中国剩余定理的中外历史发展比较

发布时间：2019-06-01 12:34

【摘要】：《孙子算经》中的“物不知数”问题在中国传统数学史上占有极为重要的地位。至南宋,秦九韶对物不知数问题做精细研究,最终创造了此题的解法,称为大衍总数术(简称大衍术),著录于《数书九章》中。现今称此术为“中国剩余定理”。中国剩余定理是举世闻名的定理,是中外任何一本基础数论教科书中不可或缺的,并被广泛应用于密码学、快速傅里叶变换理论等诸多领域中,但其历史发展的研究却较为稀少。本论文在前人研究的基础上,以中国剩余定理发展的历史为研究对象,将相关文献进行系统地梳理,尤其对南宋秦九韶《数书九章》,清代张敦仁《求一算术》、黄宗宪《求一术通解》,以及印度婆什迦罗二世的《丽罗娃底》,日本关孝和《括要算法》,德国高斯《算术探索》等数学原典;以及日本三上义夫《中国和日本的数学发展》和《中国算学之特色》、法国巴歇《数学趣味》中所记载的相关资料进行深入研究。主要完成了以下工作:首先,从大衍术产生的背景出发,以历代数学家对其的贡献为主线,梳理出国内中国剩余定理的历史发展,并结合钱宝琮的相关文献,作出了中国剩余定理在中国的历史发展演进路线简图;分印度、日本、欧洲三大板块,依次梳理出国外中国剩余定理的历史发展。其次,从研究的时间与成果、问题的起源与传播、符号的产生与使用等多个角度,将国内外对中国剩余定理的相关研究作对比。其中,国内重点讨论秦九韶和黄宗宪的研究工作,国外以欧洲且主要以高斯时期的数学家为研究对象。希望能够以多种视角全面的呈现国内外中国剩余定理研究的差异。中国剩余定理是一个旷世之作,但秦九韶在运用时出现了错误。因此,本论文还分析了秦九韶运用大衍术计算“古历会积”算题时出现的错误及其修正情况。最后,本论文参考李倍始《13世纪中国数学》中对一次同余式组解法的十种水平的分类,及其所呈现的15个有代表性的数学家或著作所达到的水平的表格,结合本论文的相关内容,按照其分类方法,补充了秦九韶之前(主要是印度)以及其后(中国、日本、欧洲)的数学家所达到的水平(中国至清末黄宗宪、日本主要是关孝和与三上义夫、欧洲至比利时赫师慎),并作出了相对完善的列表。发现印度普遍水平较低,到了婆什迦罗二世才有提升。日本关孝和仅达到印度的最高水平,但比其晚了500多年。中国清末的黄宗宪是同时代水平最高的,且最早达到十种水平。而对于欧洲,李倍始的列表中有所遗漏,早在1612年,法国巴歇便达到了高斯的水平。总之,“物不知数问题”的解法要义不明,或许是一种“缺憾”。但正是如此,才导致了秦九韶对其算法原意的探析,进而得出大衍总数术。一道数学问题最终成为了数学史上的华丽篇章,因此探究其解法背后隐藏的原理——中国剩余定理的演变源流、梳理该原理的中外历史发展,无疑是具有积极意义的。
[Abstract]:The problem of knowing the number of things in Sun Tzu's Sutra occupies a very important position in the history of traditional Chinese mathematics. To the Southern Song Dynasty, Qin Jiushao made a detailed study of the problem of unknown matter, and finally created a solution to this problem, called the total number of derivatives (Da Yan for short), which is recorded in the Nine chapters of the Book of numbers. At present, this technique is called "Chinese remainder Theorem". Chinese remainder theorem is a world-famous theorem, which is indispensable in any basic number theory textbook at home and abroad, and is widely used in cryptography, fast Fourier transform theory and many other fields. However, the study of its historical development is relatively rare. On the basis of previous studies, this paper takes the history of the development of Chinese surplus theorem as the research object, and systematically combs the relevant literature, especially for Qin Jiushao in the Southern Song Dynasty and Zhang Dunren in the Qing Dynasty. Huang Zongxian's General solution of "seeking a skill", as well as the original mathematical scriptures such as Lerova II of Indian Boruscharo II, Guan Hsiao and "including the algorithm" of Japan, Gao Si of Germany, "arithmetic Exploration", etc. As well as the mathematical development of China and Japan and the characteristics of Chinese arithmetic, the relevant data recorded in Bayer's Mathematical interest, France, are deeply studied. The main work has been completed as follows: first of all, starting from the background of the great derivative, taking the contributions of mathematicians of the past dynasties as the main line, this paper combs the historical development of the Chinese surplus theorem in China, and combines the relevant literature of Qian Baocong. A schematic diagram of the historical development and evolution of the Chinese remainder theorem in China is made. Divided into India, Japan and Europe, the historical development of foreign Chinese surplus theorem is sorted out in turn. Secondly, from the point of view of the time and achievement of the study, the origin and propagation of the problem, the generation and use of symbols, and so on, this paper compares the relevant research on the surplus theorem in China at home and abroad. Among them, the research work of Qin Jiushao and Huang Zongxian is discussed in China, and the mathematicians of Gao Si period are taken as the research objects abroad. It is hoped that the differences in the study of Chinese residual theorem at home and abroad can be presented comprehensively from a variety of perspectives. The Chinese remainder theorem is an extensive work, but Qin Jiushao made a mistake in his application. Therefore, this paper also analyzes the errors and correction of Qin Jiushao in calculating the problem of "ancient calendar confluence" by using Da Yan technique. Finally, this paper refers to the classification of ten levels of the solution of one congruence group in Li Beizhi's 13th Century Chinese Mathematics, and the table of the level reached by 15 representative mathematicians or works. Combined with the relevant contents of this paper, according to its classification method, this paper supplements the level reached by mathematicians before Qin Jiushao (mainly India) and then (China, Japan, Europe) (Huang Zongxian from China to the end of Qing Dynasty. Japan is mainly Guan Xiaohe and Sanshang Yifu, Europe to Belgium Shi Shen), and made a relatively perfect list. It was found that India was generally low and did not improve until Bashgaro II. Japan's Guan Hsiao and only reached India's highest level, but more than 500 years later. Huang Zongxian in late Qing Dynasty was the highest at the same time and reached ten levels at the earliest. For Europe, Li Bizhi's list is missing, as early as 1612, France reached the level of Gao Si. In a word, the solution of the problem of "thing does not know the number" is unclear, which may be a kind of defect. However, it was precisely this that led to Qin Jiushao's analysis of the original meaning of his algorithm, and then came to the conclusion of the total number of derivatives. A mathematical problem has finally become a gorgeous chapter in the history of mathematics, so it is undoubtedly of positive significance to explore the evolution of the Chinese surplus theorem, which is hidden behind its solution, and to sort out the historical development of the principle at home and abroad.
【学位授予单位】：四川师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O156

【参考文献】