- Teaches number theory through problem solving, making it perfect for self-study and Olympiad preparation
- Contains over 260 challenging problems and 110 homework exercises in number theory with hints and detailed solutions
- Encourages the creative applications of methods, rather than memorization
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- About this Textbook
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Through its engaging and unusual problems, this book demonstrates methods of reasoning necessary for learning number theory. Every technique is followed by problems (as well as detailed hints and solutions) that apply theorems immediately, so readers can solve a variety of abstract problems in a systematic, creative manner. New solutions often require the ingenious use of earlier mathematical concepts - not the memorization of formulas and facts. Questions also often permit experimental numeric validation or visual interpretation to encourage the combined use of deductive and intuitive thinking.
The first chapter starts with simple topics like even and odd numbers, divisibility, and prime numbers and helps the reader to solve quite complex, Olympiad-type problems right away. It also covers properties of the perfect, amicable, and figurate numbers and introduces congruence. The next chapter begins with the Euclidean algorithm, explores the representations of integer numbers in different bases, and examines continued fractions, quadratic irrationalities, and the Lagrange Theorem. The last section of Chapter Two is an exploration of different methods of proofs. The third chapter is dedicated to solving Diophantine linear and nonlinear equations and includes different methods of solving Fermat’s (Pell’s) equations. It also covers Fermat’s factorization techniques and methods of solving challenging problems involving exponent and factorials. Chapter Four reviews the Pythagorean triple and quadruple and emphasizes their connection with geometry, trigonometry, algebraic geometry, and stereographic projection. A special case of Waring’s problem as a representation of a number by the sum of the squares or cubes of other numbers is covered, as well as quadratic residuals, Legendre and Jacobi symbols, and interesting word problems related to the properties of numbers. Appendices provide a historic overview of number theory and its main developments from the ancient cultures in Greece, Babylon, and Egypt to the modern day.
Drawing from cases collected by an accomplished female mathematician, Methods in Solving Number Theory Problems is designed as a self-study guide or supplementary textbook for a one-semester course in introductory number theory. It can also be used to prepare for mathematical Olympiads. Elementary algebra, arithmetic and some calculus knowledge are the only prerequisites. Number theory gives precise proofs and theorems of an irreproachable rigor and sharpens analytical thinking, which makes this book perfect for anyone looking to build their mathematical confidence. - About the authors
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Ellina Grigorieva, PhD, is Professor of Mathematics at Texas Women's University, Denton, TX, USA.
- Table of contents (6 chapters)
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Numbers: Problems Involving Integers
Pages 1-61
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Further Study of Integers
Pages 63-139
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Diophantine Equations and More
Pages 141-244
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Pythagorean Triples, Additive Problems, and More
Pages 245-334
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Homework
Pages 335-375
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Table of contents (6 chapters)
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Bibliographic Information
- Bibliographic Information
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- Book Title
- Methods of Solving Number Theory Problems
- Authors
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- Ellina Grigorieva
- Copyright
- 2018
- Publisher
- Birkhäuser Basel
- Copyright Holder
- Springer International Publishing AG, part of Springer Nature
- eBook ISBN
- 978-3-319-90915-8
- DOI
- 10.1007/978-3-319-90915-8
- Hardcover ISBN
- 978-3-319-90914-1
- Softcover ISBN
- 978-3-030-08130-0
- Edition Number
- 1
- Number of Pages
- XXI, 391
- Number of Illustrations
- 4 b/w illustrations, 12 illustrations in colour
- Topics
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