Exploring one of the most dynamic areas of mathematics, Advanced Number Theory with Applications covers a wide range of algebraic, analytic, combinatorial, cryptographic, and geometric aspects of number theory. Written by a recognized leader in algebra and number theory, the book includes a page reference for every citing in the bibliography and more than 1,500 entries in the index so that students can easily cross-reference and find the appropriate data. With numerous examples throughout, the text begins with coverage of algebraic number theory, binary quadratic forms, Diophantine approximation, arithmetic functions, p-adic analysis, Dirichlet characters, density, and primes in arithmetic progression. It then applies these tools to Diophantine equations, before developing elliptic curves and modular forms. The text also presents an overview of Fermat’s Last Theorem (FLT) and numerous consequences of the ABC conjecture, including Thue–Siegel–Roth theorem, Hall’s conjecture, the Erd?s–Mollin-–Walsh conjecture, and the Granville–Langevin Conjecture. In the appendix, the author reviews sieve methods, such as Eratothesenes’, Selberg’s, Linnik’s, and Bombieri’s sieves. He also discusses recent results on gaps between primes and the use of sieves in factoring. By focusing on salient techniques in number theory, this textbook provides the most up-to-date and comprehensive material for a second course in this field. It prepares students for future study at the graduate level.
Author(s): Richard A. Mollin
Series: Discrete Mathematics and Its Applications
Edition: 1
Publisher: CRC Press
Year: 2010
Language: English
Pages: 481
Tags: Математика;Теория чисел;
Cover......Page 1
Title Page......Page 2
Copyright......Page 5
Contents......Page 8
Preface......Page 10
About the Author......Page 14
1.1 Algebraic Number Fields......Page 16
1.2 The Gaussian Field......Page 33
1.3 Euclidean Quadratic Fields......Page 47
1.4 Applications of Unique Factorization......Page 62
2.1 The Arithmetic of Ideals in Quadratic Fields......Page 70
Multiplication Formulas for Ideals in Quadratic Fields......Page 74
2.2 Dedekind Domains......Page 82
Pollard’s Algorithm......Page 109
2.3 Application to Factoring......Page 103
3.1 Basics......Page 112
3.2 Composition and the Form Class Group......Page 120
3.3 Applications via Ambiguity......Page 133
3.4 Genus......Page 144
3.5 Representation......Page 163
3.6 Equivalence Modulo p......Page 170
4.1 Algebraic and Transcendental Numbers......Page 174
4.2 Transcendence......Page 186
4.3 Minkowski’s Convex Body Theorem......Page 197
5.1 The Euler–Maclaurin Summation Formula......Page 206
5.2 Average Orders......Page 223
5.3 The Riemann ζ
-function......Page 233
6.1 Solving Modulo p[sup(n)]......Page 244
6.2 Introduction to Valuations......Page 248
6.3 Non-Archimedean vs. Archimedean Valuations......Page 255
6.4 Representation of p-Adic Numbers......Page 258
7.1 Dirichlet Characters......Page 262
7.2 Dirichlet’s L-Function and Theorem......Page 267
7.3 Dirichlet Density......Page 278
8.1 Lucas–Lehmer Theory......Page 286
8.2 Generalized Ramanujan–Nagell Equations......Page 291
8.3 Bachet’s Equation......Page 297
8.4 The Fermat Equation......Page 301
8.5 Catalan and the ABC Conjecture......Page 309
9.1 The Basics......Page 316
9.2 Mazur, Siegel, and Reduction......Page 325
9.3 Applications: Factoring & Primality Testing......Page 332
9.4 Elliptic Curve Cryptography (ECC)......Page 341
10.1 The Modular Group......Page 346
10.2 Modular Forms and Functions......Page 351
10.3 Applications to Elliptic Curves......Page 362
10.4 Shimura–Taniyama–Weil & FLT......Page 368
Appendix: Sieve Methods......Page 384
Bibliography......Page 408
Solutions to Odd-Numbered Exercises......Page 416
Index: List of Symbols......Page 466
Index: Subject......Page 468
