Methods of Solving Number Theory Problems

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Beschreibung

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 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.

Über den Autor

Ellina Grigorieva, PhD, is Professor of Mathematics at Texas Women's University, Denton, TX, USA.

Zusammenfassung

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

Inhaltsverzeichnis

Preface.- Numbers: Problems Involving Integers.- Further Study of Integers.- Diophantine Equations and More.- Pythagorean Triples, Additive Problems, and More.- Homework.

Details

Erscheinungsjahr:	2018
Fachbereich:	Wahrscheinlichkeitstheorie
Genre:	Mathematik
Rubrik:	Naturwissenschaften & Technik
Medium:	Buch
Seiten:	416
Inhalt:	xxi 391 S. 4 s/w Illustr. 12 farbige Illustr. 391 p. 16 illus. 12 illus. in color.
ISBN-13:	9783319909141
ISBN-10:	3319909142
Sprache:	Englisch
Herstellernummer:	978-3-319-90914-1
Ausstattung / Beilage:	HC runder Rücken kaschiert
Einband:	Gebunden
Autor:	Grigorieva, Ellina
Auflage:	1st ed. 2018
Hersteller:	Springer International Publishing
Maße:	241 x 160 x 28 mm
Von/Mit:	Ellina Grigorieva
Erscheinungsdatum:	18.07.2018
Gewicht:	0,787 kg

preigu-id: 113166332