Methods Of Solving Number Theory Problems

Author : Ellina Grigorieva
Publisher : Birkhäuser
Page : 391 pages
File Size : 55,9 Mb
Release : 2018-07-06
Category : Mathematics
ISBN : 9783319909158

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Methods of Solving Number Theory Problems by Ellina Grigorieva Pdf

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.

Author : Titu Andreescu,Dorin Andrica
Publisher : Springer Science & Business Media
Page : 383 pages
File Size : 43,7 Mb
Release : 2009-06-12
Category : Mathematics
ISBN : 9780817646455

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Number Theory by Titu Andreescu,Dorin Andrica Pdf

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.

Author : Wacław Sierpiński,Waclaw Sierpinski
Publisher : Elsevier Publishing Company
Page : 142 pages
File Size : 51,6 Mb
Release : 1970
Category : Mathematics
ISBN : UOM:49015001038042

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250 Problems in Elementary Number Theory by Wacław Sierpiński,Waclaw Sierpinski Pdf

Author : Hong-Bing Yu
Publisher : World Scientific
Page : 115 pages
File Size : 42,8 Mb
Release : 2010
Category : Mathematics
ISBN : 9789814271141

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Problems of Number Theory in Mathematical Competitions by Hong-Bing Yu Pdf

Number theory is an important research field of mathematics. In mathematical competitions, problems of elementary number theory occur frequently. These problems use little knowledge and have many variations. They are flexible and diverse. In this book, the author introduces some basic concepts and methods in elementary number theory via problems in mathematical competitions. Readers are encouraged to try to solve the problems by themselves before they read the given solutions of examples. Only in this way can they truly appreciate the tricks of problem-solving.

Author : Danyal Sadik
Publisher : Unknown
Page : 255 pages
File Size : 53,5 Mb
Release : 2016-08-01
Category : Electronic
ISBN : 1681175657

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Problems in Analytic Number Theory by Danyal Sadik Pdf

"One might have thought that number theory was simply the study of numbers, but that is too broad a definition, since numbers are almost ubiquitous in mathematics. Number theory is a vast and fascinating field of mathematics, sometimes called ""higher arithmetic,"" consisting of the study of the properties of whole numbers. Primes and prime factorization are especially important in number theory, as are a number of functions such as the divisor function, Riemann zeta function, and totient function. Analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. Analytic number theory, and its applications and interactions, are currently experiencing intensive progress, in sometimes unexpected directions. In recent years, many important classical questions have seen spectacular advances based on new techniques; conversely, methods developed in analytic number theory have led to the solution of striking problems in other fields. Recent advances in analytic number theory have had repercussions in various mathematical subjects, such as harmonic analysis, ergodic theory and dynamics, additive and multiplicative combinatorics and theoretical computer science. The biggest technical change after 1950 has been the development of sieve methods, particularly in multiplicative problems. These are combinatorial in nature, and quite varied. The extremal branch of combinatorial theory has in return been greatly influenced by the value placed in analytic number theory on quantitative upper and lower bounds. Another recent development is probabilistic number theory, which uses methods from probability theory to estimate the distribution of number theoretic functions, such as how many prime divisors a number has. Problems in Analytic Number Theory present a problem-solving approach to the difficult subject of analytic number theory. This book is focused at researchers, teachers, and graduate students interested in number theory and its links with other branches of science."

Author : Ellina Grigorieva
Publisher : Birkhäuser
Page : 327 pages
File Size : 44,5 Mb
Release : 2015-09-17
Category : Mathematics
ISBN : 9783319198873

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Methods of Solving Nonstandard Problems by Ellina Grigorieva Pdf

This book, written by an accomplished female mathematician, is the second to explore nonstandard mathematical problems – those that are not directly solved by standard mathematical methods but instead rely on insight and the synthesis of a variety of mathematical ideas. It promotes mental activity as well as greater mathematical skills, and is an ideal resource for successful preparation for the mathematics Olympiad. Numerous strategies and techniques are presented that can be used to solve intriguing and challenging problems of the type often found in competitions. The author uses a friendly, non-intimidating approach to emphasize connections between different fields of mathematics and often proposes several different ways to attack the same problem. Topics covered include functions and their properties, polynomials, trigonometric and transcendental equations and inequalities, optimization, differential equations, nonlinear systems, and word problems. Over 360 problems are included with hints, answers, and detailed solutions. Methods of Solving Nonstandard Problems will interest high school and college students, whether they are preparing for a math competition or looking to improve their mathematical skills, as well as anyone who enjoys an intellectual challenge and has a special love for mathematics. Teachers and college professors will be able to use it as an extra resource in the classroom to augment a conventional course of instruction in order to stimulate abstract thinking and inspire original thought.

Author : Richard Guy
Publisher : Springer Science & Business Media
Page : 455 pages
File Size : 55,9 Mb
Release : 2013-03-09
Category : Mathematics
ISBN : 9780387266770

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Unsolved Problems in Number Theory by Richard Guy Pdf

Mathematics is kept alive by the appearance of new, unsolved problems. This book provides a steady supply of easily understood, if not easily solved, problems that can be considered in varying depths by mathematicians at all levels of mathematical maturity. This new edition features lists of references to OEIS, Neal Sloane’s Online Encyclopedia of Integer Sequences, at the end of several of the sections.

Author : William Stein
Publisher : Springer Science & Business Media
Page : 173 pages
File Size : 51,7 Mb
Release : 2008-10-28
Category : Mathematics
ISBN : 9780387855257

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Elementary Number Theory: Primes, Congruences, and Secrets by William Stein Pdf

This is a book about prime numbers, congruences, secret messages, and elliptic curves that you can read cover to cover. It grew out of undergr- uate courses that the author taught at Harvard, UC San Diego, and the University of Washington. The systematic study of number theory was initiated around 300B. C. when Euclid proved that there are in?nitely many prime numbers, and also cleverly deduced the fundamental theorem of arithmetic, which asserts that every positive integer factors uniquely as a product of primes. Over a thousand years later (around 972A. D. ) Arab mathematicians formulated the congruent number problem that asks for a way to decide whether or not a given positive integer n is the area of a right triangle, all three of whose sides are rational numbers. Then another thousand years later (in 1976), Di?e and Hellman introduced the ?rst ever public-key cryptosystem, which enabled two people to communicate secretely over a public communications channel with no predetermined secret; this invention and the ones that followed it revolutionized the world of digital communication. In the 1980s and 1990s, elliptic curves revolutionized number theory, providing striking new insights into the congruent number problem, primality testing, publ- key cryptography, attacks on public-key systems, and playing a central role in Andrew Wiles’ resolution of Fermat’s Last Theorem.

Author : Titu Andreescu,Dorin Andrica,Zuming Feng
Publisher : Springer Science & Business Media
Page : 204 pages
File Size : 55,9 Mb
Release : 2007-04-05
Category : Mathematics
ISBN : 9780817645618

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104 Number Theory Problems by Titu Andreescu,Dorin Andrica,Zuming Feng Pdf

This challenging problem book by renowned US Olympiad coaches, mathematics teachers, and researchers develops a multitude of problem-solving skills needed to excel in mathematical contests and in mathematical research in number theory. Offering inspiration and intellectual delight, the problems throughout the book encourage students to express their ideas in writing to explain how they conceive problems, what conjectures they make, and what conclusions they reach. Applying specific techniques and strategies, readers will acquire a solid understanding of the fundamental concepts and ideas of number theory.

Author : Anonim
Publisher : Academic Press
Page : 449 pages
File Size : 45,9 Mb
Release : 1986-05-05
Category : Mathematics
ISBN : 9780080873329

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Number Theory by Anonim Pdf

This book is written for the student in mathematics. Its goal is to give a view of the theory of numbers, of the problems with which this theory deals, and of the methods that are used. We have avoided that style which gives a systematic development of the apparatus and have used instead a freer style, in which the problems and the methods of solution are closely interwoven. We start from concrete problems in number theory. General theories arise as tools for solving these problems. As a rule, these theories are developed sufficiently far so that the reader can see for himself their strength and beauty, and so that he learns to apply them. Most of the questions that are examined in this book are connected with the theory of diophantine equations - that is, with the theory of the solutions in integers of equations in several variables. However, we also consider questions of other types; for example, we derive the theorem of Dirichlet on prime numbers in arithmetic progressions and investigate the growth of the number of solutions of congruences.

Author : M. Ram Murty,Jody (Indigo) Esmonde
Publisher : Springer Science & Business Media
Page : 354 pages
File Size : 49,5 Mb
Release : 2005-09-28
Category : Mathematics
ISBN : 9780387269986

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Problems in Algebraic Number Theory by M. Ram Murty,Jody (Indigo) Esmonde Pdf

The problems are systematically arranged to reveal the evolution of concepts and ideas of the subject Includes various levels of problems - some are easy and straightforward, while others are more challenging All problems are elegantly solved

Author : Michael Th. Rassias
Publisher : Springer Science & Business Media
Page : 336 pages
File Size : 50,9 Mb
Release : 2010-12-02
Category : Mathematics
ISBN : 9781441904942

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Problem-Solving and Selected Topics in Number Theory by Michael Th. Rassias Pdf

The book provides a self-contained introduction to classical Number Theory. All the proofs of the individual theorems and the solutions of the exercises are being presented step by step. Some historical remarks are also presented. The book will be directed to advanced undergraduate, beginning graduate students as well as to students who prepare for mathematical competitions (ex. Mathematical Olympiads and Putnam Mathematical competition).

Author : U.S.R. Murty
Publisher : Springer Science & Business Media
Page : 458 pages
File Size : 44,6 Mb
Release : 2013-06-29
Category : Mathematics
ISBN : 9781475734416

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Problems in Analytic Number Theory by U.S.R. Murty Pdf

"In order to become proficient in mathematics, or in any subject," writes Andre Weil, "the student must realize that most topics in volve only a small number of basic ideas. " After learning these basic concepts and theorems, the student should "drill in routine exercises, by which the necessary reflexes in handling such concepts may be ac quired. . . . There can be no real understanding of the basic concepts of a mathematical theory without an ability to use them intelligently and apply them to specific problems. " Weil's insightfulobservation becomes especially important at the graduate and research level. It is the viewpoint of this book. Our goal is to acquaint the student with the methods of analytic number theory as rapidly as possible through examples and exercises. Any landmark theorem opens up a method of attacking other problems. Unless the student is able to sift out from the mass of theory the underlying techniques, his or her understanding will only be academic and not that of a participant in research. The prime number theorem has given rise to the rich Tauberian theory and a general method of Dirichlet series with which one can study the asymptotics of sequences. It has also motivated the development of sieve methods. We focus on this theme in the book. We also touch upon the emerging Selberg theory (in Chapter 8) and p-adic analytic number theory (in Chapter 10).

Author : Adrian Andreescu,Vinjai Vale
Publisher : Unknown
Page : 0 pages
File Size : 52,9 Mb
Release : 2016
Category : Algebra
ISBN : 099687450X

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111 Problems in Algebra and Number Theory by Adrian Andreescu,Vinjai Vale Pdf

Algebra plays a fundamental role not only in mathematics, but also in various other scientific fields. Without algebra there would be no uniform language to express concepts such as numbers' properties. Thus one must be well-versed in this domain in order to improve in other mathematical disciplines. We cover algebra as its own branch of mathematics and discuss important techniques that are also applicable in many Olympiad problems. Number theory too relies heavily on algebraic machinery. Often times, the solutions to number theory problems involve several steps. Such a solution typically consists of solving smaller problems originating from a hypothesis and ending with a concrete statement that is directly equivalent to or implies the desired condition. In this book, we introduce a solid foundation in elementary number theory, focusing mainly on the strategies which come up frequently in junior-level Olympiad problems.

Author : D.P. Parent
Publisher : Springer Science & Business Media
Page : 552 pages
File Size : 41,5 Mb
Release : 2013-03-09
Category : Mathematics
ISBN : 9781475751949

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Exercises in Number Theory by D.P. Parent Pdf

After an eclipse of some 50 years, Number Theory, that is to say the study of the properties of the integers, has regained in France a vitality worthy of its distinguished past. More 'and more researchers have been attracted by problems which, though it is possible to express in simple statements, whose solutions require all their ingenuity and talent. In so doing, their work enriches the whole of mathematics with new and fertile methods. To be in a position to tackle these problems, it is neces sary to be familiar with many specific aspects of number theory. These are very different from those encountered in analysis or geometry. The necessary know-how can only be acquired by study ing and solving numerous problems. Now it is very easy to form ulate problems whose solutions, while sometimes obvious, more often go beyond current methods. Moreover, there is no doubt that, even more than in other disciplines, in mathematics one must have exercises available whose solutions are accessible. This is the objective realised by this work. It is the collab orative work of several successful young number theorists. They have drawn these exercises from their own work, from the work of their associated research groups as well as from published work.