The Pell Equation

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Solving the Pell Equation

Author : Michael Jacobson
Publisher : Springer Science & Business Media
ISBN 13 : 038784922X
Total Pages : 504 pages
Book Rating : 4.3/5 (878 download)

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Book Synopsis Solving the Pell Equation by : Michael Jacobson

Download or read book Solving the Pell Equation written by Michael Jacobson and published by Springer Science & Business Media. This book was released on 2008-12-02 with total page 504 pages. Available in PDF, EPUB and Kindle. Book excerpt: Pell’s Equation is a very simple Diophantine equation that has been known to mathematicians for over 2000 years. Even today research involving this equation continues to be very active, as can be seen by the publication of at least 150 articles related to this equation over the past decade. However, very few modern books have been published on Pell’s Equation, and this will be the first to give a historical development of the equation, as well as to develop the necessary tools for solving the equation. The authors provide a friendly introduction for advanced undergraduates to the delights of algebraic number theory via Pell’s Equation. The only prerequisites are a basic knowledge of elementary number theory and abstract algebra. There are also numerous references and notes for those who wish to follow up on various topics.

Pell’s Equation

Author : Edward J. Barbeau
Publisher : Springer Science & Business Media
ISBN 13 : 0387226028
Total Pages : 220 pages
Book Rating : 4.3/5 (872 download)

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Book Synopsis Pell’s Equation by : Edward J. Barbeau

Download or read book Pell’s Equation written by Edward J. Barbeau and published by Springer Science & Business Media. This book was released on 2006-05-04 with total page 220 pages. Available in PDF, EPUB and Kindle. Book excerpt: Pell's equation is part of a central area of algebraic number theory that treats quadratic forms and the structure of the rings of integers in algebraic number fields. It is an ideal topic to lead college students, as well as some talented and motivated high school students, to a better appreciation of the power of mathematical technique. Even at the specific level of quadratic diophantine equations, there are unsolved problems, and the higher degree analogues of Pell's equation, particularly beyond the third, do not appear to have been well studied. In this focused exercise book, the topic is motivated and developed through sections of exercises which will allow the readers to recreate known theory and provide a focus for their algebraic practice. There are several explorations that encourage the reader to embark on their own research. A high school background in mathematics is all that is needed to get into this book, and teachers and others interested in mathematics who do not have (or have forgotten) a background in advanced mathematics may find that it is a suitable vehicle for keeping up an independent interest in the subject.

The Pell Equation

Author : Edward Everett Whitford
Publisher :
ISBN 13 :
Total Pages : 199 pages
Book Rating : 4.L/5 ( download)

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Book Synopsis The Pell Equation by : Edward Everett Whitford

Download or read book The Pell Equation written by Edward Everett Whitford and published by . This book was released on 1912 with total page 199 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Pell and Pell–Lucas Numbers with Applications

Author : Thomas Koshy
Publisher : Springer
ISBN 13 : 1461484898
Total Pages : 444 pages
Book Rating : 4.4/5 (614 download)

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Book Synopsis Pell and Pell–Lucas Numbers with Applications by : Thomas Koshy

Download or read book Pell and Pell–Lucas Numbers with Applications written by Thomas Koshy and published by Springer. This book was released on 2014-11-11 with total page 444 pages. Available in PDF, EPUB and Kindle. Book excerpt: Pell and Pell–Lucas numbers, like the well-known Fibonacci and Catalan numbers, continue to intrigue the mathematical world with their beauty and applicability. They offer opportunities for experimentation, exploration, conjecture, and problem-solving techniques, connecting the fields of analysis, geometry, trigonometry, and various areas of discrete mathematics, number theory, graph theory, linear algebra, and combinatorics. Pell and Pell–Lucas numbers belong to an extended Fibonacci family as a powerful tool for extracting numerous interesting properties of a vast array of number sequences. A key feature of this work is the historical flavor that is interwoven into the extensive and in-depth coverage of the subject. An interesting array of applications to combinatorics, graph theory, geometry, and intriguing mathematical puzzles is another highlight engaging the reader. The exposition is user-friendly, yet rigorous, so that a broad audience consisting of students, math teachers and instructors, computer scientists and other professionals, along with the mathematically curious will all benefit from this book. Finally, Pell and Pell–Lucas Numbers provides enjoyment and excitement while sharpening the reader’s mathematical skills involving pattern recognition, proof-and-problem-solving techniques.

Quadratic Number Fields

Author : Franz Lemmermeyer
Publisher : Springer Nature
ISBN 13 : 3030786528
Total Pages : 348 pages
Book Rating : 4.0/5 (37 download)

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Book Synopsis Quadratic Number Fields by : Franz Lemmermeyer

Download or read book Quadratic Number Fields written by Franz Lemmermeyer and published by Springer Nature. This book was released on 2021-09-18 with total page 348 pages. Available in PDF, EPUB and Kindle. Book excerpt: This undergraduate textbook provides an elegant introduction to the arithmetic of quadratic number fields, including many topics not usually covered in books at this level. Quadratic fields offer an introduction to algebraic number theory and some of its central objects: rings of integers, the unit group, ideals and the ideal class group. This textbook provides solid grounding for further study by placing the subject within the greater context of modern algebraic number theory. Going beyond what is usually covered at this level, the book introduces the notion of modularity in the context of quadratic reciprocity, explores the close links between number theory and geometry via Pell conics, and presents applications to Diophantine equations such as the Fermat and Catalan equations as well as elliptic curves. Throughout, the book contains extensive historical comments, numerous exercises (with solutions), and pointers to further study. Assuming a moderate background in elementary number theory and abstract algebra, Quadratic Number Fields offers an engaging first course in algebraic number theory, suitable for upper undergraduate students.

An Introduction to Diophantine Equations

Author : Titu Andreescu
Publisher : Springer Science & Business Media
ISBN 13 : 0817645497
Total Pages : 350 pages
Book Rating : 4.8/5 (176 download)

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Book Synopsis An Introduction to Diophantine Equations by : Titu Andreescu

Download or read book An Introduction to Diophantine Equations written by Titu Andreescu and published by Springer Science & Business Media. This book was released on 2010-09-02 with total page 350 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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 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.

Quadratic Diophantine Equations

Author : Titu Andreescu
Publisher : Springer
ISBN 13 : 0387541098
Total Pages : 224 pages
Book Rating : 4.3/5 (875 download)

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Book Synopsis Quadratic Diophantine Equations by : Titu Andreescu

Download or read book Quadratic Diophantine Equations written by Titu Andreescu and published by Springer. This book was released on 2015-06-29 with total page 224 pages. Available in PDF, EPUB and Kindle. Book excerpt: This text treats the classical theory of quadratic diophantine equations and guides the reader through the last two decades of computational techniques and progress in the area. The presentation features two basic methods to investigate and motivate the study of quadratic diophantine equations: the theories of continued fractions and quadratic fields. It also discusses Pell’s equation and its generalizations, and presents some important quadratic diophantine equations and applications. The inclusion of examples makes this book useful for both research and classroom settings.

Power Play

Author : Edward Barbeau
Publisher : Cambridge University Press
ISBN 13 : 9780883855232
Total Pages : 204 pages
Book Rating : 4.8/5 (552 download)

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Book Synopsis Power Play by : Edward Barbeau

Download or read book Power Play written by Edward Barbeau and published by Cambridge University Press. This book was released on 1997-07-24 with total page 204 pages. Available in PDF, EPUB and Kindle. Book excerpt: A fund of knowledge for amateur and professional mathematicians.

A Course in Number Theory

Author : H. E. Rose
Publisher : Oxford University Press
ISBN 13 : 9780198523765
Total Pages : 420 pages
Book Rating : 4.5/5 (237 download)

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Book Synopsis A Course in Number Theory by : H. E. Rose

Download or read book A Course in Number Theory written by H. E. Rose and published by Oxford University Press. This book was released on 1995 with total page 420 pages. Available in PDF, EPUB and Kindle. Book excerpt: This textbook covers the main topics in number theory as taught in universities throughout the world. Number theory deals mainly with properties of integers and rational numbers; it is not an organized theory in the usual sense but a vast collection of individual topics and results, with some coherent sub-theories and a long list of unsolved problems. This book excludes topics relying heavily on complex analysis and advanced algebraic number theory. The increased use of computers in number theory is reflected in many sections (with much greater emphasis in this edition). Some results of a more advanced nature are also given, including the Gelfond-Schneider theorem, the prime number theorem, and the Mordell-Weil theorem. The latest work on Fermat's last theorem is also briefly discussed. Each chapter ends with a collection of problems; hints or sketch solutions are given at the end of the book, together with various useful tables.

Recreations in the Theory of Numbers

Author : Albert H. Beiler
Publisher : Courier Corporation
ISBN 13 : 0486210960
Total Pages : 383 pages
Book Rating : 4.4/5 (862 download)

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Book Synopsis Recreations in the Theory of Numbers by : Albert H. Beiler

Download or read book Recreations in the Theory of Numbers written by Albert H. Beiler and published by Courier Corporation. This book was released on 1964-01-01 with total page 383 pages. Available in PDF, EPUB and Kindle. Book excerpt: Number theory proves to be a virtually inexhaustible source of intriguing puzzle problems. Includes divisors, perfect numbers, the congruences of Gauss, scales of notation, the Pell equation, more. Solutions to all problems.

Diophantine Analysis

Author : Jörn Steuding
Publisher : Birkhäuser
ISBN 13 : 3319488171
Total Pages : 239 pages
Book Rating : 4.3/5 (194 download)

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Book Synopsis Diophantine Analysis by : Jörn Steuding

Download or read book Diophantine Analysis written by Jörn Steuding and published by Birkhäuser. This book was released on 2016-12-21 with total page 239 pages. Available in PDF, EPUB and Kindle. Book excerpt: This collection of course notes from a number theory summer school focus on aspects of Diophantine Analysis, addressed to Master and doctoral students as well as everyone who wants to learn the subject. The topics range from Baker’s method of bounding linear forms in logarithms (authored by Sanda Bujačić and Alan Filipin), metric diophantine approximation discussing in particular the yet unsolved Littlewood conjecture (by Simon Kristensen), Minkowski’s geometry of numbers and modern variations by Bombieri and Schmidt (Tapani Matala-aho), and a historical account of related number theory(ists) at the turn of the 19th Century (Nicola M.R. Oswald). Each of these notes serves as an essentially self-contained introduction to the topic. The reader gets a thorough impression of Diophantine Analysis by its central results, relevant applications and open problems. The notes are complemented with many references and an extensive register which makes it easy to navigate through the book.

Elliptic Diophantine Equations

Author : Nikos Tzanakis
Publisher : Walter de Gruyter
ISBN 13 : 3110281147
Total Pages : 196 pages
Book Rating : 4.1/5 (12 download)

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Book Synopsis Elliptic Diophantine Equations by : Nikos Tzanakis

Download or read book Elliptic Diophantine Equations written by Nikos Tzanakis and published by Walter de Gruyter. This book was released on 2013-08-29 with total page 196 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents in a unified and concrete way the beautiful and deep mathematics - both theoretical and computational - on which the explicit solution of an elliptic Diophantine equation is based. It collects numerous results and methods that are scattered in the literature. Some results are hidden behind a number of routines in software packages, like Magma and Maple; professional mathematicians very often use these routines just as a black-box, having little idea about the mathematical treasure behind them. Almost 20 years have passed since the first publications on the explicit solution of elliptic Diophantine equations with the use of elliptic logarithms. The "art" of solving this type of equation has now reached its full maturity. The author is one of the main persons that contributed to the development of this art. The monograph presents a well-balanced combination of a variety of theoretical tools (from Diophantine geometry, algebraic number theory, theory of linear forms in logarithms of various forms - real/complex and p-adic elliptic - and classical complex analysis), clever computational methods and techniques (LLL algorithm and de Weger's reduction technique, AGM algorithm, Zagier's technique for computing elliptic integrals), ready-to-use computer packages. A result is the solution in practice of a large general class of Diophantine equations.

Cubic Fields with Geometry

Author : Samuel A. Hambleton
Publisher : Springer
ISBN 13 : 9783030014025
Total Pages : 493 pages
Book Rating : 4.0/5 (14 download)

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Book Synopsis Cubic Fields with Geometry by : Samuel A. Hambleton

Download or read book Cubic Fields with Geometry written by Samuel A. Hambleton and published by Springer. This book was released on 2018-11-19 with total page 493 pages. Available in PDF, EPUB and Kindle. Book excerpt: The objective of this book is to provide tools for solving problems which involve cubic number fields. Many such problems can be considered geometrically; both in terms of the geometry of numbers and geometry of the associated cubic Diophantine equations that are similar in many ways to the Pell equation. With over 50 geometric diagrams, this book includes illustrations of many of these topics. The book may be thought of as a companion reference for those students of algebraic number theory who wish to find more examples, a collection of recent research results on cubic fields, an easy-to-understand source for learning about Voronoi’s unit algorithm and several classical results which are still relevant to the field, and a book which helps bridge a gap in understanding connections between algebraic geometry and number theory. The exposition includes numerous discussions on calculating with cubic fields including simple continued fractions of cubic irrational numbers, arithmetic using integer matrices, ideal class group computations, lattices over cubic fields, construction of cubic fields with a given discriminant, the search for elements of norm 1 of a cubic field with rational parametrization, and Voronoi's algorithm for finding a system of fundamental units. Throughout, the discussions are framed in terms of a binary cubic form that may be used to describe a given cubic field. This unifies the chapters of this book despite the diversity of their number theoretic topics.

Elements of Mathematics

Author : John Stillwell
Publisher : Princeton University Press
ISBN 13 : 0691171688
Total Pages : 440 pages
Book Rating : 4.6/5 (911 download)

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Book Synopsis Elements of Mathematics by : John Stillwell

Download or read book Elements of Mathematics written by John Stillwell and published by Princeton University Press. This book was released on 2016 with total page 440 pages. Available in PDF, EPUB and Kindle. Book excerpt: An exciting look at the world of elementary mathematics Elements of Mathematics takes readers on a fascinating tour that begins in elementary mathematics--but, as John Stillwell shows, this subject is not as elementary or straightforward as one might think. Not all topics that are part of today's elementary mathematics were always considered as such, and great mathematical advances and discoveries had to occur in order for certain subjects to become "elementary." Stillwell examines elementary mathematics from a distinctive twenty-first-century viewpoint and describes not only the beauty and scope of the discipline, but also its limits. From Gaussian integers to propositional logic, Stillwell delves into arithmetic, computation, algebra, geometry, calculus, combinatorics, probability, and logic. He discusses how each area ties into more advanced topics to build mathematics as a whole. Through a rich collection of basic principles, vivid examples, and interesting problems, Stillwell demonstrates that elementary mathematics becomes advanced with the intervention of infinity. Infinity has been observed throughout mathematical history, but the recent development of "reverse mathematics" confirms that infinity is essential for proving well-known theorems, and helps to determine the nature, contours, and borders of elementary mathematics. Elements of Mathematics gives readers, from high school students to professional mathematicians, the highlights of elementary mathematics and glimpses of the parts of math beyond its boundaries.

Continued Fractions

Author : Andrew M Rockett
Publisher : World Scientific Publishing Company
ISBN 13 : 9813103418
Total Pages : 200 pages
Book Rating : 4.8/5 (131 download)

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Book Synopsis Continued Fractions by : Andrew M Rockett

Download or read book Continued Fractions written by Andrew M Rockett and published by World Scientific Publishing Company. This book was released on 1992-08-08 with total page 200 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents the arithmetic and metrical theory of regular continued fractions and is intended to be a modern version of A. Ya. Khintchine's classic of the same title. Besides new and simpler proofs for many of the standard topics, numerous numerical examples and applications are included (the continued fraction of e, Ostrowski representations and t-expansions, period lengths of quadratic surds, the general Pell's equation, homogeneous and inhomogeneous diophantine approximation, Hall's theorem, the Lagrange and Markov spectra, asymmetric approximation, etc). Suitable for upper level undergraduate and beginning graduate students, the presentation is self-contained and the metrical results are developed as strong laws of large numbers.

Mathematics and Its History

Author : John Stillwell
Publisher : Springer Nature
ISBN 13 : 3030551938
Total Pages : 400 pages
Book Rating : 4.0/5 (35 download)

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Book Synopsis Mathematics and Its History by : John Stillwell

Download or read book Mathematics and Its History written by John Stillwell and published by Springer Nature. This book was released on 2020-11-07 with total page 400 pages. Available in PDF, EPUB and Kindle. Book excerpt: This textbook provides a unified and concise exploration of undergraduate mathematics by approaching the subject through its history. Readers will discover the rich tapestry of ideas behind familiar topics from the undergraduate curriculum, such as calculus, algebra, topology, and more. Featuring historical episodes ranging from the Ancient Greeks to Fermat and Descartes, this volume offers a glimpse into the broader context in which these ideas developed, revealing unexpected connections that make this ideal for a senior capstone course. The presentation of previous versions has been refined by omitting the less mainstream topics and inserting new connecting material, allowing instructors to cover the book in a one-semester course. This condensed edition prioritizes succinctness and cohesiveness, and there is a greater emphasis on visual clarity, featuring full color images and high quality 3D models. As in previous editions, a wide array of mathematical topics are covered, from geometry to computation; however, biographical sketches have been omitted. Mathematics and Its History: A Concise Edition is an essential resource for courses or reading programs on the history of mathematics. Knowledge of basic calculus, algebra, geometry, topology, and set theory is assumed. From reviews of previous editions: “Mathematics and Its History is a joy to read. The writing is clear, concise and inviting. The style is very different from a traditional text. I found myself picking it up to read at the expense of my usual late evening thriller or detective novel.... The author has done a wonderful job of tying together the dominant themes of undergraduate mathematics.” Richard J. Wilders, MAA, on the Third Edition "The book...is presented in a lively style without unnecessary detail. It is very stimulating and will be appreciated not only by students. Much attention is paid to problems and to the development of mathematics before the end of the nineteenth century.... This book brings to the non-specialist interested in mathematics many interesting results. It can be recommended for seminars and will be enjoyed by the broad mathematical community." European Mathematical Society, on the Second Edition

Graphs on Surfaces and Their Applications

Author : Sergei K. Lando
Publisher : Springer Science & Business Media
ISBN 13 : 3540383611
Total Pages : 463 pages
Book Rating : 4.5/5 (43 download)

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Book Synopsis Graphs on Surfaces and Their Applications by : Sergei K. Lando

Download or read book Graphs on Surfaces and Their Applications written by Sergei K. Lando and published by Springer Science & Business Media. This book was released on 2013-04-17 with total page 463 pages. Available in PDF, EPUB and Kindle. Book excerpt: Graphs drawn on two-dimensional surfaces have always attracted researchers by their beauty and by the variety of difficult questions to which they give rise. The theory of such embedded graphs, which long seemed rather isolated, has witnessed the appearance of entirely unexpected new applications in recent decades, ranging from Galois theory to quantum gravity models, and has become a kind of a focus of a vast field of research. The book provides an accessible introduction to this new domain, including such topics as coverings of Riemann surfaces, the Galois group action on embedded graphs (Grothendieck's theory of "dessins d'enfants"), the matrix integral method, moduli spaces of curves, the topology of meromorphic functions, and combinatorial aspects of Vassiliev's knot invariants and, in an appendix by Don Zagier, the use of finite group representation theory. The presentation is concrete throughout, with numerous figures, examples (including computer calculations) and exercises, and should appeal to both graduate students and researchers.