Diophantine Equations And Inequalities In Algebraic Number Fields

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Diophantine Equations and Inequalities in Algebraic Number Fields

Yuan Wang
Diophantine Equations and Inequalities in Algebraic Number Fields Book PDF

Author : Yuan Wang
Publisher : Springer Science & Business Media
Page : 185 pages
File Size : 40,5 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783642581717

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Diophantine Equations and Inequalities in Algebraic Number Fields by Yuan Wang Pdf

The circle method has its genesis in a paper of Hardy and Ramanujan (see [Hardy 1])in 1918concernedwiththepartitionfunction andtheproblemofrep resenting numbers as sums ofsquares. Later, in a series of papers beginning in 1920entitled "some problems of'partitio numerorum''', Hardy and Littlewood (see [Hardy 1]) created and developed systematically a new analytic method, the circle method in additive number theory. The most famous problems in ad ditive number theory, namely Waring's problem and Goldbach's problem, are treated in their papers. The circle method is also called the Hardy-Littlewood method. Waring's problem may be described as follows: For every integer k 2 2, there is a number s= s(k) such that every positive integer N is representable as (1) where Xi arenon-negative integers. This assertion wasfirst proved by Hilbert [1] in 1909. Using their powerful circle method, Hardy and Littlewood obtained a deeper result on Waring's problem. They established an asymptotic formula for rs(N), the number of representations of N in the form (1), namely k 1 provided that 8 2 (k - 2)2 - +5. Here

Discriminant Equations in Diophantine Number Theory

Jan-Hendrik Evertse,Klmn Gyory
Discriminant Equations in Diophantine Number Theory Book PDF

Author : Jan-Hendrik Evertse,Klmn Gyory
Publisher : Cambridge University Press
Page : 477 pages
File Size : 47,5 Mb
Release : 2016-11-03
Category : Mathematics
ISBN : 9781107097612

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Discriminant Equations in Diophantine Number Theory by Jan-Hendrik Evertse,Klmn Gyory Pdf

The first comprehensive and up-to-date account of discriminant equations and their applications. For graduate students and researchers.

Unit Equations in Diophantine Number Theory

Jan-Hendrik Evertse,Klmn Gyory
Unit Equations in Diophantine Number Theory Book PDF

Author : Jan-Hendrik Evertse,Klmn Gyory
Publisher : Cambridge University Press
Page : 381 pages
File Size : 50,8 Mb
Release : 2015-12-30
Category : Mathematics
ISBN : 9781107097605

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Unit Equations in Diophantine Number Theory by Jan-Hendrik Evertse,Klmn Gyory Pdf

A comprehensive, graduate-level treatment of unit equations and their various applications.

Number Theory

Daniel Duverney
Number Theory Book PDF

Author : Daniel Duverney
Publisher : World Scientific
Page : 348 pages
File Size : 50,7 Mb
Release : 2010
Category : Mathematics
ISBN : 9789814307451

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Number Theory by Daniel Duverney Pdf

This textbook presents an elementary introduction to number theory and its different aspects: approximation of real numbers, irrationality and transcendence problems, continued fractions, diophantine equations, quadratic forms, arithmetical functions and algebraic number theory. These topics are covered in 12 chapters and more than 200 solved exercises. Clear, concise, and self-contained, this textbook may be used by undergraduate and graduate students, as well as highschool mathematics teachers. More generally, it will be suitable for all those who are interested in number theory, this fascinating branch of mathematics.

Equations and Inequalities

Jiri Herman,Radan Kucera,Jaromir Simsa
Equations and Inequalities Book PDF

Author : Jiri Herman,Radan Kucera,Jaromir Simsa
Publisher : Springer Science & Business Media
Page : 362 pages
File Size : 50,5 Mb
Release : 2000-03-23
Category : Education
ISBN : 0387989420

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Equations and Inequalities by Jiri Herman,Radan Kucera,Jaromir Simsa Pdf

A look at solving problems in three areas of classical elementary mathematics: equations and systems of equations of various kinds, algebraic inequalities, and elementary number theory, in particular divisibility and diophantine equations. In each topic, brief theoretical discussions are followed by carefully worked out examples of increasing difficulty, and by exercises which range from routine to rather more challenging problems. While it emphasizes some methods that are not usually covered in beginning university courses, the book nevertheless teaches techniques and skills which are useful beyond the specific topics covered here. With approximately 330 examples and 760 exercises.

Quadratic Number Fields

Franz Lemmermeyer
Quadratic Number Fields Book PDF

Author : Franz Lemmermeyer
Publisher : Springer Nature
Page : 348 pages
File Size : 48,7 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.

A Brief Guide to Algebraic Number Theory

H. P. F. Swinnerton-Dyer
A Brief Guide to Algebraic Number Theory Book PDF

Author : H. P. F. Swinnerton-Dyer
Publisher : Cambridge University Press
Page : 164 pages
File Size : 55,5 Mb
Release : 2001-02-22
Category : Mathematics
ISBN : 0521004233

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A Brief Guide to Algebraic Number Theory by H. P. F. Swinnerton-Dyer Pdf

Broad graduate-level account of Algebraic Number Theory, first published in 2001, including exercises, by a world-renowned author.

Pell’s Equation

Edward J. Barbeau
Pell s Equation Book PDF

Author : Edward J. Barbeau
Publisher : Springer Science & Business Media
Page : 220 pages
File Size : 43,8 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.

Diophantus and Diophantine Equations

Izabella Grigorʹevna Bashmakova,Joseph H. Silverman
Diophantus and Diophantine Equations Book PDF

Author : Izabella Grigorʹevna Bashmakova,Joseph H. Silverman
Publisher : Cambridge University Press
Page : 110 pages
File Size : 53,6 Mb
Release : 1997
Category : Mathematics
ISBN : 0883855267

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Diophantus and Diophantine Equations by Izabella Grigorʹevna Bashmakova,Joseph H. Silverman Pdf

Semi-popular maths on an area of number theory related to Fermat.

Solving the Pell Equation

Michael Jacobson,Hugh Williams
Solving the Pell Equation Book PDF

Author : Michael Jacobson,Hugh Williams
Publisher : Springer Science & Business Media
Page : 504 pages
File Size : 47,5 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.

Quadratic Diophantine Equations

Titu Andreescu,Dorin Andrica
Quadratic Diophantine Equations Book PDF

Author : Titu Andreescu,Dorin Andrica
Publisher : Springer
Page : 211 pages
File Size : 43,5 Mb
Release : 2015-06-29
Category : Mathematics
ISBN : 9780387541099

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Quadratic Diophantine Equations by Titu Andreescu,Dorin Andrica Pdf

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.

Algebraic Number Theory

Aneta Hajek
Algebraic Number Theory Book PDF

Author : Aneta Hajek
Publisher : Unknown
Page : 0 pages
File Size : 43,8 Mb
Release : 2015-08
Category : Electronic
ISBN : 168117183X

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Algebraic Number Theory by Aneta Hajek Pdf

Algebraic number theory is the branch of number theory that deals with algebraic numbers. Historically, algebraic number theory developed as a set of tools for solving problems in elementary number theory, namely Diophantine equations (i.e., equations whose solutions are integers or rational numbers). More recently, algebraic number theory has developed into the abstract study of algebraic numbers and number fields themselves, as well as their properties. Algebraic number theory is a major branch of number theory that studies algebraic structures related to algebraic integers. This is generally accomplished by considering a ring of algebraic integers O in an algebraic number field K/Q, and studying their algebraic properties such as factorization, the behaviour of ideals, and field extensions. In this setting, the familiar features of the integerssuch as unique factorizationneed not hold. The virtue of the primary machinery employedGalois theory, group cohomology, group representations, and L-functionsis that it allows one to deal with new phenomena and yet partially recover the behaviour of the usual integers. The higher reaches of algebraic number theory are now one of the crown jewels of mathematics. But algebraic number theory is not merely interesting in itself. It has become an important tool over a wide range of pure mathematics, and many of ideas involved generalize, for example to algebraic geometry. This book is intended both for number theorist and more generally for working algebraists.

An Introduction to Diophantine Equations

Titu Andreescu,Dorin Andrica,Ion Cucurezeanu
An Introduction to Diophantine Equations Book PDF

Author : Titu Andreescu,Dorin Andrica,Ion Cucurezeanu
Publisher : Springer Science & Business Media
Page : 350 pages
File Size : 45,7 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.

The Story of Algebraic Numbers in the First Half of the 20th Century

Władysław Narkiewicz
The Story of Algebraic Numbers in the First Half of the 20th Century Book PDF

Author : Władysław Narkiewicz
Publisher : Springer
Page : 443 pages
File Size : 51,5 Mb
Release : 2019-01-18
Category : Mathematics
ISBN : 9783030037543

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The Story of Algebraic Numbers in the First Half of the 20th Century by Władysław Narkiewicz Pdf

The book is aimed at people working in number theory or at least interested in this part of mathematics. It presents the development of the theory of algebraic numbers up to the year 1950 and contains a rather complete bibliography of that period. The reader will get information about results obtained before 1950. It is hoped that this may be helpful in preventing rediscoveries of old results, and might also inspire the reader to look at the work done earlier, which may hide some ideas which could be applied in contemporary research.

Analytic Methods for Diophantine Equations and Diophantine Inequalities

H. Davenport
Analytic Methods for Diophantine Equations and Diophantine Inequalities Book PDF

Author : H. Davenport
Publisher : Cambridge University Press
Page : 164 pages
File Size : 50,5 Mb
Release : 2005-02-07
Category : Mathematics
ISBN : 113944123X

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Analytic Methods for Diophantine Equations and Diophantine Inequalities by H. Davenport Pdf

Harold Davenport was one of the truly great mathematicians of the twentieth century. Based on lectures he gave at the University of Michigan in the early 1960s, this book is concerned with the use of analytic methods in the study of integer solutions to Diophantine equations and Diophantine inequalities. It provides an excellent introduction to a timeless area of number theory that is still as widely researched today as it was when the book originally appeared. The three main themes of the book are Waring's problem and the representation of integers by diagonal forms, the solubility in integers of systems of forms in many variables, and the solubility in integers of diagonal inequalities. For the second edition of the book a comprehensive foreword has been added in which three prominent authorities describe the modern context and recent developments. A thorough bibliography has also been added.