Overview
- 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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Table of contents (6 chapters)
Keywords
About this book
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 therepresentations 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.
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Bibliographic Information
Book Title: Methods of Solving Number Theory Problems
Authors: Ellina Grigorieva
DOI: https://doi.org/10.1007/978-3-319-90915-8
Publisher: Birkhäuser Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer International Publishing AG, part of Springer Nature 2018
Hardcover ISBN: 978-3-319-90914-1Published: 18 July 2018
Softcover ISBN: 978-3-030-08130-0Published: 05 January 2019
eBook ISBN: 978-3-319-90915-8Published: 06 July 2018
Edition Number: 1
Number of Pages: XXI, 391
Number of Illustrations: 4 b/w illustrations, 12 illustrations in colour
Topics: Number Theory, Mathematics Education, Mathematical Logic and Foundations