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E-grāmata: Advanced Number Theory with Applications

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(University of Calgary, Alberta, Canada)
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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 Fermats Last Theorem (FLT) and numerous consequences of the ABC conjecture, including ThueSiegelRoth theorem, Halls conjecture, the ErdösMollin-Walsh conjecture, and the GranvilleLangevin Conjecture. In the appendix, the author reviews sieve methods, such as Eratothesenes, Selbergs, Linniks, and Bombieris 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.

Recenzijas

The reader following this book will obtain a thorough overview of some very deep mathematics which is still in active research today. I readily recommend this book to advanced undergraduates and beginning graduate students interested in advanced number theory. This book can also be read by the enthusiast who is well-acquainted with the author's previous book Fundamental Number Theory with Applications. IACR Book Reviews, May 2011

each section comes with a large number of illustrating examples and accompanying exercises. The rich bibliography contains 106 references, where maximum information is imparted by explicit page reference for each citing of a given item within the text. The carefully compiled index has more than 1,500 entries presented for maximum cross-referencing. Overall, this excellent textbook bespeaks the authors outstanding expository mastery just as much as his mathematical erudition and elevated taste. Presenting a wide panorama of topics in advanced classical and contemporary number theory, and that in an utmost lucid and comprehensible style of writing, the author takes the reader to the forefront of research in the field, and on a truly exciting journey over and above. Werner Kleinert, Zentralblatt MATH, 2010

When I was looking over books for my course, I was very pleased by yours, and look forward to teaching from it. after much thought I found that I liked yours best for its completeness, its problems, and for the way you weave current results and conjectures into the text. Among other things that pleased me about your book, Im so glad continued fractions come where they do. a worthy book David Barth-Hart, Associate Head, School of Mathematical Sciences, Rochester Institute of Technology, New York, USA

This terrific book is testimony to Richard Mollins mathematical erudition, wonderful taste, and also his breadth of culture. Mollins treatment of elliptic curves is a model of clear exposition [ It] succeeds very well in its goal of providing a means of transition from more or less foundational material to papers and advanced monographs, i.e., research in the field. a wondrous book, successfully fulfilling the authors purpose of effecting a bridge to modern number theory for the somewhat initiated. its very nice to find in Mollins book a high quality and coherent treatment of this beautiful material and pointers in abundance to where to go next. Michael Berg, Loyola Marymount University, MAA Review, 2009

Preface ix
About the Author xiii
Algebraic Number Theory and Quadratic Fields
1(54)
Algebraic Number Fields
1(17)
The Gaussian Field
18(14)
Euclidean Quadratic Fields
32(15)
Applications of Unique Factorization
47(8)
Ideals
55(42)
The Arithmetic of Ideals in Quadratic Fields
55(12)
Dedekind Domains
67(21)
Application to Factoring
88(9)
Binary Quadratic Forms
97(62)
Basics
97(8)
Composition and the Form Class Group
105(13)
Applications via Ambiguity
118(11)
Genus
129(19)
Representation
148(7)
Equivalence Modulo p
155(4)
Diophantine Approximation
159(32)
Algebraic and Transcendental Numbers
159(12)
Transcendence
171(11)
Minkowski's Convex Body Theorem
182(9)
Arithmetic Functions
191(38)
The Euler-Maclaurin Summation Formula
191(17)
Average Orders
208(10)
The Riemann ζ-function
218(11)
Introduction to p-Adic Analysis
229(18)
Solving Modulo pn
229(4)
Introduction to Valuations
233(7)
Non-Archimedean vs. Archimedean Valuations
240(3)
Representation of p-Adic Numbers
243(4)
Dirichlet: Characters, Density, and Primes in Progression
247(24)
Dirichlet Characters
247(5)
Dirichlet's L-Function and Theorem
252(11)
Dirichlet Density
263(8)
Applications to Diophantine Equations
271(30)
Lucas-Lehmer Theory
271(5)
Generalized Ramanujan-Nagell Equations
276(6)
Bachet's Equation
282(4)
The Fermat Equation
286(8)
Catalan and the ABC Conjecture
294(7)
Elliptic Curves
301(30)
The Basics
301(9)
Mazur, Siegel, and Reduction
310(7)
Applications: Factoring & Primality Testing
317(9)
Elliptic Curve Cryptography (ECC)
326(5)
Modular Forms
331(38)
The Modular Group
331(5)
Modular Forms and Functions
336(11)
Applications to Elliptic Curves
347(6)
Shimura---Taniyama---Weil & FLT
353(16)
Appendix: Sieve Methods 369(24)
Bibliography 393(8)
Solutions to Odd-Numbered Exercises 401(50)
Index: List of Symbols 451(2)
Index: Subject 453
Richard A. Mollin is a professor in the Department of Mathematics and Statistics at the University of Calgary. In the past twenty-three years, Dr. Mollin has founded the Canadian Number Theory Association and has been awarded six Killam Resident Fellowships. Over the past thirty-three years, he has written more than 190 publications.
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