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Topic: Legendre symbol

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Quadratic residue

Quadratic reciprocity

Lenstra elliptic curve factorization

Totient function

Prime number

Generalized Riemann hypothesis

Artin conjecture

Quadratic sieve

Legendre polynomials

Adrien Marie Legendre

Congruence of squares

Natural number

Riemann zeta function

Modular arithmetic

Chinese remainder theorem

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	Adrien-Marie Legendre
		Legendre's researches connected with the gamma function are of importance, and are well known; the subject was also treated by Carl Friedrich Gauss in his memoir Disquisitiones Generales Circa Series Infinitas (1816), but in a very different manner.
		Legendre's theorem is a fundamental one in geodesy, and his contributions to the subject are of the greatest importance.
		Legendre's name is most widely known on account of his Eléments de Géométrie, the most successful of the numerous attempts that have been made to supersede Euclid as a text-book on geometry.
	www.nndb.com /people/891/000093612 (1750 words)

	Adrien-Marie Legendre
		Most of his work was brought to perfection by others: his work on roots of polynomials inspired Galois theory; Abel's work on elliptic functions was built on Legendre's; some of Gauss' work in statistics and number theory completed that of Legendre.
		Legendre did an impressive amount of work on elliptic functions, including the classification of elliptic integrals, but it took Abel's stroke of genius to study the inverses of Jacobi's functions and solve the problem completely.
		In theoretical mechanics, he is known for the Legendre transform[?], which is used to go from the Lagrangian to the Hamiltonian formulation of mechanics.
	www.ebroadcast.com.au /lookup/encyclopedia/le/Legendre.html (189 words)

	Legendre symbol - Wikipedia, the free encyclopedia
		The Legendre symbol is a special case of the Jacobi symbol.
		The Legendre symbol is related to Euler's criterion and Euler proved that
		The Jacobi symbol is a generalization of the Legendre symbol that allows composite bottom numbers.
	en.wikipedia.org /wiki/Legendre_symbol (265 words)

	Adrien-Marie Legendre Summary
		Legendre's interest in celestial mechanics eventually led to two further papers, one on the attraction of certain ellipsoids, and the other on the form and density of fluid planets.
		Legendre began both investigations in the mid-1780s, although it was not until later that he made his most significant contributions.
		Legendre succeeded Laplace as the examiner in mathematics of students assigned to the artillery in 1799, a position he held until 1815.
	www.bookrags.com /Adrien-Marie_Legendre (2649 words)

	PlanetMath: Legendre symbol
		The Legendre symbol can be computed by means of Euler's criterion or Gauss' lemma.
		Generalizations of this symbol are the Jacobi Symbol and the Kronecker symbol.
		This is version 9 of Legendre symbol, born on 2001-10-08, modified 2008-01-25.
	planetmath.org /encyclopedia/LegendreSymbol.html (110 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)

	Legendre symbol
		The Legendre symbol is used by mathematicians in the theory of numbers, particularly in the fields of factorization and quadratic residues.
		Thus we can see that the Legendre symbol is completely multiplicative, i.e.
		The Legendre symbol can be used to compactly formulate the law of quadratic reciprocity.
	www.ebroadcast.com.au /lookup/encyclopedia/ja/Jacobi_symbol.html (194 words)

	Scientific Symbols, Icons, Mathematical Symbols - Numericana
		The Tai-Chi Mandala: The taiji (Yin-Yang) symbol was Bohr's coat-of-arms.
		William Oughtred (1574-1660) was instrumental in the subsequent popularization of the symbol, which appears next in 1618, in the appendix [attributed to him] of the English translation by Edward Wright of John Napier's Descriptio (where early logarithms were first described in 1614).
		The symbol itself is properly called a lemniscus, a latin noun which means "pendant ribbon" and was first used in 1694 by Jacob Bernoulli (1654-1705) to describe a planar curve now called the Lemniscate of Bernoulli.
	home.att.net /~numericana/answer/symbol.htm (2623 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.htm
		His 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.
		Legendre's major work on elliptic functions in Exercises du Calcul Intégral appeared in three volumes in 1811, 1817, and 1819.
		Legendre polynomials form an orthonormal basis for the vector space of polynomials.
	www.cse.ohio-state.edu /~brinkmei/math/Legendre.htm (136 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)

	The Jacobe Symbol (Site not responding. Last check: )
		If n is odd, the jacobe 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 jacobe symbol does not tell you whether a is a square mod n.
		If the jacobe 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 (275 words)

	Squares in Arithmetic Progression (mod p)
		This is to be expected, because our Legendre symbols are unavoidable for ALL primes, so the only reason we do not have non-trivial progressions for p = 2, 3, 5, 7, or 11 is that in these cases the applicable common difference equals the prime itself.
		Our objective is to minimize the total number of Legendre symbols, and to include cases that involve both signs of these symbols, in order to maximize the coverage.
		The task is to find conditions, probably involving one or two more Legendre symbols, and progressions with step sizes of 13, 19, 29, 31, and 37 to complete the unavoidable set.
	www.mathpages.com /home/kmath291.htm (1063 words)

	Jacobi symbol - Wikipedia, the free encyclopedia
		The last property is known as reciprocity, similar to the law of quadratic reciprocity for Legendre symbols.
		The general statements about quadratic residuals with respect to the Legendre symbol cannot be made with the Jacobi symbol.
		Since the Jacobi symbol is a product of Legendre symbols, there are cases where two Legendre symbols evaluate to −1 and the Jacobi symbol evaluates to 1.
	en.wikipedia.org /wiki/Jacobi_symbol (247 words)

	The Jewel of Arithmetic: Quadratic Reciprocity
		It's just an abstract symbol equal to either +1 or -1 depending on whether p is or isn't a square modulo q.
		Legendre succeeded in proving somes special cases of this, and also gave what he thought was a complete proof, but his argument relied on the premise that every arithmetic progression contains infinitely many primes.
		This is in fact true, as subsequently shown by Dirichlet, but at the time of Legendre's proof it wasn't known, so Gauss pointed out that Legendre's proof was not valid.
	www.mathpages.com /home/kmath075.htm (1007 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)

	[No title] (Site not responding. Last check: )
		I don't know the implementation of maple's Legendre function, but if it is the same Legendre/Jacobi function that I know, it does _not_ require factorization, and should be of the same or slightly easier complexity than gcd(p,q).
		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)

	Jacobi's Symbol
		Jacobi (in 1846) published a paper in which he defined his own symbol by extending the Legendre symbol (a/p).
		For Legendre, p is restricted to being an odd prime - Jacobi allows p to be any odd positive integer.
		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)

	Unified Analysis of Euclidean Algorithms
		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
	pauillac.inria.fr /algo/seminars/sem98-99/vallee.html (814 words)

	The Jacobe Prime Test (Site not responding. Last check: )
		If n is prime the jacobe symbol is the legendre symbol, which is 1 if x is a square mod n, and -1 otherwise.
		Raise a square to the (n-1)/2 and you are raising something else to the n-1, which becomes 1, just like the legendre symbol.
		Modular exponentiation is tractable, and the jacobe symbol [x\n] is computed easily using the gcd algorithm.
	www.mathreference.com /num-pfac,jacobe.html (589 words)

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