同余及其应用
CONGRUENCE AND ITS
APPLICATIONS
专 业：信息与计算科学
姓 名： X X X 指 导 教 师： X X X 申请学位级别： X X 论文提交日期： XXXXXXXX 学位授予单位： XXXX大学
摘 要
本论文归纳总结了同余的相关性质定理，如Wilson定理，Fermat小定理以及Euler定理，以及集合论中的等价关系、商集等相关知识、数论中关于同余的一些性质，并熟悉剩余环相关的知识。还研究了含有未知数的同余方程，例如线性同余方程，多项式同余方程，线性同余方程组等。学习同余在实际和理论中的应用，结合实际探究了同余性质在整除性校验，万年历，散列函数上的应用以及构造校验位等方面的应用。这些应用体现了用同余性质解决问题的简洁性。在文章的最后，研究了当今在数论中最流行的工具Maple语言，学习其如何执行数论中关于同余的计算，并且编程计算相关的问题。
关键词：同余； 同余方程； 剩余环； 欧拉定理； 同余的应用； Maple语言
ABSTRACT
The article summarizes the related theorems of congruence, such as Wilson theorem, Fermat theorem and Euler theorem, some properties of the congruence equivalence relations, quotient set and other related knowledge, number theory as well as in set theory, and familiar with the relevant knowledge of the remaining ring. Also study the congruence equation containing the unknown number, such as linear congruence equation, polynomial congruence equation, linear congruence equations and so on. Learning the application of congruence in practical and theory, combined with the actual research in the congruence properties of divisibility checking, calendar, a hash function is applied on the application and construction check etc.Embodies simplicity to solve problems with congruence properties. At the end of the article, studying the most popular theory in today's tools of Maple language and learning how to perform the calculation about congruence in number theory, and programming computing the related problem of congruence.
Key words: Congruences; congruence equation; the remaining ring; Euler theorem; congruence application; Maple language
目 录
1. 前言 ........................................................................................................................ 1 2. 同余 ......................................................................................................................... 3 2.1 同余引言............................................................................................ 3 2.1.1 相关定义 ........................................................................................ 3 2.1.2 相关性质定理 ................................................................................ 3 2.2 线性同余方程 ................................................................................... 5 2.3 中国剩余定理 ................................................................................... 6 2.4 求解多项式同余方程 ....................................................................... 8 2.5 线性同余方程组 ............................................................................. 9 2.6 利用波拉德方法分解整数 ............................................................. 13 3. 同余的应用 ........................................................................................................ 16 3.1 整除性检验 ..................................................................................... 16 3.2 万年历............................................................................................ 19 3.4 散列函数 ....................................................................................... 24 3.5 校验位........................................................... 错误！未定义书签。 4. 特殊的同余式 ................................................................................................... 31 4.1 威尔逊定理和费马小定理 ............................................................. 31 4.2 欧拉定理.......................................................................................... 32 5. Maple 语言或Mathematic语言 ................................................................. 35 5.1 求解多项式同余方程 ..................................................................... 35
5.2 求解同余方程组 ............................................................................. 35 5.3 求解中国剩余定理 ......................................................................... 36 结 论 .................................................................................................................. 37 参考文献 .................................................................................................................. 38 致 谢 .................................................................................................................. 39