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 Erdoes-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.
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