Advanced Number Theory with Applications

ISBN-13: 9781420083286 / Angielski / Twarda / 2009 / 440 str.

Richard A. Mollin

Opis

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 Erdos-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.

Spis treści

Algebraic Number Theory and Quadratic Fields. Ideals. Binary Quadratic Forms. Diophantine Approximation. Arithmetic Functions. Introduction to p-Adic Analysis. Dirichlet: Characters, Density, and Primes in Progression. Applications to Diophantine Equations. Elliptic Curves. Modular Forms. Appendix. Bibliography. Solutions to Odd-Numbered Exercises. Indices.

O autorze

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.