Number Theory : Structures, Examples, and Problems

Number theory, an ongoing rich area of mathematical exploration, is noted for its theoretical depth, with connections and applications to other fields from representation theory, to physics, cryptography, and more. While the forefront of number theory is replete with sophisticated and famous open problems, at its foundation are basic, elementary ideas that can stimulate and challenge beginning students. This lively introductory text focuses on a problem-solving approach to the subject. Key features of Number Theory: Structures, Examples, and Problems: \* A rigorous exposition starts with the natural numbers and the basics. \* Important concepts are presented with an example, which may also emphasize an application. The exposition moves systematically and intuitively to uncover deeper properties. \* Topics include divisibility, unique factorization, modular arithmetic and the Chinese Remainder Theorem, Diophantine equations, quadratic residues, binomial coefficients, Fermat and Mersenne primes and other special numbers, and special sequences. Sections on mathematical induction and the pigeonhole principle, as well as a discussion of other number systems are covered. \* Unique exercises reinforce and motivate the reader, with selected solutions to some of the problems. \* Glossary, bibliography, and comprehensive index round out the text. Written by distinguished research mathematicians and renowned teachers, this text is a clear, accessible introduction to the subject and a source of fascinating problems and puzzles, from advanced high school students to undergraduates, their instructors, and general readers at all levels. While The Forefront Of Number Theory Is Replete With Sophisticated And Famous Open Problems, At Its Foundation Are Basic, Elementary Ideas That Can Stimulate And Challenge Beginning Students. This Introductory Text Focuses On A Problem-solving Approach To The Subject, Situating Each Concept Within The Framework Of An Example For Readers To Solve. Fundamentals. Divisibility -- Powers Of Integers -- Floor Function And Fractional Part -- Digits Of Numbers -- Basic Principles In Number Theory -- Arithmetic Functions -- More On Divisibility -- Diophantine Equations -- Some Special Problems In Number Theory -- Problems Involving Binomial Coefficients -- Miscellaneous Problems -- Solutions To Additional Problems. Divisibility -- Powers Of Integers -- Floor Function And Fractional Part -- Digits Of Numbers -- Basic Principles In Number Theory -- Arithmetic Functions -- More On Divisibility -- Diophantine Equations -- Some Special Problems In Number Theory -- Problems Involving Binomial Coefficients -- Miscellaneous Problems. Titu Andreescu, Dorin Andrica. Includes Bibliographical References (pages 377-380) And Indexes. This introductory textbook takes a problem-solving approach to number theory, situating each concept within the framework of an example or a problem for solving. Starting with the essentials, the text covers divisibility, unique factorization, modular arithmetic and the Chinese Remainder Theorem, Diophantine equations, binomial coefficients, Fermat and Mersenne primes and other special numbers, and special sequences. Included are sections on mathematical induction and the pigeonhole principle, as well as a discussion of other number systems. By emphasizing examples and applications the authors motivate and engage readers.