LYCOS RETRIEVER 
Chinese Remainder Theorem
built 478 days ago
Chinese Remainder Theorem, CRT, is one of the jewels of mathematics. It is a perfect combination of beauty and utility or, in the words of Horace, [O]mne tulit punctum qui miscuit utile dulci. Known already for ages, CRT continues to present itself in new contexts and open vistas for new types of applications. So far, its usefulness has been obvious within the realm of "three C's". Computing was its original field of application, and continues to be important as regards various aspects of algorithmics and modular computations. Theory of codes and cryptography are two more recent fields of application.
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The Chinese Remainder Theorem is one of the oldest theorems in number theory. Your first job is to discover the right statement of this theorem. Most of the statement of the theorem is provided in the next section - you just need to fill in the missing part.
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A message which is coded with a public key, which is based on Chinese Remainder Theorem, is virtually impossible to solve without the private key which is based on prime numbers. If the First World War was the chemists war (chlorine gas used) and if the Second World War was the physicist war (atomic bombs used), that would suggest that the Third World War will be the mathematicians?s war, because they master the new weapon which is called information.
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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.
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Recall that the cases where there is a unique solution modulo m1m2 is covered by the Chinese Remainder Theorem for two congruences. Suppose instead you had three or more congruences. Is there a version of the Chinese Remainder Theorem in this case? There is indeed, and (surprise, surprise) it's your job to find it.
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Chinese Remainder Theorem