41. Chinese Remainder Theorem --  Encyclopædia Britannica chinese remainder theorem ancient theorem that gives the conditions necessaryfor multiple equations to have a simultaneous integer solution. http://www.britannica.com/eb/article-9384390

Home Browse Newsletters Store ... Subscribe Already a member? Log in Content Related to this Topic This Article's Table of Contents Chinese remainder theorem Print this Table of Contents Shopping Price: USD $1495 Revised, updated, and still unrivaled. The Official Scrabble Players Dictionary (Hardcover) Price: USD $15.95 The Scrabble player's bible on sale! Save 30%. Merriam-Webster's Collegiate Dictionary Price: USD $19.95 Save big on America's best-selling dictionary. Discounted 38%! More Britannica products Chinese remainder theorem Page 1 of 1 ancient theorem that gives the conditions necessary for multiple equations to have a simultaneous integer solution. The theorem has its origin in the work of the 3rd-century- AD Chinese mathematician Sun Zi, although the complete theorem was first given in 1247 by Qin Jiushao Chinese remainder theorem... (75 of 250 words) var mm = [["Jan.","January"],["Feb.","February"],["Mar.","March"],["Apr.","April"],["May","May"],["June","June"],["July","July"],["Aug.","August"],["Sept.","September"],["Oct.","October"],["Nov.","November"],["Dec.","December"]];

42. Encyclopædia Britannica congruence chinese remainder theorem division chinese remainder theorem.CURRENT SUBJECT. chinese remainder theorem. Main Article Index Entry http://www.britannica.com/eb/subject?subjectId=41821

43. NewOrder - Computer Security And Networking Portal Applications to chinese remainder theorem @ Articles Encryption Mar 21 2005,0744 (UTC+0). data writes Difficulty Lever Novice/Intermediate http://neworder.box.sk/newsread.php?newsid=13394

44. Chapter 7: The Chinese Remainder Theorem Chapter 7. The chinese remainder theorem. 7.1 Solving Two Congruences 7.2 A MoreGeneral Theorem 7.3 Solving Lots of Congruences 7.4 Explicit Formulas http://www.math.mtu.edu/mathlab/COURSES/holt/dnt/chinese.html

Chapter 7 The Chinese Remainder Theorem Solving Two Congruences A More General Theorem Solving Lots of Congruences Explicit Formulas ... DNT Table of Contents

45. Math Forum Discussions Re chinese remainder theorem Posted Jul 30, 2005 512 PM I m trying tounderstand the chinese remainder theorem (crt). http://mathforum.org/kb/message.jspa?messageID=3868310&tstart=0

46. Math Forum Discussions Re chinese remainder theorem Posted Jul 31, 2005 450 AM I m trying tounderstand the chinese remainder theorem. let m_1, , m_r be integers http://mathforum.org/kb/message.jspa?messageID=3869009&tstart=0

47. Chinese Remainder Theorem Math reference, the chinese remainder theorem. chinese remainder theorem.Given a set of values c1 c2 cn, and a set of mutually coprime moduli m1 m2 http://www.mathreference.com/num-mod,chr.html

Modular Mathematics, Chinese Remainder Theorem Search Site map Contact us Main Page Numbers Modular Mathematics Use the arrows at the bottom to step through Modular Math. Chinese Remainder Theorem Given a set of values c c ... c n , and a set of mutually coprime moduli m m ... m n , is there an integer x such that x = c i mod m i for each i in 1 through n? Let z be the product of all the moduli. If x is a solution then so is x plus any multiple of z. If w is not a multiple of z, say w is not divisible by m , then x+w will not equal c mod m , and x+w will not be a solution. The solution, if it exists, is well defined mod z. To show that a solution exists, we simply construct one. Let a i be the product of all the moduli other than m i . Verify that a i and m i are coprime. Let b i be the inverse of a i mod m i . Finally, let x be the sum of a i b i c i for all i in 1 ... n. Verify that x satisfies all n equations simultaneously. If the original moduli are not coprime, split each equation up into a set of equations by factoring the composite modulus into prime powers. Then consider all the equations together. Equations sharing a common prime modulus are either redundant or inconsistent. An inconsistent example is x = 4 mod 6 and x = 11 mod 15. This would force x = 1 mod 3 and x = 2 mod 3, which is impossible. The example x = 4 mod 6 and x = 7 mod 15 has the solution x = 22 mod 30.

48. Chinese Remainder Theorem Math reference, chinese remainder theorem for rings. http://www.mathreference.com/ring,chr.html

Rings, Chinese Remainder Theorem Search Site map Contact us Main Page Rings Use the arrows at the bottom to step through Ring Theory. Chinese Remainder Theorem The chinese remainder theorem was developed for modular arithmetic , but it generalizes to ideals in a commutative ring r. Let h h ... h n be a set of coprime ideals. By coprime, we mean the sum of any two ideals spans the ring. Let j be the product of all these ideals. We will prove r/j is isomorphic to the direct product of the quotient rings r/h i , as i runs from 1 to n. An element in r/j can be mapped to the i th component in the direct product via r/h i . This is a well defined ring homomorphism, since each h i wholly contains j. We need to show it is 1-1 and onto. Focus on h . We know x + y = 1 for some x in h and y in h . Do the same for each h i in the set. Multiply all these equations together, and something in h + something in the product of the other ideals gives 1. Write this as x + y = 1. Reduce mod h , and y = -1. If y is mapped into any other quotient ring, other than r/h

49. Time Efficient Chinese Remainder Theorem Algorithm For Full-field Fringe Phase A The Optics InfoBase is OSA s online repository containing fulltext articles ofall peer-reviewed journals, meetings proceedings, and other publications of http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-6-1136

50. AoPS Math Forum :: View Topic - Chinese Remainder Theorem All times are GMT 7 Hours, chinese remainder theorem Post Posted Tue May24, 2005 713 am Post subject chinese remainder theorem, 1 Mark this post http://www.artofproblemsolving.com/Forum/topic-38722.html

Check out the AoPS Fall/Winter Class Schedule. Font Size: The time now is Fri Sep 16, 2005 4:16 pm All times are GMT - 7 Hours Chinese Remainder theorem Mark the topic unread Math Forum Olympiad Section ... Number Theory Theorems and Formulas Moderators: amfulger Arne harazi Megus ... pbornsztein Page of [2 Posts] Printer View Tell a Friend View previous topic View next topic Author Message realaec New Member Joined: 02 Mar 2005 Posts: 5 Posted: Tue May 24, 2005 7:13 am Post subject: Chinese Remainder theorem I know how to solve a basic equation with the CRT however, I heard from a friend that there is a wierd exception when x in a problem is raised to a power x^28 is congruent to two seperate numbers in deifferent mods. Do you know what this exception is, he said it had something to do with adding the power and the mod time n where n is any integer, to find solutions. Back to top ajorza P versus NP Joined: 17 Feb 2003 Posts: 36 Posted: Wed Jun 01, 2005 12:25 pm Post subject: Well, unfortunately what you ask is a little imprecise . Do you recall more info about what you are asking?

51. Solving Congruences: The Chinese Remainder Theorem Solving Congruences The chinese remainder theorem. http://www.math.okstate.edu/~wrightd/crypt/lecnotes/node21.html

Next: Challenges! Up: Cryptology Class Notes Previous: Square roots Solving Congruences: The Chinese Remainder Theorem In considering the problem of finding modular square roots, we found that the problem for a general modulus m could be reduced to that for a prime power modulus. The next problem would be how to piece the solutions for prime powers together to solve the original congruence. This is done by the Chinese Remainder Theorem, so-called because it appeared in ancient Chinese manuscripts. A typical problem is to find integers x that simultaneously solve 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. The Chinese Remainder Theorem has a very simple answer: Chinese Remainder Theorem: For relatively prime moduli m and n , the congruences have a unique solution x modulo mn Our example problem would have a unique solution modulo It's better than this; there is a relatively simple algorithm to find the solution. Since all number theory algorithms ultimately come down to Euclid's algorithm, you can be sure it happens here as well. First let's consider an even simpler example. Suppose we want all numbers

52. Chinese Remainder Theorem This directory contains an ACL2 proof of the chinese remainder theorem, asdescribed in a paper presented at ACL2 Workshop 2000. http://www.cs.utexas.edu/users/moore/acl2/workshop-2000/final/russinoff-short/cr

This directory contains an ACL2 proof of the Chinese Remainder Theorem, as described in a paper presented at ACL2 Workshop 2000. The entire proof is contained in the single event file crt.lisp , except that it depends on some lemmas from the author's library of floating-point arithmetic . In order to certify this file (after obtaining and certifying the library), first replace each of the two occurrences of " /u/druss/ " with the path to the directory under which your copy of the library resides. A second event file, summary.lisp , which contains the definitions and main lemmas involved in the proof, may then be certified. You can download a gzipped tar file containing this file together with the two event files discussed above.

53. Chinese Remainder Theorem chinese remainder theorem. To demonstrate the principle I will use an example.Consider finding the unique modular solution to 7 mod 11 and 3 mod 29. http://www.users.globalnet.co.uk/~perry/maths/crt/crt.htm

Chinese Rem'der Theorem [Home] [Other Maths] Chinese Remainder Theorem To demonstrate the principle I will use an example: Consider finding the unique modular solution to 7 mod 11 and 3 mod 29. If x is the solution then we have a pair of equations x = 7 + 11a and x = 3 + 29b. By inspection, the solution to this is x == 293 mod319, giving a=26, b=10. The two equations yield 7 + 11a = 3 + 29b, so 11a = -4 + 29b [1]. We can see that 29 is 7 mod 11, so we may reduce [1] to 11a = -4 + 7b [2] in the sense that the first time 7b-4 is a multiple of 11, 29b-4 will also be. The first solution to this is given by b=10 and a=6. Note how the value of b has unaltered - this is easy to prove. The next step is to see that [2] is merely another form of [1], and hence we may repeat the process. Now we see that 11 == 4 mod 7, so we get 4a = -4 + 7b. The first solution to this is a=6, and with [2], and b=4. From here we conclude that this process, handled with care, allows any pair of modular equations to be reduced to a vastly simpler version, solved, and then the original solution may be reconstructed. Please address questions/comments/suggestions to : Jon Perry

54. The Chinese Remainder Theorem And The Prime Memory System The chinese remainder theorem and the prime memory system. Full text, pdf formatPdf (287 KB). Source, International Conference on Computer Architecture http://portal.acm.org/citation.cfm?id=165172

55. Chinese Remainder Theorem - Summary Source Proving the chinese remainder theorem by the Cover Set Induction, H.Zhang and X. Hua, CADE11, D. Kapur (ed), 1992. Springer-Verlag. http://www.cs.nott.ac.uk/~lad/research/challenges/IWC008a.txt

Springer-Verlag.Chinese Remainder Theorem - Summary: Needs many lemmas (some of whose proofs are also challenging) An Existential Witness has to be Provided Definitions: allcongruent (num, list) -> bool allcongruent(x, nil) = true allcongruent(x, y::z) = allcongruent(x, z) and (rem(x, first(y))=rem(second(y), first(y))) allpositive - list -> bool true if all members of the list are greater than 0. allprime2 - list -> bool allprime2(nil) = true allprime2(y::z) = prime2list(x, first(y)) and allprime2(x, z) prime2 is true iff its argumens are relatively prime prime2list (num, list) -> bool prime2list(x, nil) = true prime2list(x, y::z) = prime2(x, first(y)) and prime2list(x, z) products1 - product of list. rem - returns the remainder of x by y Theorem: forall l:(list (pair nat)). exists x:nat. (allpositive(l) and allprime2(l)) -> allcongruent(x,l) forall l:(list (pair nat)). forall x,y:nat. ((allpositive(l) and allprime2(l)) and allcongruent(x,l) and allcongruent(y,l)) -> (mod (x - y),products(l)) Comments Actually two proof, existance and uniqueness. Proved in RRL by Zhang and Hua and there is a good deal of comment of the proof in their CADE-11 paper on the subject. They provide by hand the existential witness needed for the existance part of the proof. Source Proving the Chinese Remainder Theorem by the Cover Set Induction, H. Zhang and X. Hua, CADE-11, D. Kapur (ed), 1992.

56. Chinese Remainder Theorem A Mechanical Proof of the chinese remainder theorem. David M. Russinoff.This paper (ps, pdf), which was presented at ACL2 Workshop 2000 (see slides ps, http://nitro.xyzdns.net/~russ/papers/crt.html

A Mechanical Proof of the Chinese Remainder Theorem David M. Russinoff This paper ( ps pdf ), which was presented at ACL2 Workshop 2000 (see slides: ps pdf ), describes an ACL2 proof of the Chinese Remainder Theorem: If m m k are pairwise relatively prime moduli and a a k are natural numbers, then there exists a natural number x that simultaneously satisfies x a i (mod m i i k The entire proof is contained in the single event file crt.lisp , except that it depends on some lemmas from the author's library of floating-point arithmetic . In order to certify this file (after obtaining and certifying the library), first replace each of the two occurrences of " /u/druss/ " with the path to the directory under which your copy of the library resides. A second event file, summary.lisp , which contains the definitions and main lemmas involved in the proof, may then be certified.

57. Encyclopedia: Chinese Remainder Theorem Other descriptions of chinese remainder theorem. The chinese remainder theoremis any of a number of related results in Abstract algebra is the field of http://www.nationmaster.com/encyclopedia/Chinese-remainder-theorem

Supporter Benefits Signup Login Sources ... Pies Related Articles People who viewed "Chinese remainder theorem" also viewed: Chinese remainder theorem Eulers phi function Dirichlet series RSA ... Multiplicative group of integers modulo n What's new? Our next offering Latest newsletter Student area Lesson plans Recent Updates John W. Donaldson John Stossel John Smith of Jamestown John McDermott (footballer) ... More Recent Articles Top Graphs Richest Most Murderous Most Taxed Most Populous ... More Stats Encyclopedia: Chinese remainder theorem Updated 31 days 23 hours 47 minutes ago. Other descriptions of Chinese remainder theorem The Chinese remainder theorem is the name for several related results in abstract algebra and number theory Abstract algebra is the field of mathematics concerned with the study of algebraic structures such as groups, rings and fields. ... Traditionally, number theory is that branch of pure mathematics concerned with the properties of integers. ... Contents Simultaneous congruences of integers Statement for principal ideal domains Statement for general rings See also ... External links Simultaneous congruences of integers The original form of the theorem, contained in a book by the Chinese mathematician

58. Multidigit Modular Multiplication With The Explicit Chinese Our main tool is the Explicit chinese remainder theorem, 0.3 A MechanicalProof of the chinese remainder theorem Russinoff (2000) (Correct) http://citeseer.ist.psu.edu/bernstein95multidigit.html

59. Chinese Remainder Theorem Solve the following linear congruences x = ai mod mi x =. live web statistics. http://www.math.rutgers.edu/~erowland/chineseremaindertheorem.html

header("Chinese Remainder Theorem"); Solve the following linear congruences x a i mod m i x var sc_project=371526; var sc_partition=1;

60. Chinese Mathematics : Rebecca And Tommy The chinese remainder theorem (TaYen) This meant find N such that whendivided by b gives a remainder of a and when divided by d gives a remainder of c http://www.roma.unisa.edu.au/07305/remain.htm

The Chinese Remainder Theorem (Ta-Yen) It was not until 1247 that Qin Jiushao (c 1202-1261) published a general method for solving systems of linear congruence's in his book called ' Shushu jiuzhang (Mathematical Treatise in Nine Sections)' (Katz, 1992, p188). A book clearly influenced by the old chiu chang suan shu , as were a majority of Chinese mathematical works. Before this time only specific problems had been solved, by people such as Shu Zi (late third century). This method became known as the Ta-Yen. The basic format of problems it was to solve was ; N = a(mod b) = c(mod d) = ... This meant find N such that when divided by b gives a remainder of a and when divided by d gives a remainder of c. Throughout this page we will use the example N = 10(mod 12) = 0(mod 11) = 0(mod 10) = 4(mod 9) = 6(mod 8) = 0(mod 7) = 4(mod 6) In simple terms the method goes like this : 1. Find the least common multiple of the moduli. In our example the moduli are 12,11,10,9,8,7 and 6. How was this done? Reduce all moduli to a multiplication of prime numbers or their powers, unless they are already prime or a power of a prime alone. x x 3, 11 = 11 (already a prime), 10 = 2

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