初等数论及其应用

YUI posted @ 2010年9月08日 20:10 in 未分类 , 1752 阅读

 

目录  · · · · · ·

前言
符号表
何谓数论
第1章 整数
1.1 数和序列    良序性质    丢番图逼近
1.2 和与积
1.3 数学归纳法
1.4 斐波那契数
1.5 整除性  1.若c|a,c|b,则c|(ma+nb)  带余除法a=bq+r  用良序性质证明带余除法  用q=[a/b],r=a-b[a/b]来证明关于最大整数函数的一个有用的性质
第2章 整数的表示法和运算
 2.1 整数的表示法
 2.2 整数的计算机运算
 2.3 整数运算的复杂度
第3章 素数和最大公因子
3.1 素数    最基本的素性检验是试除法
3.2 素数的分布
3.3 最大公因子  互素  (a+cb,b)=(a,b)  线性组合ma+nb,基于1.5整除性的1  最大公因子是线性组合中最小的正整数(基于良序性质)的证明  两两互素
3.4 欧几里得算法  拉梅定理
3.5 算术基本定理
3.6 因子分解法和费马数  奇数n=ab=s^2-t^2  费马数Fn=2^(2^n)+1的每个素因子都形如2^(n+2)k+1  费马数互素  F0F1F2···Fn-1=Fn-2
3.7 线性丢番图方程  d=(a,b),若d|c,则ax+by=c有无穷整数解
第4章 同余
 4.1 同余引言  m|(a-b),则a和b模m同余,记a≡b(mod m)  定理4.4 ac≡bc(mod m),d=(c,m),则a≡b(mod m/d)    
(c,m)=1,且ac≡bc(mod m),则a≡b(mod m)  a≡b(mod m1),···,则a≡b(mod [m1,m2,···,mk])
 4.2 线性同余方程
 4.3 中国剩余定理
 4.4 求解多项式同余方程
 4.5 线性同余方程组
 4.6 利用波拉德方法分解整数
第5章 同余的应用
 5.1 整除性检验
 5.2 万年历
5.3 循环赛赛程
5.4 散列函数
5.5 校验位
第6章 特殊的同余式
6.1 威尔逊定理和费马小定理
6.2 伪素数
6.3 欧拉定理
第7章 乘性函数
7.1 欧拉函数
7.2 因子和与因子个数
7.3 完全数和梅森素数
7.4 莫比乌斯反演
第8章 密码学
8.1 字符密码
8.2 分组密码和流密码
8.3 取幂密码
8.4 公钥密码
8.5 背包密码
8.6 密码协议及应用
第9章 原根
9.1 整数的阶和原根
9.2 素数的原根
9.3 原根的存在性
9.4 指数的算术
9.5 用整数的阶和原根进行素性检验
9.6 通用指数
第10章 原根与整数的阶的应用
 10.1 伪随机数
 10.2 埃尔伽莫密码系统
 10.3 电话线缆绞接中的一个应用 
第11章 二次剩余
 11.1 二次剩余与二次非剩余
……
第12章 十进制分数与连分数
第13章 某些非线性丢番图方程
第14章 高斯整数
附录
参考文献
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