數(shù)論概論

內(nèi)容概要

　　《數(shù)論概論(英文版.第4版)》面向非數(shù)學(xué)專業(yè)學(xué)生，講述了有關(guān)數(shù)論的知識，教給他們?nèi)绾斡脭?shù)學(xué)方法思考問題，同時(shí)介紹了目前數(shù)論研究的某些前沿課題。本書采用輕松的寫作風(fēng)格，引領(lǐng)讀者進(jìn)入美妙的數(shù)論世界，不斷激發(fā)讀者的好奇心，并通過一些精心設(shè)計(jì)的練習(xí)來培養(yǎng)讀者的探索精神與創(chuàng)新能力。對于定理的證明，則強(qiáng)調(diào)證明方法而不僅僅是得到特定的結(jié)果。
　　與第3版相比，本版的具體更新如下：
　　新增一章，詳細(xì)介紹數(shù)學(xué)歸納法（第26章）。
　　前言部分給出了各章之間依賴關(guān)系的流程圖，便于讀者選擇閱讀。
　　調(diào)整了內(nèi)容的組織結(jié)構(gòu)，將反證法的相關(guān)材料前移至第8章，原根的相關(guān)章節(jié)移至二次互反律與平方和之后，上一版第47~50章的內(nèi)容移至網(wǎng)上。
　　給出了二次互反律的完整證明，以及雅可比符號二次互反律的部分證明。
　　更新了書中的實(shí)例及章后練習(xí)題。

作者簡介

　　　　Joseph H．Silverman
擁有哈佛大學(xué)博士學(xué)位。他目前為布朗大學(xué)數(shù)學(xué)教授，之前曾任教于麻省理工學(xué)院和波士頓大學(xué)。1998年，他獲得了美國數(shù)學(xué)會Steele獎(jiǎng)的著述獎(jiǎng)，獲獎(jiǎng)著作為《The
Arithmetic of Elliptic Curves》和《Advanced Topics in the Arithmetic
of Elliptic Curves》。 他的研究興趣是數(shù)論、橢圓曲線和密碼學(xué)等。

書籍目錄

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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