Mollin Advanced Number Theory with Applications

1. Auflage 2011
ISBN: 978-1-4200-8329-3
Verlag: Taylor & Francis
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 440 Seiten

Reihe: Discrete Mathematics and Its Applications

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.

Zielgruppe

Graduate and senior undergraduate students in mathematics and computer science; mathematicians, cryptographers, and computer scientists.

Autoren/Hrsg.

Mollin, Richard A.

Fachgebiete

• Mathematik | Informatik Mathematik Mathematik Allgemein Diskrete Mathematik, Kombinatorik
• Mathematik | Informatik EDV | Informatik Betriebssysteme
• Mathematik | Informatik Mathematik Algebra Zahlentheorie

Inhaltsverzeichnis

Algebraic Number Theory and Quadratic Fields
Algebraic Number Fields
The Gaussian Field
Euclidean Quadratic Fields
Applications of Unique Factorization
Ideals
The Arithmetic of Ideals in Quadratic Fields
Dedekind Domains
Application to Factoring
Binary Quadratic Forms
Basics
Composition and the Form Class Group
Applications via Ambiguity
Genus
Representation
Equivalence Modulo p
Diophantine Approximation
Algebraic and Transcendental Numbers
Transcendence
Minkowski’s Convex Body Theorem
Arithmetic Functions
The Euler–Maclaurin Summation Formula
Average Orders
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 Characters
Dirichlet’s L-Function and Theorem
Dirichlet Density
Applications to Diophantine Equations
Lucas–Lehmer Theory
Generalized Ramanujan–Nagell Equations
Bachet’s Equation
The Fermat Equation
Catalan and the ABC-Conjecture
Elliptic Curves
The Basics
Mazur, Siegel, and Reduction
Applications: Factoring and Primality Testing
Elliptic Curve Cryptography (ECC)
Modular Forms
The Modular Group
Modular Forms and Functions
Applications to Elliptic Curves
Shimura–Taniyama–Weil and FLT
Appendix: Sieve Methods
Bibliography
Solutions to Odd-Numbered Exercises
Index: List of Symbols
Index: Alphabetical Listing

Über Autor(innen)

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.