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Author : Koblitz
Publisher : Springer Science & Business Media
Page : 366 pages
File Size : 47,5 Mb
Release : 2013-11-21
Category : Science
ISBN : 9781489966995
Author : Ian Stewart,David Tall
Publisher : CRC Press
Page : 334 pages
File Size : 46,5 Mb
Release : 2001-12-12
Category : Mathematics
ISBN : 9781439864081
First published in 1979 and written by two distinguished mathematicians with a special gift for exposition, this book is now available in a completely revised third edition. It reflects the exciting developments in number theory during the past two decades that culminated in the proof of Fermat's Last Theorem. Intended as a upper level textbook, it
Author : Harold M. Edwards
Publisher : Springer Science & Business Media
Page : 436 pages
File Size : 46,8 Mb
Release : 2000-01-14
Category : Mathematics
ISBN : 0387950028
This introduction to algebraic number theory via the famous problem of "Fermats Last Theorem" follows its historical development, beginning with the work of Fermat and ending with Kummers theory of "ideal" factorization. The more elementary topics, such as Eulers proof of the impossibilty of x+y=z, are treated in an uncomplicated way, and new concepts and techniques are introduced only after having been motivated by specific problems. The book also covers in detail the application of Kummers theory to quadratic integers and relates this to Gauss'theory of binary quadratic forms, an interesting and important connection that is not explored in any other book.
Author : Ian Stewart
Publisher : Springer
Page : 257 pages
File Size : 50,9 Mb
Release : 1979-05-31
Category : Science
ISBN : 0412138409
The title of this book may be read in two ways. One is 'algebraic number-theory', that is, the theory of numbers viewed algebraically; the other, 'algebraic-number theory', the study of algebraic numbers. Both readings are compatible with our aims, and both are perhaps misleading. Misleading, because a proper coverage of either topic would require more space than is available, and demand more of the reader than we wish to; compatible, because our aim is to illustrate how some of the basic notions of the theory of algebraic numbers may be applied to problems in number theory. Algebra is an easy subject to compartmentalize, with topics such as 'groups', 'rings' or 'modules' being taught in comparative isolation. Many students view it this way. While it would be easy to exaggerate this tendency, it is not an especially desirable one. The leading mathematicians of the nineteenth and early twentieth centuries developed and used most of the basic results and techniques of linear algebra for perhaps a hundred years, without ever defining an abstract vector space: nor is there anything to suggest that they suf fered thereby. This historical fact may indicate that abstrac tion is not always as necessary as one commonly imagines; on the other hand the axiomatization of mathematics has led to enormous organizational and conceptual gains.
Author : Michal Krizek,Florian Luca,Lawrence Somer
Publisher : Springer Science & Business Media
Page : 280 pages
File Size : 54,7 Mb
Release : 2013-03-14
Category : Mathematics
ISBN : 9780387218502
The pioneering work of Pierre de Fermat has attracted the attention of mathematicians for over 350 years. This book provides an overview of the many properties of Fermat numbers and demonstrates their applications in areas such as number theory, probability theory, geometry, and signal processing. It is an ideal introduction to the basic mathematical ideas and algebraic methods connected with the Fermat numbers.
Author : Gary Cornell,Joseph H. Silverman,Glenn Stevens
Publisher : Springer Science & Business Media
Page : 592 pages
File Size : 53,8 Mb
Release : 2013-12-01
Category : Mathematics
ISBN : 9781461219743
This volume contains the expanded lectures given at a conference on number theory and arithmetic geometry held at Boston University. It introduces and explains the many ideas and techniques used by Wiles, and to explain how his result can be combined with Ribets theorem and ideas of Frey and Serre to prove Fermats Last Theorem. The book begins with an overview of the complete proof, followed by several introductory chapters surveying the basic theory of elliptic curves, modular functions and curves, Galois cohomology, and finite group schemes. Representation theory, which lies at the core of the proof, is dealt with in a chapter on automorphic representations and the Langlands-Tunnell theorem, and this is followed by in-depth discussions of Serres conjectures, Galois deformations, universal deformation rings, Hecke algebras, and complete intersections. The book concludes by looking both forward and backward, reflecting on the history of the problem, while placing Wiles'theorem into a more general Diophantine context suggesting future applications. Students and professional mathematicians alike will find this an indispensable resource.
Author : Paulo Ribenboim
Publisher : Springer Science & Business Media
Page : 418 pages
File Size : 46,7 Mb
Release : 2000-03-10
Category : Mathematics
ISBN : 9780387985084
In 1995, Andrew Wiles completed a proof of Fermat's Last Theorem. Although this was certainly a great mathematical feat, one shouldn't dismiss earlier attempts made by mathematicians and clever amateurs to solve the problem. In this book, aimed at amateurs curious about the history of the subject, the author restricts his attention exclusively to elementary methods that have produced rich results.
Author : Paulo Ribenboim
Publisher : Springer Science & Business Media
Page : 306 pages
File Size : 42,6 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781468493429
Lecture I The Early History of Fermat's Last Theorem.- 1 The Problem.- 2 Early Attempts.- 3 Kummer's Monumental Theorem.- 4 Regular Primes.- 5 Kummer's Work on Irregular Prime Exponents.- 6 Other Relevant Results.- 7 The Golden Medal and the Wolfskehl Prize.- Lecture II Recent Results.- 1 Stating the Results.- 2 Explanations.- Lecture III B.K. = Before Kummer.- 1 The Pythagorean Equation.- 2 The Biquadratic Equation.- 3 The Cubic Equation.- 4 The Quintic Equation.- 5 Fermat's Equation of Degree Seven.- Lecture IV The Naïve Approach.- 1 The Relations of Barlow and Abel.- 2 Sophie Germain.- 3 Co.
Author : Ian Stewart,David Tall
Publisher : CRC Press
Page : 338 pages
File Size : 53,9 Mb
Release : 2015-10-14
Category : Mathematics
ISBN : 9781498738408
Updated to reflect current research, Algebraic Number Theory and Fermat’s Last Theorem, Fourth Edition introduces fundamental ideas of algebraic numbers and explores one of the most intriguing stories in the history of mathematics—the quest for a proof of Fermat’s Last Theorem. The authors use this celebrated theorem to motivate a general study of the theory of algebraic numbers from a relatively concrete point of view. Students will see how Wiles’s proof of Fermat’s Last Theorem opened many new areas for future work. New to the Fourth Edition Provides up-to-date information on unique prime factorization for real quadratic number fields, especially Harper’s proof that Z(√14) is Euclidean Presents an important new result: Mihăilescu’s proof of the Catalan conjecture of 1844 Revises and expands one chapter into two, covering classical ideas about modular functions and highlighting the new ideas of Frey, Wiles, and others that led to the long-sought proof of Fermat’s Last Theorem Improves and updates the index, figures, bibliography, further reading list, and historical remarks Written by preeminent mathematicians Ian Stewart and David Tall, this text continues to teach students how to extend properties of natural numbers to more general number structures, including algebraic number fields and their rings of algebraic integers. It also explains how basic notions from the theory of algebraic numbers can be used to solve problems in number theory.
Author : Simon Singh
Publisher : HarperCollins UK
Page : 368 pages
File Size : 49,6 Mb
Release : 2012-11-22
Category : Science
ISBN : 9780007381999
‘I have a truly marvellous demonstration of this proposition which this margin is too narrow to contain.’
Author : N. Koblitz
Publisher : Unknown
Page : 380 pages
File Size : 47,8 Mb
Release : 2014-09-01
Category : Electronic
ISBN : 1489967001
Author : Paulo Ribenboim
Publisher : Springer Science & Business Media
Page : 336 pages
File Size : 41,9 Mb
Release : 1979-12-18
Category : Computers
ISBN : 0387904328
Fermat's problem, also ealled Fermat's last theorem, has attraeted the attention of mathematieians far more than three eenturies. Many clever methods have been devised to attaek the problem, and many beautiful theories have been ereated with the aim of proving the theorem. Yet, despite all the attempts, the question remains unanswered. The topie is presented in the form of leetures, where I survey the main lines of work on the problem. In the first two leetures, there is a very brief deseription of the early history , as well as a seleetion of a few of the more representative reeent results. In the leetures whieh follow, I examine in sue eession the main theories eonneeted with the problem. The last two lee tu res are about analogues to Fermat's theorem. Some of these leetures were aetually given, in a shorter version, at the Institut Henri Poineare, in Paris, as well as at Queen's University, in 1977. I endeavoured to produee a text, readable by mathematieians in general, and not only by speeialists in number theory. However, due to a limitation in size, I am aware that eertain points will appear sketehy. Another book on Fermat's theorem, now in preparation, will eontain a eonsiderable amount of the teehnieal developments omitted here. It will serve those who wish to learn these matters in depth and, I hope, it will clarify and eomplement the present volume.
Author : Vijaya Kumar Murty
Publisher : American Mathematical Soc.
Page : 278 pages
File Size : 54,9 Mb
Release : 1995
Category : Mathematics
ISBN : 9780821803134
The most significant recent development in number theory is the work of Andrew Wiles on modular elliptic curves. Besides implying Fermat's Last Theorem, his work establishes a new reciprocity law. Reciprocity laws lie at the heart of number theory. Wiles' work draws on many of the tools of modern number theory and the purpose of this volume is to introduce readers to some of this background material. Based on a seminar held during 1993-1994 at the Fields Institute for Research in Mathematical Sciences, this book contains articles on elliptic curves, modular forms and modular curves, Serre's conjectures, Ribet's theorem, deformations of Galois representations, Euler systems, and annihilators of Selmer groups. All of the authors are well known in their field and have made significant contributions to the general area of elliptic curves, Galois representations, and modular forms. Features: Brings together a unique collection of number theoretic tools. Makes accessible the tools needed to understand one of the biggest breakthroughs in mathematics. Provides numerous references for further study.
Author : A. J. Van Der Poorten
Publisher : Wiley-Interscience
Page : 248 pages
File Size : 46,6 Mb
Release : 1996-02-16
Category : Mathematics
ISBN : UOM:39015036039314
Around 1637, the French jurist Pierre de Fermat scribbled in the margin of his copy of the book Arithmetica what came to be known as Fermat's Last Theorem, the most famous question in mathematical history. Stating that it is impossible to split a cube into two cubes, or a fourth power into two fourth powers, or any higher power into two like powers, but not leaving behind the marvelous proof he claimed to have had, Fermat prompted three and a half centuries of mathematical inquiry which culminated only recently with the proof of the theorem by Andrew Wiles. This book offers the first serious treatment of Fermat's Last Theorem since Wiles's proof. It is based on a series of lectures given by the author to celebrate Wiles's achievement, with each chapter explaining a separate area of number theory as it pertains to Fermat's Last Theorem. Together, they provide a concise history of the theorem as well as a brief discussion of Wiles's proof and its implications. Requiring little more than one year of university mathematics and some interest in formulas, this overview provides many useful tips and cites numerous references for those who desire more mathematical detail. The book's most distinctive feature is its easy-to-read, humorous style, complete with examples, anecdotes, and some of the lesser-known mathematics underlying the newly discovered proof. In the author's own words, the book deals with "serious mathematics without being too serious about it." Alf van der Poorten demystifies mathematical research, offers an intuitive approach to the subject-loosely suggesting various definitions and unexplained facts-and invites the reader to fill in the missing links in some of the mathematical claims. Entertaining, controversial, even outrageous, this book not only tells us why, in all likelihood, Fermat did not have the proof for his last theorem, it also takes us through historical attempts to crack the theorem, the prizes that were offered along the way, and the consequent motivation for the development of other areas of mathematics. Notes on Fermat's Last Theorem is invaluable for students of mathematics, and of real interest to those in the physical sciences, engineering, and computer sciences-indeed for anyone who craves a glimpse at this fascinating piece of mathematical history. An exciting introduction to modern number theory as reflected by the history of Fermat's Last Theorem This book displays the unique talents of author Alf van der Poorten in mathematical exposition for mathematicians. Here, mathematics' most famous question and the ideas underlying its recent solution are presented in a way that appeals to the imagination and leads the reader through related areas of number theory. The first book to focus on Fermat's Last Theorem since Andrew Wiles presented his celebrated proof, Notes on Fermat's Last Theorem surveys 350 years of mathematical history in an amusing and intriguing collection of tidbits, anecdotes, footnotes, exercises, references, illustrations, and more. Proving that mathematics can make for lively reading as well as intriguing thought, this thoroughly accessible treatment Helps students and professionals develop a background in number theory and provides introductions to the various fields of theory that are touched upon * Offers insight into the exciting world of mathematical research * Covers a number of areas appropriate for classroom use * Assumes only one year of university mathematics background even for the more advanced topics * Explains why Fermat surely did not have the proof to his theorem * Examines the efforts of mathematicians over the centuries to solve the problem * Shows how the pursuit of the theorem contributed to the greater development of mathematics
Author : Michal Křížek,Lawrence Somer,Alena Šolcová
Publisher : Springer Nature
Page : 342 pages
File Size : 47,7 Mb
Release : 2021-09-21
Category : Mathematics
ISBN : 9783030838997
This book provides an overview of many interesting properties of natural numbers, demonstrating their applications in areas such as cryptography, geometry, astronomy, mechanics, computer science, and recreational mathematics. In particular, it presents the main ideas of error-detecting and error-correcting codes, digital signatures, hashing functions, generators of pseudorandom numbers, and the RSA method based on large prime numbers. A diverse array of topics is covered, from the properties and applications of prime numbers, some surprising connections between number theory and graph theory, pseudoprimes, Fibonacci and Lucas numbers, and the construction of Magic and Latin squares, to the mathematics behind Prague’s astronomical clock. Introducing a general mathematical audience to some of the basic ideas and algebraic methods connected with various types of natural numbers, the book will provide invaluable reading for amateurs and professionals alike.
