| Since the publication of the first edition, I have received many communications from readers all over the world. It is my great pleasure to thank the following people for their comments, corrections and encouragements: Prof. Jim Austin, Prof. Friedrich L. Bauer, Dr. Hassan Daghigh Dr. Deniz Deveci, Mr. Rich Fearn, Prof. Martin Hellman, Prof. Zixin Hou, Mr. Waseem Hus- sain, Dr. Gerard R. Maze, Dr. Paul Maguire, Dr. Helmut Meyn, Mr. Robert Pargeter, Mr. Mok-Kong Shen, Dr. Peter Shiu, Prof. Jonathan P. Sorenson, and Dr. David L. Stern. Special thanks must be given to Prof. Martin Hellman of Stanford University for writing the kind Foreword to this edition and also for his helpful advice and kind guidance, to Dr. Hans WSssner, Mr. Alfred Hofmann, Mrs. Ingeborg Mayer, Mrs. Ulrike Stricker, and Mr. Frank Holzwarth of Springer-Verlag for their kind help and encouragements during the preparation of this edition, and to Dr. Rodney Coleman, Prof. Glyn |
1. Elementary Number Theory 1.1 Introduction 1.1.1 What is Number Theory? 1.1.2 Applications of Number Theory 1.1.3 Algebraic Preliminaries 1.2 Theory of Divisibility 1.2.1 Basic Concepts and Properties of Divisibility 1.2.2 Fundamental Theorem of Arithmetic 1.2.3 Mersenne Primes and Fermat Numbers 1.2.4 Euclid‘s Algorithm 1.2.5 Continued Fractions 1.3 Diophantine Equations 1.3.1 Basic Concepts of Diophantine Equations 1.3.2 Linear Diophantine Equations 1.3.3 Pell‘s Equations 1.4 Arithmetic Functions 1.4.1 Multiplicative Functions 1.4.2 Functions (n)， (n) and s(n) 1.4.3 Perfect， Amicable and Sociable Numbers 1.4.4 Functions (n)， (n) and (n) 1.5 Distribution of Prime Numbers 1.5.1 Prime Distributionj Function π(x) 1.5.2 Approximations of π(x)by x/ln x 1.5.3 Approximationa of π(x)by Li(x) 1.5.4 The Riemann s-Functions 1.5.5 The nth Prime 1.5.6 Distribution of Twin Primes 1.5.7 The Arithmetic Progression of Primes 1.6 Theory of Congruences …… 1.7 Arthmetic of Elliptic Curves 1.8 Bibliographic Notes and Further Reading 2 Computational/Algorthmic Number Theory 2.1 introduction 2.2 ALgorithms for Primality Testing 2.3 Algorithms for Integer Factorization 2.4 Algorithms for Discrete Logarithms 2.5 Quantum Nuber-Theoretic Algorithms 2.6 Miscellaneous Algorithms in Number Theory 2.7 Bibliographic Notes and Further Reading 3 Applied Nuber Theory in Computing/Cryptography 3.1 Why Applied Nuber Theory? 3.2 Computer Systems Design 3.3 Cryptography and Information Security 3.4 Bibliographic Notes and Further Reading Bibliography Index |
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