Advanced Number Theory with Applications / Edition 1

Advanced Number Theory with Applications / Edition 1

by Richard A. Mollin
     
 

ISBN-10: 1420083287

ISBN-13: 9781420083286

Pub. Date: 08/18/2009

Publisher: Taylor & Francis

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

Overview

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.

Product Details

ISBN-13:
9781420083286
Publisher:
Taylor & Francis
Publication date:
08/18/2009
Series:
Discrete Mathematics and Its Applications Series
Pages:
480
Product dimensions:
6.40(w) x 9.30(h) x 1.20(d)

Related Subjects

Table of Contents

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

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