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Topic: Chinese remainder theorem

Related Topics

Modular arithmetic theory

Remainder

Exponent

Modular arithmetic

Binary relation

Modulo operation

Equivalence relation

Multiplicative group of integers modulo n

Coprime

Formal language

Permutation

Primality test

Quadratic reciprocity

Ring theory

Dirichlet series

	Encyclopedia: Chinese remainder theorem (Site not responding. Last check: 2007-11-07)
		Typical statements are Fermat's little theorem and Euler's theorem extending it, the Chinese remainder theorem and the law of quadratic reciprocity.
		The prime number theorem and the related Riemann hypothesis are examples.
		Riemann (1859) conjectured the limit of the number of primes not exceeding a given number (the prime number theorem), introduced complex analysis into the theoryof the Riemann zeta function, and derived the explicit formulae of prime number theory from its zeroes.
	www.nationmaster.com /encyclopedia/Chinese-remainder-theorem (255 words)

	Chinese remainder theorem -- Encyclopædia Britannica (Site not responding. Last check: 2007-11-07)
		Chinese mathematician Sun Zi, although the complete theorem was first given in 1247 by Qin Jiushao.
		The British officer known as Chinese Gordon was famous for his romantic adventures in Asian countries and for his dramatic death at the siege of Khartoum.
		The Pythagorean Theorem is used to calculate the relationship between the legs and angles of a triangle.
	www.britannica.com /eb/article-9384390 (729 words)

	Chinese Remainder Theorem
		Chinese Remainder Theorem (CRT) The following problem was posed by Sunzi [Sun Tsu] (4th century AD) in the book Sunzi Suanjing: There are certain things whose number is unknown.
		Problems of this kind are all examples of what universally became known as the Chinese Remainder Theorem.
		Theorem Two simultaneous congruences n = n1 (mod m1) and n = n2 (mod m2) are only solvable when n1 = n2 (mod gcd(m1,m2)).
	www.chinapage.com /math/crt.html (863 words)

	CHINESE REMAINDER THEOREM
		Chinese Remainder Theorem, CRT, is one of the jewels of mathematics.
		Known already for ages, CRT continues to present itself in new contexts and open vistas for new types of applications.
		The individual chapters are largely independent and, consequently, the book can be used as supplementary material for courses in algorithmics, coding theory, cryptography or theory of computing.
	www.worldscibooks.com /compsci/3254.html (328 words)

	The Prime Glossary: Chinese remainder theorem (Site not responding. Last check: 2007-11-07)
		The following theorem is traditionally known as the Chinese remainder theorem (though there is some evidence that it was known to the Greeks before the Chinese).
		It is said that the ancient Chinese used a variant of this theorem to count their soldiers by having them line up in rectangles of 7 by 7, 11 by 11,...
		After counting only the remainders, they solved the associated system of equations for the smallest positive solution.
	primes.utm.edu /glossary/page.php?sort=ChineseRemainderTheorem (93 words)

	Chinese Remainder Theorem
		This is the only problem in the book relating to the CRT, so we don't know if he made a general method to solve these types of problems.
		The CRT one of the methods used to break the code and figure out the message.
		The CRT is in the category of Indeterminate Analysis.
	www.andrews.edu /~calkins/math/biograph/199899/topcrt.htm (587 words)

	Chinese Remainder Problem (Site not responding. Last check: 2007-11-07)
		Now that you know what a Chinese Remainder Problem is, you must be wondering why or what has this particular kind of problem to do with Chinese Mathematical History.
		The reason why it is called the Chinese Remainder Problem is because the earliest versions of these congruence problems occured in early Chinese mathematical works.
		The earliest of such works that contains the Chinese Remainder Problem is the Sun Tzu Suan Ching (also known as Sunzi suanjing) written in approximately late third century by Sun Zi.
	www.math.sfu.ca /histmath/China/3rdCenturyBC/CRP1.html (371 words)

	Math 5410 Chinese Remainder Theorem (Site not responding. Last check: 2007-11-07)
		Find the smallest multiple of 10 which has remainder 2 when divided by 3, and remainder 3 when divided by 7.
		Since, 2, 3, 5 and 7 are all relatively prime in pairs, the Chinese Remainder Theorem tells us that there is a unique solution modulo 210 (= 2x3x5x7).
		Remark 1: The theorem is valid in much more general situations than we have presented here.
	www-math.cudenver.edu /~wcherowi/courses/m5410/ctccrt.html (230 words)

	Chinese remainder theorem (Site not responding. Last check: 2007-11-07)
		In other words, given the remainders an integer gets when it's divided by an arbitrary set of divisors, you can uniquely determine the integer's remainder when it is divided by the least common multiple of those divisors.
		Note: For example, knowing the remainder of n when it's divided by 3 and the remainder when it's divided by 5 allows you to determine the remainder of n when it's divided by LCM(3,5) = 15.
		Paul E. Black, "Chinese remainder theorem", from Dictionary of Algorithms and Data Structures, Paul E. Black, ed., NIST.
	www.nist.gov /dads/HTML/chineseRmndr.html (148 words)

	math_class: Number Theory 101 (Chinese Remainder Theorem)
		I had forgotten, at the time, that I wanted to hit on the Chinese Remainder Theorem.
		The linear congruence a * x == b (mod n) has d unique (as unique as one can be with modulo arithmetic, meaning unique equivalence classes from the modulo) solutions where d = gcd(a,n) if b is divisible by d and no solutions if b is not divisble by d.
		I think that the story of the name of the Chinese Remainder Theorem is, by far, the best introduction that one can have to it.
	www.csh.rit.edu /~pat/math/series/nt/20020926 (2306 words)

	Chinese Remainder Theorem (Site not responding. Last check: 2007-11-07)
		CSC 372 - Assignment 3 - Chinese Remainder Theorem...
		Welcome to IEEE Xplore 2.0: Rings, fields, the Chinese remainder theorem and an...
		ACM SIGARCH: Volume 21, Issue 2, The Chinese remainder theorem......
	www.scienceoxygen.com /math/305.html (135 words)

	Solving Congruences: The Chinese Remainder Theorem
		This is done by the Chinese Remainder Theorem, so-called because it appeared in ancient Chinese manuscripts.
		It's important in our applications that the two moduli be relatively prime; otherwise, we would have to check that the two congruences are consistent.
		Chinese Remainder Theorem: For relatively prime moduli m and n, the congruences
	www.math.okstate.edu /~wrightd/crypt/lecnotes/node21.html (364 words)

	Chinese Remainder Theorem
		Repeatedly divided by 3, the remainder is 2; by 5 the remainder is 3; and by 7 the remainder is 2.
		In mathematical parlance the problems can be stated as finding n, given its remainders of division by several numbers
		The modern day theorem is best stated with a couple of useful notations.
	www.cut-the-knot.org /blue/chinese.shtml (550 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-11-07)
		That is the problem, but it seems too easy.
		Let the number of apples the teacher has be n.
		This is the criterion for using the Chinese Remainder Theorem.
	mathforum.org /library/drmath/view/55857.html (257 words)

	The Chinese Remainder Theorem
		In terms of rings, the Chinese Remainder Theorem asserts that the natural map
		The element found by the Chinese Remainder Theorem algorithm in this case is
		The following lemma is a nice application of the Chinese Remainder Theorem.
	modular.fas.harvard.edu /papers/ant/html/node31.html (336 words)

	Multidigit Modular Multiplication With The Explicit Chinese Remainder Theorem - Bernstein (ResearchIndex) (Site not responding. Last check: 2007-11-07)
		This method involves no multiprecision arithmetic, except in an easy precomputation; it is practical in software and extremely well suited for hardware.
		Our main tool is the Explicit Chinese Remainder Theorem, which says exactly how u diers from a particular linear...
		Multidigit modular multiplication with the explicit chinese remainder theorem.
	citeseer.ist.psu.edu /453093.html (415 words)

	Wikinfo \| Chinese remainder theorem
		The original form of the theorem, contained in a book by the Chinese mathematician Ch'in Chiu-Shao published in 1247, is a statement about simultaneous congruences (see modular arithmetic).
		For a principal ideal domain R the Chinese remainder theorem takes the following form:
		Images, some of which are used under the doctrine of Fair use or used with permission, may not be available.
	www.wikinfo.org /wiki.php?title=Chinese_remainder_theorem (500 words)

	Chinese Remainder Theorem
		The chinese remainder theorem was developed for modular arithmetic, but it generalizes to ideals in a commutative ring r.
		By coprime, we mean the sum of any two ideals spans the ring.
		By the binomial theorem, every term winds up in one of the two exponentiated ideals.
	www.mathreference.com /ring,chr.html (388 words)

	Chinese remainder theorem (Site not responding. Last check: 2007-11-07)
		Author: hasinoff What is the Chinese remainder theorem as it applies to solving equations involving the modulus operator?
		"The Chinese Remainder Theorem: if (Mi,Mj) = 1 for i != j, then the system x == C1 (mod M1), x == C2 (mod M2),.
		Such problems were studied in antiquity, particularly by ancient Chinese mathematicians, so the solution to the problem is called the Chinese remainder theorem.
	www.newton.dep.anl.gov /newton/askasci/1995/math/MATH056.HTM (156 words)

	Citebase - Poisson statistics via the Chinese remainder theorem (Site not responding. Last check: 2007-11-07)
		Authors: Granville, A. Kurlberg, P. We consider the distribution of spacings between consecutive elements in subsets of Z/qZ where q is highly composite and the subsets are defined via the Chinese remainder theorem.
		We give a sufficient criterion for the spacing distribution to be Poissonian as the number of prime factors of q tends to infinity, and as an application we show that the value set of a generic polynomial modulo q have Poisson spacings.
		Z that are created via the Chinese remainder theorem from subsets of Z/q
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0412135 (439 words)

	The Chinese Remainder Theorem (Site not responding. Last check: 2007-11-07)
		The Chinese Remainder Theorem (CRT) gives the answer to the problem:
		Find the number x, that satisfies all the n equations simultaneously:
		When using the CRT in a number theoretic transform, the algorithm can be implemented very efficiently using only single-precision arithmetic when rk
	www.apfloat.org /crt.html (179 words)

	Citebase - Around the Chinese Remainder Theorem
		We prove an explicit Chinese Remainder Theorem for one variable polynomials with complex coefficients, and derive some consequences.
		@misc{didry-2004-, author = {Jean-Marie Didry and Pierre-Yves Gaillard}, title = {Around the Chinese Remainder Theorem}, url = {http://www.citebase.org/cgi-bin/citations?id=oai:arXiv.org:math/0412133}, year = {2004} }
		Use the Correlation Generator to explore the correlation between download impact ("hits") and citation impact.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0412133 (213 words)

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