数论概论
2012-6
机械工业出版社
（美）Joseph H. Silverman
409
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《数论概论(英文版.第4版)》面向非数学专业学生，讲述了有关数论的知识，教给他们如何用数学方法思考问题，同时介绍了目前数论研究的某些前沿课题。本书采用轻松的写作风格，引领读者进入美妙的数论世界，不断激发读者的好奇心，并通过一些精心设计的练习来培养读者的探索精神与创新能力。对于定理的证明，则强调证明方法而不仅仅是得到特定的结果。
与第3版相比，本版的具体更新如下：
新增一章，详细介绍数学归纳法（第26章）。
前言部分给出了各章之间依赖关系的流程图，便于读者选择阅读。
调整了内容的组织结构，将反证法的相关材料前移至第8章，原根的相关章节移至二次互反律与平方和之后，上一版第47~50章的内容移至网上。
给出了二次互反律的完整证明，以及雅可比符号二次互反律的部分证明。
更新了书中的实例及章后练习题。
Joseph H．Silverman
拥有哈佛大学博士学位。他目前为布朗大学数学教授，之前曾任教于麻省理工学院和波士顿大学。1998年，他获得了美国数学会Steele奖的著述奖，获奖著作为《The
Arithmetic of Elliptic Curves》和《Advanced Topics in the Arithmetic
of Elliptic Curves》。 他的研究兴趣是数论、椭圆曲线和密码学等。
Preface
Flowchart of Chapter Dependencies
Introduction
1 What Is Number Theory?
2 Pythagorean Triples
3 Pythagorean Triples and the Unit Circle
4 Sums of Higher Powers and Fermat's Last Theorem
5 Divisibility and the Greatest Common Divisor
6 Linear Equations and the Greatest Common Divisor
7 Factorization and the Fundamental Theorem of
Arithmetic..
8 Congruences
9 Congruences, Powers, and Fermat's Little Theorem
10 Congruences, Powers, and Euler's Formula
11 Euler's Phi Function and the Chinese Remainder Theorem..
12 Prime Numbers
13 Counting Primes
14 Mersenne Primes
15 Mersenne Primes and Perfect Numbers
16 Powers Modulo m and Successive Squaring
17 Computing k th Roots Modulo rn
18 Powers, Roots, and "Unbreakable" Codes
19 Primality Testing and Carmichael Numbers
20 Squares Modulo p
21 Is --1 a Square Modulop? Is 2?
22 Quadratic Reciprocity
23 Proof of Quadratic Reciprocity
24 Which Primes Are Sums of Two Squares?
25 Which Numbers Are Sums of Two Squares?
26 As Easy as One, Two, Three
27 Euler's Phi Function and Sums of Divisors
28 Powers Modulo p and Primitive Roots
29 Primitive Roots and Indices
30 The Equation X4 + y4 =z4
31 Square-Triangular Numbers Revisited'.
32 Pell's Equation
33 Diophantine Approximation
34 Diophantine Approximation and Pell's Equation
35 Number Theory and Imaginary Numbers
36 The Gaussian Integers and Unique Factorization
37 Irrational Numbers and Transcendental Numbers
38 Binomial Coefficients and Pascal's Triangle
39 Fibonacci's Rabbits and Linear Recurrence Sequences
40 Oh, What a Beautiful Function
41 Cubic Curves and Elliptic Curves
42 Elliptic Curves with Few Rational Points
43 Points on Elliptic Curves Modulo p
44 Torsion Collections Modulo p and Bad Primes
45 Defect Bounds and Modularity Patterns
46 Elliptic Curves and Fermat's Last Theorem
Further Reading
Index
47 The Topsy-Turvy World of Continued Fractions [online]
48 Continued Fractions and Pell's Equation [online]
49 Generating Functions [online]
50 Sums of Powers [online]
A Factorization of Small Composite Integers [online]
B A List of Primes [online]
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不过确实是非数学专业参考用书，数学专业的同学看就比较浅了
A book for your reference shelf