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 Erdös–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.
Table of Contents
Algebraic Number Theory and Quadratic Fields
Algebraic Number Fields
The Gaussian Field
Euclidean Quadratic Fields
Applications of Unique Factorization
The Arithmetic of Ideals in Quadratic Fields
Application to Factoring
Binary Quadratic Forms
Composition and the Form Class Group
Applications via Ambiguity
Equivalence Modulo p
Algebraic and Transcendental Numbers
Minkowski’s Convex Body Theorem
The Euler–Maclaurin Summation Formula
The Riemann zeta-function
Introduction to p-Adic Analysis
Solving Modulo pn
Introduction to Valuations
Non-Archimedean vs. Archimedean Valuations
Representation of p-Adic Numbers
Dirichlet: Characters, Density, and Primes in Progression
Dirichlet’s L-Function and Theorem
Applications to Diophantine Equations
Generalized Ramanujan–Nagell Equations
The Fermat Equation
Catalan and the ABC-Conjecture
Mazur, Siegel, and Reduction
Applications: Factoring and Primality Testing
Elliptic Curve Cryptography (ECC)
The Modular Group
Modular Forms and Functions
Applications to Elliptic Curves
Shimura–Taniyama–Weil and FLT
Appendix: Sieve Methods
Solutions to Odd-Numbered Exercises
Index: List of Symbols
Index: Alphabetical Listing
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.
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 author’s 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, I’m 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 Mollin’s mathematical erudition, wonderful taste, and also his breadth of culture. … Mollin’s 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 author’s purpose of effecting a bridge to modern number theory for the somewhat initiated. … it’s very nice to find in Mollin’s 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