數(shù)論導(dǎo)引

內(nèi)容概要

埃弗里斯特編著的《數(shù)論導(dǎo)引（影印版）》是“國(guó)外數(shù)學(xué)名著系列”之一，從最初等的數(shù)論知識(shí)談起，一直講到解析數(shù)論、代數(shù)數(shù)論、橢圓曲線以及數(shù)論在密碼理論中的應(yīng)用等，涉及范圍很廣闊，而且內(nèi)容并不膚淺。書(shū)中還有不少練習(xí)題，以及歷史的評(píng)注等?？晒?shù)論及相關(guān)專(zhuān)業(yè)研究生、教師及科研人員等學(xué)習(xí)參考。

書(shū)籍目錄

Introduction
1  A Brief History of Prime
  1.1  Euclid and Primes
  1.2  Summing Over the Primes
  1.3  Listing the Primes
  1.4  Fermat Numbers
  1.5 Primality Testing
  1.6  Proving the Fundamental Theorem of Arithmetic
  1.7  Euclid's Theorem Revisited
2  Diophantine Equations
  2.1  Pythagoras
  2.2  The Fundamental Theorem of Arithmetic in Other Contexts
  2.3  Sums of Squares
  2.4  Siegel's Theorem
  2.5  Fermat, Catalan, and Euler
3  Quadratic Diophantine Equations
  3.1  Quadratic Congruences
  3.2  Euler's Criterion
  3.3  The Quadratic Reciprocity Law
  3.4  Quadratic Rings
  3.5 Units in Z
  3.6  Quadratic Forms
4  Recovering the Fundamental Theorem of Arithmetic
  4.1  Crisis
  4.2  An Ideal Solution
  4.3  Fundamental Theorem of Arithmetic for Ideals
  4.4  The Ideal Class Group
5  Elliptic Curves
  5.1  Rational Points
  5.2  The Congruent Number Problem
  5.3  Explicit Formulas
  5.4  Points of Order Eleven
  5.5  Prime Values of Elliptic Divisibility Sequences
  5.6  Ramanujan Numbers and the Taxicab Problem
6  Elliptic Functions
  6.1  Elliptic Functions
  6.2  Parametrizing an Elliptic Curve
  6.3  Complex Torsion
  6.4  Partial Proof of Theorem 6.5
7  Heights
  7.1  Heights on Elliptic Curves
  7.2  Mordell's Theorem
  7.3  The Weak Mordell Theorem: Congruent Number Curve
  7.4  The Parallelogram Law and the Canonical Height
  7.5  Mahler Measure and the Naive Parallelogram Law
8  The Riemann Zeta Function
  8.1  Euler's Summation Formula
  8.2  Multiplicative Arithmetic Functions
  8.3  Dirichlet Convolution
  8.4  Euler Products
  8.5  Uniform Convergence
  8.6  The Zeta Function Is Analytic
  8.7  Analytic Continuation of the Zeta Function
9  The Functional Equation of the Riemann Zeta Function
  9.1  The Gamma Function
  9.2  The Functional Equation
  9.3  Fourier Analysis on Schwartz Spaces
  9.4  Fourier Analysis of Periodic Functions
  9.5  The Theta Function
  9.6  The Gamma Function Revisited
10 Primes in an Arithmetic Progression
  10.1 A New Method of Proof
  10.2 Congruences Modulo 3
  10.3 Characters of Finite Abelian Groups
  10.4 Dirichlet Characters and L-Functions
  10.5 Analytic Continuation and Abel's Summation Formula
  10.6 Abel's Limit Theorem
11  Converging Streams
  11.1 The Class Number Formula
  11.2 The Dedekind Zeta Function
  11.3 Proof of the Class Number Formula
  11.4 The Sign of the Gauss Sum
  11.5 The Conjectures of Birch and Swinnerton-Dyer
12  Computational Number Theory
  12.1 Complexity of Arithmetic Computations
  12.2 Public-key Cryptography
  12.3 Primality Testing: Euclidean Algorithm
  12.4 Primality Testing: Pseudoprimes
  12.5 Carmichael Numbers
  12.6 Probabilistic Primality Testing
  12.7 The Agrawal-Kayal-Saxena Algorithm
  12.8 Factorizing
  12.9 Complexity of Arithmetic in Finite Fields
References
Index

編輯推薦

　　An Introduction to Number Theory provides an introduction to themain streams of number theory.Starting with the unique factorizationproperty of the integers， the theme of factorization is revisited severaltimes throughout the book to illustrate how the ideas handed downfrom Euclid continue to reverberate through the subject. In particular，the book shows how the Fundamental Theorem of Arithmetic， handeddown from antiquity， informs much of the teaching of modem numbertheory. The result is that number theory will be understood， not asa collection of tricks and isolated results， but as a coherent andinterconnected theory. A number of different approaches to numbertheory are presented， and the different streams in the book are broughttogether in a chapter that describes the class number formula forquadratic fields and the famous conjectures of Birch and Swinnerton-Dyer. The final chapter introduces some of the main ideas behindmodern computational number theory and its applications incryptography. Written for graduate and advanced undergraduatestudents 'of mathematics， this text will also appeal to students incognate subjects who wish to learn some of the big ideas in numbertheory.

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