An Introduction to Diophantine Equations: A Problem-Based Approach [1 ed.]

Description:

This problem-solving book is an introduction to the study of Diophantine equations, a class of equations in which only integer solutions are allowed. The material is organized in two parts: Part I introduces the reader to elementary methods necessary in solving Diophantine equations, such as the decomposition method, inequalities, the parametric method, modular arithmetic, mathematical induction, Fermat's method of infinite descent, and the method of quadratic fields; Part II contains complete solutions to all exercises in Part I. The presentation features some classical Diophantine equations, including linear, Pythagorean, and some higher degree equations, as well as exponential Diophantine equations. Many of the selected exercises and problems are original or are presented with original solutions. An Introduction to Diophantine Equations: A Problem-Based Approach is intended for undergraduates, advanced high school students and teachers, mathematical contest participants — including Olympiad and Putnam competitors — as well as readers interested in essential mathematics. The work uniquely presents unconventional and non-routine examples, ideas, and techniques.

Title : An Introduction to Diophantine Equations - A Problem-Based Approach [1 ed.]

author(s) : Titu Andreescu, Dorin Andrica, Ion Cucurezeanu

Publisher: Birkhäuser Basel

Year: 2010

size : 2 Mb

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Mathematics, number theory,

The Mathematics of Infinity: A Guide to Great Ideas [1 ed.]

Description:

A balanced and clearly explained treatment of infinity in mathematics.The concept of infinity has fascinated and confused mankind for centuries with concepts and ideas that cause even seasoned mathematicians to wonder. For instance, the idea that a set is infinite if it is not a finite set is an elementary concept that jolts our common sense and imagination. the Mathematics of Infinity: A guide to Great Ideas uniquely explores how we can manipulate these ideas when our common sense rebels at the conclusions we are drawing.Writing with clear knowledge and affection for the subject, the author introduces and explores infinite sets, infinite cardinals, and ordinals, thus challenging the readers' intuitive beliefs about infinity. Requiring little mathematical training and a healthy curiosity, the book presents a user-friendly approach to ideas involving the infinite. readers will discover the main ideas of infinite cardinals and ordinal numbers without experiencing in-depth mathematical rigor. Classic arguments and illustrative examples are provided throughout the book and are accompanied by a gradual progression of sophisticated notions designed to stun your intuitive view of the world.With a thoughtful and balanced treatment of both concepts and theory, The Mathematics of Infinity focuses on the following topics:* Sets and Functions* Images and Preimages of Functions* Hilbert's Infinite Hotel* Cardinals and Ordinals* The Arithmetic of Cardinals and Ordinals* the Continuum Hypothesis* Elementary Number Theory* The Riemann Hypothesis* The Logic of ParadoxesRecommended as recreational reading for the mathematically inquisitive or as supplemental reading for curious college students, the Mathematics of Infinity: A Guide to Great Ideas gently leads readers into the world of counterintuitive mathematics.

Title : The Mathematics of Infinity: A Guide to Great Ideas [1 ed.]

author(s) : Theodore G. Faticoni

Publisher: Wiley-Interscience

Year: 2006

size : 7 Mb

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Mathematics, number theory,

Partitions, q-series, and modular forms [1 ed.]

Description:

Partitions, q-Series, and Modular Forms contains a collection of research and survey papers that grew out of a Conference on Partitions, q-Series and Modular Forms at the University of Florida, Gainesville in March 2008. It will be of interest to researchers and graduate students that would like to learn of recent developments in the theory of q-series and modular and how it relates to number theory, combinatorics and special functions.

Title : Partitions, q-series, and modular forms [1 ed.]

author(s) : George E. Andrews, Peter Paule (auth.), Krishnaswami Alladi, Frank Garvan

Publisher: Springer-Verlag New York

Year: 2012

size : 2 Mb

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Mathematics, number theory,

Arithmetics [1 ed.]

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Number theory is a branch of mathematics which draws its vitality from a rich historical background. It is also traditionally nourished through interactions with other areas of research, such as algebra, algebraic geometry, topology, complex analysis and harmonic analysis. More recently, it has made a spectacular appearance in the field of theoretical computer science and in questions of communication, cryptography and error-correcting codes. Providing an elementary introduction to the central topics in number theory, this book spans multiple areas of research. The first part corresponds to an advanced undergraduate course. All of the statements given in this part are of course accompanied by their proofs, with perhaps the exception of some results appearing at the end of the chapters. A copious list of exercises, of varying difficulty, are also included here. The second part is of a higher level and is relevant for the first year of graduate school. It contains an introduction to elliptic curves and a chapter entitled “Developments and Open Problems”, which introduces and brings together various themes oriented toward ongoing mathematical research. Given the multifaceted nature of number theory, the primary aims of this book are to:
- provide an overview of the various forms of mathematics useful for studying numbers
- demonstrate the necessity of deep and classical themes such as Gauss sums
- highlight the role that arithmetic plays in modern applied mathematics
- include recent proofs such as the polynomial primality algorithm
- approach subjects of contemporary research such as elliptic curves
- illustrate the beauty of arithmetic The prerequisites for this text are undergraduate level algebra and a little topology of Rn. It will be of use to undergraduates, graduates and phd students, and may also appeal to professional mathematicians as a reference text.

Title : Arithmetics [1 ed.]

author(s) : Marc Hindry

Publisher: Springer-Verlag London

Year: 2011

size : 11 Mb

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Mathematics, number theory,

Fundamental Number Theory with Applications [2 ed.]

Description:

An update of the most accessible introductory number theory text available, Fundamental Number Theory with Applications, Second Edition presents a mathematically rigorous yet easy-to-follow treatment of the fundamentals and applications of the subject. The substantial amount of reorganizing makes this edition clearer and more elementary in its coverage. New to the Second Edition
• Removal of all advanced material to be even more accessible in scope
• New fundamental material, including partition theory, generating functions, and combinatorial number theory
• Expanded coverage of random number generation, Diophantine analysis, and additive number theory
• More applications to cryptography, primality testing, and factoring
• An appendix on the recently discovered unconditional deterministic polynomial-time algorithm for primality testing Taking a truly elementary approach to number theory, this text supplies the essential material for a first course on the subject. Placed in highlighted boxes to reduce distraction from the main text, nearly 70 biographies focus on major contributors to the field. The presentation of over 1,300 entries in the index maximizes cross-referencing so students can find data with ease.

Title : Fundamental Number Theory with Applications [2 ed.]

author(s) : Richard A. Mollin

Publisher: CRC Press

Year: 2008

size : 6 Mb

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Mathematics, number theory,

An Invitation to Q-Series

Description:

The aim of these lecture notes is to provide a self-contained exposition of several fascinating formulas discovered by Srinivasa Ramanujan. Two central results in these notes are: (1) the evaluation of the Rogers-Ramanujan continued fraction -- a result that convinced G H Hardy that Ramanujan was a "mathematician of the highest class", and (2) what G. H. Hardy called Ramanujan's "Most Beautiful Identity". This book covers a range of related results, such as several proofs of the famous Rogers-Ramanujan identities and a detailed account of Ramanujan's congruences. It also covers a range of techniques in q-series.

Title : An Invitation to Q-Series - From Jacobi's Triple Product Identity to Ramanujan's Most Beautiful Identity

author(s) : Hei-chi Chan

Publisher: World Scientific

Year: 2011

size : 1 Mb

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Mathematics, number theory,

Advanced Number Theory with Applications

Description:

Exploring one of the most dynamic areas of mathematics, Advanced Number Theory with Applications covers a wide range of algebraic, analytic, combinatorial, cryptographic, and geometric aspects of number theory. Written by a recognized leader in algebra and number theory, the book includes a page reference for every citing in the bibliography and more than 1,500 entries in the index so that students can easily cross-reference and find the appropriate data. With numerous examples throughout, the text begins with coverage of algebraic number theory, binary quadratic forms, Diophantine approximation, arithmetic functions, p-adic analysis, Dirichlet characters, density, and primes in arithmetic progression. It then applies these tools to Diophantine equations, before developing elliptic curves and modular forms. The text also presents an overview of Fermat’s Last Theorem (FLT) and numerous consequences of the ABC conjecture, including Thue–Siegel–Roth theorem, Hall’s conjecture, the Erd?s–Mollin-–Walsh conjecture, and the Granville–Langevin Conjecture. In the appendix, the author reviews sieve methods, such as Eratothesenes’, Selberg’s, Linnik’s, and Bombieri’s sieves. He also discusses recent results on gaps between primes and the use of sieves in factoring. By focusing on salient techniques in number theory, this textbook provides the most up-to-date and comprehensive material for a second course in this field. It prepares students for future study at the graduate level.

Title : Advanced Number Theory with Applications

author(s) : Richard A. Mollin

Publisher: CRC Press

Year: 2010

size : 4 Mb

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Mathematics, number theory,

Analytic Number Theory : An Introductory Course

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This valuable book focuses on a collection of powerful methods of analysis that yield deep number-theoretical estimates. Particular attention is given to counting functions of prime numbers and multiplicative arithmetic functions. Both real variable ("elementary") and complex variable ("analytic") methods are employed. The reader is assumed to have knowledge of elementary number theory (abstract algebra will also do) and real and complex analysis. Specialized analytic techniques, including transform and Tauberian methods, are developed as needed.

Title : Analytic Number Theory - An Introductory Course

author(s) : Paul T. Bateman, Harold G. Diamond

Publisher: World Scientific Publishing Company

Year: 2004

size : 3 Mb

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Mathematics, number theory,

104 number theory problems : from the training of the USA IMO team

Description:

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 research in number theory. Offering inspiration and intellectual delight, the problems throughout the book encourage students to express their ideas, conjectures, and conclusions in writing. Applying specific techniques and strategies, readers will acquire a solid understanding of the fundamental concepts and ideas of number theory.

Title : 104 number theory problems - from the training of the USA IMO team

author(s) : Titu Andreescu, Dorin Andrica, Zuming Feng

Publisher: Birkhäuser

Year: 2007

size : 1 Mb

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Mathematics, number theory,

Complex numbers from A to--Z

Description:

It is impossible to imagine modern mathematics without complex numbers. "Complex Numbers from A to ...Z" introduces the reader to this fascinating subject which, from the time of L. Euler, has become one of the most utilized ideas in mathematics.
The exposition concentrates on key concepts and then elementary results concerning these numbers. The reader learns how complex numbers can be used to solve algebraic equations and to understand the geometric interpretation of complex numbers and the operations involving them.
The theoretical parts of the book are augmented with rich exercises and problems at various levels of difficulty. A special feature of the book is the last chapter, a selection of real outstanding Olympiad and other important mathematical contest problems solved by employing the methods already presented.
The book reflects the unique experience of the authors. It distills a vast mathematical literature, most of which is unknown to the western public, and captures the essence of an abundant problem culture.

Title : Complex numbers from A to--Z

author(s) : Titu Andreescu, Dorin Andrica

Publisher: Birkhäuser

Year: 2006

size : 2 Mb

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Mathematics, number theory,