Pell S Equation

Author : Edward J. Barbeau
Publisher : Springer Science & Business Media
Page : 220 pages
File Size : 41,5 Mb
Release : 2006-05-04
Category : Mathematics
ISBN : 9780387226026

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Pell’s Equation by Edward J. Barbeau Pdf

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.

Author : Michael Jacobson,Hugh Williams
Publisher : Springer Science & Business Media
Page : 504 pages
File Size : 44,6 Mb
Release : 2008-12-02
Category : Mathematics
ISBN : 9780387849225

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Solving the Pell Equation by Michael Jacobson,Hugh Williams Pdf

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.

Author : Edward Everett Whitford
Publisher : Unknown
Page : 204 pages
File Size : 48,6 Mb
Release : 2008
Category : Mathematics
ISBN : UOM:39076002780828

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The Pell Equation by Edward Everett Whitford Pdf

An Unabridged, Digitally Enlarged History Of The Equation; Table Of Solutions From A = 1,5001 To A = 1,700, Bibliography With References To Over 300 Authors, and Table Of Continued Fractions For Square Root Of A.

Author : Edward J Barbeau
Publisher : Unknown
Page : 228 pages
File Size : 47,9 Mb
Release : 2003-01-14
Category : Electronic
ISBN : 1468492411

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Pell's Equation by Edward J Barbeau Pdf

Author : Franz Lemmermeyer
Publisher : Springer Nature
Page : 348 pages
File Size : 43,7 Mb
Release : 2021-09-18
Category : Mathematics
ISBN : 9783030786526

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Quadratic Number Fields by Franz Lemmermeyer Pdf

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.

Author : Thomas Koshy
Publisher : Springer
Page : 444 pages
File Size : 49,9 Mb
Release : 2014-11-11
Category : Mathematics
ISBN : 9781461484899

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Pell and Pell–Lucas Numbers with Applications by Thomas Koshy Pdf

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.​

Author : Neville Robbins
Publisher : Jones & Bartlett Learning
Page : 368 pages
File Size : 46,5 Mb
Release : 2006
Category : Computers
ISBN : 0763737682

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Beginning Number Theory by Neville Robbins Pdf

Thoroughly Revised And Updated, The New Second Edition Of Neville Robbins' Beginning Number Theory Includes All Of The Major Topics Covered In A Classic Number Theory Course And Blends In Numerous Applications And Specialized Treatments Of Number Theory, Including Cryptology, Fibonacci Numbers, And Computational Number Theory. The Text Strikes A Balance Between Traditional And Algorithmic Approaches To Elementary Number Theory And Is Supported With Numerous Exercises, Applications, And Case Studies Throughout. Computer Exercises For CAS Systems Are Also Included.

Author : Titu Andreescu,Dorin Andrica,Ion Cucurezeanu
Publisher : Springer Science & Business Media
Page : 350 pages
File Size : 42,9 Mb
Release : 2010-09-02
Category : Mathematics
ISBN : 9780817645496

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An Introduction to Diophantine Equations by Titu Andreescu,Dorin Andrica,Ion Cucurezeanu Pdf

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.

Author : Edward Everett Whitford
Publisher : Forgotten Books
Page : 203 pages
File Size : 47,6 Mb
Release : 2015-06-15
Category : Mathematics
ISBN : 1330096673

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The Pell Equation by Edward Everett Whitford Pdf

Excerpt from The Pell Equation Edward Everett Whitford was born in Brookfield, N. Y., January 31, 1865; graduated from Brookfield Academy in 1881; received the degree of A.B. from Colgate University in 1886 and of A.M. in 1890. He taught in Colby Academy, New London, N. H., Keystone Academy, Factoryville, Pa., Shamokin (Pa.) High School, Commercial High School and Pratt Institute, Brooklyn, N. Y. He was principal of Brookfield High School, 1900-1. He is now instructor in mathematics in the College of the City of New York with which institution he has been connected since 1905. He has been a graduate student in Columbia University since February, 1904, and is a member of the American Mathematical Society. The writer takes this opportunity of expressing his thanks to Professor David Eugene Smith for fruitful suggestions and able and helpful criticism. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works.

Author : Samuel A. Hambleton,Hugh C. Williams
Publisher : Springer
Page : 493 pages
File Size : 43,7 Mb
Release : 2018-11-07
Category : Mathematics
ISBN : 9783030014049

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Cubic Fields with Geometry by Samuel A. Hambleton,Hugh C. Williams Pdf

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.

Author : Nikos Tzanakis
Publisher : Walter de Gruyter
Page : 196 pages
File Size : 45,8 Mb
Release : 2013-08-29
Category : Mathematics
ISBN : 9783110281149

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Elliptic Diophantine Equations by Nikos Tzanakis Pdf

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.

Author : Jörn Steuding
Publisher : Birkhäuser
Page : 232 pages
File Size : 41,5 Mb
Release : 2016-12-21
Category : Mathematics
ISBN : 9783319488172

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Diophantine Analysis by Jörn Steuding Pdf

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.

Author : Dušan Djukić,Vladimir Janković,Ivan Matić,Nikola Petrović
Publisher : Springer Science & Business Media
Page : 819 pages
File Size : 52,7 Mb
Release : 2011-05-05
Category : Mathematics
ISBN : 9781441998545

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The IMO Compendium by Dušan Djukić,Vladimir Janković,Ivan Matić,Nikola Petrović Pdf

"The IMO Compendium" is the ultimate collection of challenging high-school-level mathematics problems and is an invaluable resource not only for high-school students preparing for mathematics competitions, but for anyone who loves and appreciates mathematics. The International Mathematical Olympiad (IMO), nearing its 50th anniversary, has become the most popular and prestigious competition for high-school students interested in mathematics. Only six students from each participating country are given the honor of participating in this competition every year. The IMO represents not only a great opportunity to tackle interesting and challenging mathematics problems, it also offers a way for high school students to measure up with students from the rest of the world. Until the first edition of this book appearing in 2006, it has been almost impossible to obtain a complete collection of the problems proposed at the IMO in book form. "The IMO Compendium" is the result of a collaboration between four former IMO participants from Yugoslavia, now Serbia and Montenegro, to rescue these problems from old and scattered manuscripts, and produce the ultimate source of IMO practice problems. This book attempts to gather all the problems and solutions appearing on the IMO through 2009. This second edition contains 143 new problems, picking up where the 1959-2004 edition has left off.

Author : David A. Cox
Publisher : John Wiley & Sons
Page : 372 pages
File Size : 43,8 Mb
Release : 2011-10-24
Category : Mathematics
ISBN : 9781118031001

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Primes of the Form x2 + ny2 by David A. Cox Pdf

Modern number theory began with the work of Euler and Gauss to understand and extend the many unsolved questions left behind by Fermat. In the course of their investigations, they uncovered new phenomena in need of explanation, which over time led to the discovery of field theory and its intimate connection with complex multiplication. While most texts concentrate on only the elementary or advanced aspects of this story, Primes of the Form x2 + ny2 begins with Fermat and explains how his work ultimately gave birth to quadratic reciprocity and the genus theory of quadratic forms. Further, the book shows how the results of Euler and Gauss can be fully understood only in the context of class field theory. Finally, in order to bring class field theory down to earth, the book explores some of the magnificent formulas of complex multiplication. The central theme of the book is the story of which primes p can be expressed in the form x2 + ny2. An incomplete answer is given using quadratic forms. A better though abstract answer comes from class field theory, and finally, a concrete answer is provided by complex multiplication. Along the way, the reader is introduced to some wonderful number theory. Numerous exercises and examples are included. The book is written to be enjoyed by readers with modest mathematical backgrounds. Chapter 1 uses basic number theory and abstract algebra, while chapters 2 and 3 require Galois theory and complex analysis, respectively.

Author : Claude Brezinski
Publisher : Springer Science & Business Media
Page : 556 pages
File Size : 50,8 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783642581694

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History of Continued Fractions and Padé Approximants by Claude Brezinski Pdf

The history of continued fractions is certainly one of the longest among those of mathematical concepts, since it begins with Euclid's algorithm for the great est common divisor at least three centuries B.C. As it is often the case and like Monsieur Jourdain in Moliere's "Ie bourgeois gentilhomme" (who was speak ing in prose though he did not know he was doing so), continued fractions were used for many centuries before their real discovery. The history of continued fractions and Pade approximants is also quite im portant, since they played a leading role in the development of some branches of mathematics. For example, they were the basis for the proof of the tran scendence of 11' in 1882, an open problem for more than two thousand years, and also for our modern spectral theory of operators. Actually they still are of great interest in many fields of pure and applied mathematics and in numerical analysis, where they provide computer approximations to special functions and are connected to some convergence acceleration methods. Con tinued fractions are also used in number theory, computer science, automata, electronics, etc ...