爱德华·卢卡斯与素性判定
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作者(加)休·C.威廉姆斯 著
出版社哈尔滨工业大学出版社
ISBN9787560392660
出版时间2021-03
装帧平装
开本16开
定价78元
货号1202369719
上书时间2024-11-25
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目录
Table of symbols
Preface
Preliminaries
1.1 Results from elementary number theory
1.2 Algorithms and complexity
1.3 Continued fractions
1.4 Some arithmetic functions
1.5 Results concerning binomial congruences
Notes for Chapter 1
2 The Beginnings
2.1 Antiquity
2.2 From the Middle Ages to Mersenne
2.3 Fermat and Euler
2.4 Lagrange, Legendre, and Gauss
2.5 The early tablemakers
Notes for Chapter 2
3 Lucas Early Work
3.1 Lucas earliest primality tests
3.2 Lucas and M127
Notes for Chapter 3
4 The Lucas Functions
4.1 Definition of the Lucas functions
4.2 Identity properties of the Lucas functions
4.3 Arithmetic properties
4.4 Computation of the Lucas functions
Notes for Chapter 4
5 Lucas Tests
5.1 Lucas early tests for Mersenne primes
5.2 The Fermat numbers
5.3 Landry and F6
5.4 Lucas and necessity
5.5 Lucas extended tests
Notes for Chapter 5
Later Developments
6.1 The work of Proth and Pocklington
6.2 Early factoring methods
6.3 Coles example
6.4 The Fermat numbers
Notes for Chapter 6
Early Devices
7.1 The beginnings of mechanization
7.2 Mechanisms for testing Mersenne numbers
7.3 The number sieve
Notes for Chapter 7
Kraitchik and Lehmer
8.1 Kraitchik
8.2 D.H. Lehmer
8.3 Some special numbers
8.4 The Lehmer functions
8.5 Some tables
Notes for Chapter 8
9 Finite Fields
9.1 Groups and rings
9.2 Polynomials and fields
9.3 Finite fields
9.4 Some polynomial congruences
Notes for Chapter 9
10 Lucas Functions Generalized
10.1 A generalization of the Lucas functions
10.2 Some identity properties
10.3 Some arithmetic properties
10.4 Some results on Cn
10.5 Primality tests
Notes for Chapter 10
11 Special Tests for Primality
11.1 Gauss and Jacobi sums
11.2 A primality test
11.3 Some special cases
Notes for Chapter 11
12 The Influence of the Computer
12.1 Some observations
12.2 Some primality tests
12.3 Some further tests
12.4 A result of Lenstra and R1031
Notes for Chapter 12
13 Results from the Computer
13.1 The Cunningham project
13.2 Primes of the form k2n + 1
13.3 Fast multiplication
13.4 Fermat and Mersenne numbers
Notes for Chapter 13
14 Primality Proofs
14.1 Factoring and primality tests
14.2 The complexity of primality testing
14.3 Certificates of primality
14.4 Introduction to elliptic curves
14.5 Very short certificates of primality
Notes for Chapter 14
15 Probabilistic Primality Tests
15.1 Pseudoprimes and Carmichael numbers
15.2 Stronger pseudoprimes
15.3 Cryptography
15.4 Probabilistic methods
Notes for Chapter 15
16 Recent Sieve Devices
16.1 Modern sieve devices
16.2 Pseudosquares and primality testing
16.3 The search for pseudosquares
16.4 Application to testing primality
Notes for Chapter 16
17 Primality Proving Today
17.1 The APR test
17.2 The Jacobi sums test
17.3 Primality testing with elliptic curves
17.4 The Lucas connection
17.5 Conclusion
Notes for Chapter 17
Bibliography
Index
编辑手记
内容摘要
本书主要包括初等数论的结果，算法与复杂性，卢卡斯的早期工作，卢卡斯函数的定义、等同性，卢卡斯测试，普罗斯和波金顿的工作，机械化的开始，有限域，卢卡斯函数的推广，素性的特殊测定，计算机的影响，素性证明，最近的筛分设备等内容。素数的数学思想具有悠久的历史，在人类意识中占据了一个重要的位置。本书深入浅出的讲解了如何判断一个给定的整数是否是素数，叙述时列举了许多经典题例与图片，定理叙述简洁明了。本书可供相关专业师生使用，也可供相关爱好者阅读。
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