Number Theory
Structures, Examples, and Problems
(Sprache: Englisch)
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...
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Klappentext zu „Number Theory “
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.
One of the oldest, liveliest branches of mathematics, number theory is noted for its theoretical depth and applications to other fields, including representation theory, physics, and cryptography. The forefront of number theory is replete with sophisticated and famous open problems; at its foundation, however, are basic, elementary ideas that can stimulate and challenge beginning students.
This introductory textbook takes a problem-solving approach to number theory, situating each concept within the framework of an example or a problem for readers to solve. 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, special sequences, and problems of density. Included are sections on mathematical induction and the pigeonhole principle, as well as a discussion of other number systems.
By emphasizing examples and applications, and by introducing and reinforcing every idea with an exercise, the authors motivate and engage readers. The exposition proceeds incrementally from first principles, starting with the natural numbers and then intuitively and rigorously uncovering deeper properties. A comprehensive index and selected solutions complete the work.
Written by distinguished research mathematicians and renowned teachers, "Number Theory: A Problem-Solving Approach" will appeal to senior high school and undergraduate students and instructors. It is a clear, accessible introduction to the subject and a source of fascinating problems and puzzles for readers at all levels.
This introductory textbook takes a problem-solving approach to number theory, situating each concept within the framework of an example or a problem for readers to solve. 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, special sequences, and problems of density. Included are sections on mathematical induction and the pigeonhole principle, as well as a discussion of other number systems.
By emphasizing examples and applications, and by introducing and reinforcing every idea with an exercise, the authors motivate and engage readers. The exposition proceeds incrementally from first principles, starting with the natural numbers and then intuitively and rigorously uncovering deeper properties. A comprehensive index and selected solutions complete the work.
Written by distinguished research mathematicians and renowned teachers, "Number Theory: A Problem-Solving Approach" will appeal to senior high school and undergraduate students and instructors. It is a clear, accessible introduction to the subject and a source of fascinating problems and puzzles for readers at all levels.
Inhaltsverzeichnis zu „Number Theory “
- The Natural Numbers: An Introduction - Prime Numbers, Divisibility and the Euclidean Algorithm
- Parity and Modular Systems
- Linear Diophantine Equations
- Some Classic Results: Fermat, Euler, and Wilson
- Quadratic Residues
- Quadratic Diophantine Equations
- Some More Advanced Topics
Autoren-Porträt von Titu Andreescu, Dorin Andrica
Titu Andreescu received his Ph.D. from the West University of Timisoara, Romania. The topic of his dissertation was "Research on Diophantine Analysis and Applications." Professor Andreescu currently teaches at The University of Texas at Dallas. He is past chairman of the USA Mathematical Olympiad, served as director of the MAA American Mathematics Competitions (1998-2003), coach of the USA International Mathematical Olympiad Team (IMO) for 10 years (1993-2002), director of the Mathematical Olympiad Summer Program (1995-2002), and leader of the USA IMO Team (1995-2002). In 2002 Titu was elected member of the IMO Advisory Board, the governing body of the world's most prestigious mathematics competition. Titu co-founded in 2006 and continues as director of the AwesomeMath Summer Program (AMSP). He received the Edyth May Sliffe Award for Distinguished High School Mathematics Teaching from the MAA in 1994 and a "Certificate of Appreciation" from the president of the MAA in 1995 for his outstanding service as coach of the Mathematical Olympiad Summer Program in preparing the US team for its perfect performance in Hong Kong at the 1994 IMO. Titu's contributions to numerous textbooks and problem books are recognized worldwide.Dorin Andrica received his Ph.D in 1992 from "Babes¿-Bolyai" University in Cluj-Napoca, Romania; his thesis treated critical points and applications to the geometry of differentiable submanifolds. Professor Andrica has been chairman of the Department of Geometry at "Babes¿-Bolyai" since 1995. He has written and contributed to numerous mathematics textbooks, problem books, articles and scientific papers at various levels. He is an invited lecturer at university conferences around the world: Austria, Bulgaria, Czech Republic, Egypt, France, Germany, Greece, Italy, the Netherlands, Portugal, Serbia, Turkey, and the USA. Dorin is a member of the Romanian Committee for the Mathematics Olympiad and is a member on the editorial boards of several
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international journals. Also, he is well known for his conjecture about consecutive primes called "Andrica's Conjecture." He has been a regular faculty member at the Canada-USA Mathcamps between 2001-2005 and at the AwesomeMath Summer Program (AMSP) since 2006.
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Bibliographische Angaben
- Autoren: Titu Andreescu , Dorin Andrica
- 2009, 2009, 384 Seiten, 150 Abbildungen, Maße: 16 x 24,1 cm, Gebunden, Englisch
- Verlag: Springer
- ISBN-10: 081763245X
- ISBN-13: 9780817632458
- Erscheinungsdatum: 11.03.2009
Sprache:
Englisch
Rezension zu „Number Theory “
From the reviews:"The book is a collection of number theory problems chosen from various national and international Mathematical Olympiads. ... Each chapter of this book starts a brief ... review of concepts, with various solved examples, mainly selected from Olympiad exams. ... The book could be used as a text for undergraduates ... . The main audience will consist of Olympiad-level students ... . I recommend this friendly volume for students looking for challenging problems in number theory and teachers of number theory for undergraduates ... ." (Mehdi Hassani, The Mathematical Association of America, June, 2009)
"This is a collection of elementary number theory problems taken mainly from mathematical olympiads and other contests held in different countries, mainly in recent years. ... This makes the book a useful source of material for tests, homeworks, projects, and classroom discussion. ... The book is a welcome addition to the library of Problem Books, and will undoubtedly prove valuable to coaches and teams involved in mathematical competitions as well as to university instructors and students." (Mowaffaq Hajja, Zentralblatt MATH, Vol. 1182, 2010)
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