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 Fermatโs Last Theorem (FLT) and numerous consequences of the ABC conjecture, including ThueโSiegelโRoth theorem, Hallโs conjecture, the ErdรถsโMollin-โWalsh conjecture, and the GranvilleโLangevin Conjecture. In the appendix, the author reviews sieve methods, such as Eratothesenesโ, Selbergโs, Linnikโs, and Bombieriโs 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.