Advanced Number Theory with Applications

Richard A. Mollin

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August 26, 2009 by Chapman and Hall/CRC
Textbook - 440 Pages - 6 B/W Illustrations
ISBN 9781420083286 - CAT# C8328
Series: Discrete Mathematics and Its Applications

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Features

  • Applies tools, such as algebraic number theory, to Diophantine equations
  • Presents the application of elliptic curve cryptography
  • Discusses modular forms and functions, including applications to elliptic curves used to prove FLT—topics not found in similar books
  • Describes sieve methods, including Bombieri’s asymptotic sieve and the number field sieve
  • Offers an accessible overview of the proof of FLT
  • Contains nearly 50 mini-bios of relevant mathematicians, more than 330 problems, and solutions to odd-numbered exercises

Solutions manual available for qualifying instructors

Summary

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.