同余理论和应用 论文.doc

毕业（设计）论文 题 目 学 院 专业班级 学生姓名 指导教师 成 绩 2013年 6月20日摘 要 同余理论在数论中是非常重要的一个部分, 由整除概念的延拓得到了同余理论, 引进了同余之后就简化了数论中的很多问题, 它的应用十分广泛, 在生活中也经常遇到, 这也极大地推动和丰富了它们的发展, 许多新的理论、技巧和方法的诞生与发展都是同余理论的应用与推广. 本文首先归纳总结了同余的性质及其应用, 在同余的定义的基础上, 介绍了同余的中几个重要的性质, 并且侧重讨论了同余的几个重要的应用, 剩余类和剩余系也是它的研究对象.然后, 我们又归纳总结了同余方程的解及应用, 例如: 一次同余式的形式、解同余式以及解一次同余式组和中国剩余定理的应用等问题, 讨论的过程中由浅入深, 层层推进, 对相关知识的总结形成了较为完整的知识体系, 对解同余方程的一些特殊的性质理解更为深刻.最后, 结合同余理论的概念与性质, 给出常见的习题及解题技巧, 并举例说明了几个重要的同余定理和它们的应用. 关键词： 同余；同余方程；剩余系；一次同余式 Abstract Congruence in number theory is an important part of, It is the concept of divisible extension, after the introduction of congruence simplifies many of the problems in number theory, it is widely used, often encountered in life, it also greatly facilitate and enrich their development, many new theories, techniques and methods are the birth and development of the theory of congruence application and promotion. This paper briefly describes the properties and applications of congruence, on this basis, given the definition of congruence theory, and pointed out several important congruence theory of the nature and focused on the congruence of several applications of the remaining classes and surplus lines; secondly, summarized congruence equations and applications, such as a congruence in the form of a solution of the congruence reconciliation congruence group and the Chinese Remainder Theorem and other issues, discussions Deep layers of advancing knowledge on the formation of a more complete knowledge of the system, the solution of some special congruence equation profound understanding of the nature; Finally, combining the concept of congruence theory and nature gives common exercises and problem-solving skills, and an example illustrates several important congruence theorems and their applications. Key words: Congruence; Congruence equation; Residue system; Once congruence目 录 摘要 I Abstract II 绪论 1 第1章