孙子定理用英语怎么说
孙子定理用英语怎么说
孙子定理用英语怎么说
孙子定理用英语怎么说
孙子定理,即中国剩余定理、孙子剩余定理.英文为【Chinese Remainder Theorem】
「孙子算经」︰
\x09「今有物,不知其数,三三数之,剩二,五五数之,剩三,七七数之,剩二,问物几何?」
\x09答曰：「二十三」
\x09解曰：「三三数之剩二,置一百四十,五五数之剩三,置六十三,七七数之剩二,置三十,并之,得二百三十三,以二百一十减之,即得.凡三三数之剩一,则置七十,五五数之剩一,则置二十一,七七数之剩一,则置十五,即得.」
【Chinese Remainder Theorem】
Theorem statement
\x09The original form of the theorem,contained in a third-century AD book Sun Zi suanjing (孙子算经 The Mathematical Classic by Sun Zi) by Chinese mathematician Sun Tzu and later republished in a 1247 book by Qin Jiushao,the Shushu Jiuzhang (数书九章 Mathematical Treatise in Nine Sections) is a statement about simultaneous congruences (see modular arithmetic).
Suppose n1,n2,…,nk are positive integers which are pairwise coprime.Then,for any given set of integers a1,a2,…,ak,there exists an integer x solving the system of simultaneous congruences
\x09\x09x ≡ a1 (mod n1)
\x09\x09x ≡ a2 (mod n2)
\x09\x09...
\x09\x09x ≡ ak (mod nk)
Furthermore,all solutions x to this system are congruent modulo the product N = n1n2…nk.
Hence
\x09\x09x ≡ y (mod ni) for all 1≤i≤k ,if and only if x ≡ y (mod N).
Sometimes,the simultaneous congruences can be solved even if the ni's are not pairwise coprime.A solution x exists if and only if:
\x09\x09ai ≡ aj (mod gcd(ni,nj)) for all i and j.
All solutions x are then congruent modulo the least common multiple of the ni.
孙子定理 [sūn zǐ dìng lǐ]
基本翻译
Chinese remainder theorem