).
	en.wikipedia.org /wiki/Number_theory (1344 words)

	Encyclopedia: Jacobi symbol (Site not responding. Last check: 2007-11-03)
		Karl Gustav Jacob Jacobi (Potsdam December 10, 1804 - Berlin February 18, 1851), was not only a great German mathematician but also considered by many as the most inspiring teacher of his time (Bell, p.
		It is defined as follows: The Legendre symbol is used by mathematicians in the area of number theory, particularly in the fields of factorization and quadratic residues.
		In mathematics, in number theory, the law of quadratic reciprocity connects the solvability of two related quadratic equations in modular arithmetic.
	www.nationmaster.com /encyclopedia/Jacobi-symbol (402 words)

	Primality Testing
		The Legendre symbol can be generalized to the Jacobi symbol.
		Note that the Jacobi symbol, since it is the product of Legendre symbols, can only have the values of 0, +1 or -1.
		There are several properties of the Jacobi symbol that make its computation fairly easy and, most importantly, do not require that n be factored.
	www-math.cudenver.edu /~wcherowi/courses/m5410/ctcprime.html (610 words)

	Legendre symbol : Jacobi symbol (Site not responding. Last check: 2007-11-03)
		terms defined : Legendre symbol : Jacobi symbol
		All is still licensed under the GNU FDL.
		He must sit and rest, he decided, and next time he would noted that he was feeling quite warm and comfortable.
	www.termsdefined.net /ja/jacobi-symbol.html (477 words)

	Unified Analysis of Euclidean Algorithms (Site not responding. Last check: 2007-11-03)
		The average behavior of nine algorithms derived from the Euclidean Algorithm is analysed.
		Vallée and her student, C. Lemée, gave some new results for the analysis of the average complexity of the computation of a fundamental function in number theory: the Jacobi symbol, which allows to determine whether a number is a square in a given modular arithmetic or not.
		The Jacobi symbol extends the Legendre symbol and is defined as
	www-lipn.univ-paris13.fr /~banderier/Seminar/vallee99.html (814 words)

	LASEC: Student Project (Site not responding. Last check: 2007-11-03)
		In 2004, a demonstrator for this scheme using the quartic residue symbol as a homomorphism has been implemented.
		The goal of this project was to optimize the existing implementation and to implement 3 additional homomorphisms (Jacobi symbol, discrete logarithm, RSA exponentiation) and compare them to each other.
		By optimizing and reimplementing we achieved to reduce the running time of the existing implementation of the basic algorithm for the quartic residue symbol by half.
	lasecwww.epfl.ch /abstract_demonstrator.shtml (132 words)

	Number Theory Glossary
		) is the Jacobi symbol.) This test is also called the Solovay-Strassen test for its original proposers.
		The ring of Gaussian Integers is the extension of the integers with a symbol
		), is a function which is defined in terms of the Jacobi symbol.
	www.math.umbc.edu /~campbell/NumbThy/Class/Glossary.html (827 words)

	Amazon.com: Complex/Archetype/Symbol in the Psychology of C.G. Jung: Books: Jolande Jacobi,Ralph Manheim (Site not responding. Last check: 2007-11-03)
		In this volume, Dr. Jacobi presents a study of three central, interrelated concepts in analytical psychology: the individual complex, the universal archetype, and the dynamic symbol.
		Jacobi has made it her task, in this book, to expound the important connection on the one hand between the individual complex and the universal, instinctual archetype, and on the other hand between this and the symbol.
		The appearance of her study is the more welcome to me in that the concept of the archetype has given rise to the greatest misunderstandings and -- if one may judge by the adverse criticisms -- must be presumed to be very difficult to comprehend.
	www.amazon.com /Complex-Archetype-Symbol-Psychology-C-G/dp/0691017743 (1012 words)

	Euler-Jacobi pseudoprime - Wikipedia, the free encyclopedia
		In number theory, an odd composite integer n is called an Euler-Jacobi pseudoprime to base a, if a and n are coprime, and
		The motivation for this definition is the fact that all prime numbers n satisfy the above equation, as explained in the Legendre symbol article.
		The equation can be tested rather quickly, which can be used for probabilistic primality testing.
	en.wikipedia.org /wiki/Euler-Jacobi_pseudoprime (433 words)

	[No title] (Site not responding. Last check: 2007-11-03)
		// Calculate the Jacobi Symbol // // This is used in many prime-testing algorithms.
		The Jacobi symbol is a // generalization of the Legendre symbol, and this function can be used to // calculate the Legendre symbol as well.
		// // The Legendre symbol and Jacobi symbol are often written as (n/m) which is // horrible notation that's indistinguishable from a parenthesized division.
	futureboy.homeip.net /frinksamp/JacobiSymbol.frink (181 words)

	[No title] (Site not responding. Last check: 2007-11-03)
		Among the integers in Z.sub.N.sup.* is defined a function, the Jacobi symbol, that evaluates easily to either 1 or -1.
		Thus, because of Fact 1, if one is given a Jacobi symbol 1 root and a Jacobi symbol -1 root of any square, he can easily factor N. With this background, the following describes how the RSA cryptosystem can be made fair in a simple way.
		One square root of Z mod N is X, which has Jacobi symbol equal to 1 (since the Jacobi symbol is multiplicative).
	www.eff.org /Privacy/Key_escrow/Clipper/key_escrow.patent (6759 words)

	Mathematica Usage Messages
		Constant is an attribute which indicates zero derivative of a symbol with respect to all parameters.
		Indeterminate is a symbol that represents a numerical quantity whose magnitude cannot be determined.
		Null is a symbol used to indicate the absence of an expression or a result.
	vlado.fmf.uni-lj.si /vlado/symbol/usage.htm (11884 words)

	Historia Matematica Mailing List Archive: Re: [HM] Gauss and the Jacobi symbol
		Next message: William C Waterhouse: "Re: [HM] Gauss and the Jacobi symbol"
		In reply to: William C Waterhouse: "Re: [HM] Gauss and the Jacobi symbol"
		Next in thread: William C Waterhouse: "Re: [HM] Gauss and the Jacobi symbol"
	sunsite.utk.edu /math_archives/.http/hypermail/historia/sep99/0158.html (489 words)

	Number theory algorithms
		The Legendre symbol (m/ n) is defined as +1 if m is a quadratic residue modulo n and -1 if it is a non-residue.
		The Jacobi symbol and the Legendre symbol have values +1, -1 or 0.
		Using these identities, we can recursively reduce the computation of the Jacobi symbol [a/b;] to the computation of the Jacobi symbol for numbers that are on the average half as large.
	homepage.mac.com /yacas/manual/Algochapter2.html (3947 words)

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