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Number Theory is more than a comprehensive treatment of the subject. It is an introduction to topics in higher level mathematics, and unique in its scope; topics from analysis, modern algebra, and discrete mathematics are all included.

The book is divided into two parts. Part A covers key concepts of number theory and could serve as a first course on the subject. Part B delves into more advanced topics and an exploration of related mathematics. Part B contains, for example, complete proofs of the Hasse-Minkowski theorem and the prime number theorem, as well as self-contained accounts of the character theory of finite groups and the theory of elliptic functions.

The prerequisites for this self-contained text are elements from linear algebra. Valuable references for the reader are collected at the end of each chapter. It is suitable as an introduction to higher level mathematics for undergraduates, or for self-study.

From the reviews:

"This is a book which many mathematicians could enjoy browsing, and one which a good undergraduate could be encouraged to read to learn something of the interconnections, which exist between apparently disparate parts of mathematics."

--Canadian Mathematical Society

"As a source for information on the 'reach' of number theory into other areas of mathematics, it is an excellent work."

--Mathematical Association of America

Product Description
This book is divided into two parts. The first part is preliminary and consists of algebraic number theory and the theory of semisimple algebras. There are two principal topics: classification of quadratic forms and quadratic Diophantine equations. The second topic is a new framework which contains the investigation of Gauss on the sums of three squares as a special case.To make the bookconcise, the author proves some basic theorems in number theory only in some special cases. However, the book is self-contained when the base field is the rational number field, and the main theorems are statedwith an arbitrary number field as the base field. So the reader familiar with class field theory will be able to learn the arithmetic theory of quadratic forms with no further references. ... Read more

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The four contributions collected in this volume deal with several advanced results in analytic number theory. Friedlander’s paper contains some recent achievements of sieve theory leading to asymptotic formulae for the number of primes represented by suitable polynomials. Heath-Brown's lecture notes mainly deal with counting integer solutions to Diophantine equations, using among other tools several results from algebraic geometry and from the geometry of numbers. Iwaniec’s paper gives a broad picture of the theory of Siegel’s zeros and of exceptional characters of L-functions, and gives a new proof of Linnik’s theorem on the least prime in an arithmetic progression. Kaczorowski’s article presents an up-to-date survey of the axiomatic theory of L-functions introduced by Selberg, with a detailed exposition of several recent results.

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The subject of the book is Diophantine approximation and Nevanlinna theory. Not only does the text provide new results and directions, it also challenges open problems and collects latest research activities on these subjects made by the authors over the past eight years. Some of the significant findings are the proof of the Green-Griffiths conjecture by using meromorphic connections and Jacobian sections, and a generalized abc-conjecture. The book also presents the state of the art in the studies of the analogues between Diophantine approximation (in number theory) and value distribution theory (in complex analysis), with a method based on Vojta’s dictionary for the terms of these two fields. The approaches are relatively natural and more effective than existing methods.

The book is self-contained and appended with a comprehensive and up-to-date list of references. It is of interest to a broad audience of graduate students and researchers specialized in pure mathematics.

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This book constitutes the refereed proceedings of the 8th International Algorithmic Number Theory Symposium, ANTS 2008, held in Banff, Canada, in May 2008.

The 28 revised full papers presented together with 2 invited papers were carefully reviewed and selected for inclusion in the book. The papers are organized in topical sections on elliptic curves cryptology and generalizations, arithmetic of elliptic curves, integer factorization, K3 surfaces, number fields, point counting, arithmetic of function fields, modular forms, cryptography, and number theory.

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Number theory as studied by the logician is the subject matter of the book. This first volume can stand on its own as a somewhat unorthodox introduction to mathematical logic for undergraduates, dealing with the usual introductory material: recursion theory, first-order logic, completeness, incompleteness, and undecidability. In addition, its second chapter contains the most complete logical discussion of Diophantine Decision Problems available anywhere, taking the reader right up to the frontiers of research (yet remaining accessible to the undergraduate). The first and third chapters also offer greater depth and breadth in logico-arithmetical matters than can be found in existing logic texts. Each chapter contains numerous exercises, historical and other comments aimed at developing the student's perspective on the subject, and a partially annotated bibliography. ... Read more

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Many of the important and creative developments in modern mathematics resulted from attempts to solve questions that originate in number theory. The publication of Emil Grosswald’s classic text presents an illuminating introduction to number theory. Combining the historical developments with the analytical approach, Topics from the Theory of Numbers offers the reader a diverse range of subjects to investigate, including: (1) divisibility, (2) congruences, (3) the Riemann zeta function, (4) Diophantine equations and Fermat’s conjecture, (5) the theory of partitions.

Comprehensive in nature, Topics from the Theory of Numbers is an ideal text for advanced undergraduates and graduate students alike.

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Eugène Charles Catalan made his famous conjecture – that 8 and 9 are the only two consecutive perfect powers of natural numbers – in 1844 in a letter to the editor of Crelle’s mathematical journal. One hundred and fifty-eight years later, Preda Mihailescu proved it.

Catalan’s Conjecture presents this spectacular result in a way that is accessible to the advanced undergraduate. The first few sections of the book require little more than a basic mathematical background and some knowledge of elementary number theory, while later sections involve Galois theory, algebraic number theory and a small amount of commutative algebra. The prerequisites, such as the basic facts from the arithmetic of cyclotomic fields, are all discussed within the text.

The author dissects both Mihailescu’s proof and the earlier work it made use of, taking great care to select streamlined and transparent versions of the arguments and to keep the text self-contained. Only in the proof of Thaine’s theorem is a little class field theory used; it is hoped that this application will motivate the interested reader to study the theory further.

Beautifully clear and concise, this book will appeal not only to specialists in number theory but to anyone interested in seeing the application of the ideas of algebraic number theory to a famous mathematical problem.

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This digital document is an article from Science and Its Times, brought to you by Gale®, a part of Cengage Learning, a world leader in e-research and educational publishing for libraries, schools and businesses.The length of the article is 619 words.The article is delivered in HTML format and is available in your Amazon.com Digital Locker immediately after purchase.You can view it with any web browser.The histories of science, technology, and mathematics merge with the study of humanities and social science in this interdisciplinary reference work. Essays on people, theories, discoveries, and concepts are combined with overviews, bibliographies of primary documents, and chronological elements to offer students a fascinating way to understand the impact of science on the course of human history and how science affects everyday life. Entries represent people and developments throughout the world, from about 2000 B.C. through the end of the twentieth century. ... Read more

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Arithmetic Geometry can be defined as the part of Algebraic Geometry connected with the study of algebraic varieties through arbitrary rings, in particular through non-algebraically closed fields. It lies at the intersection between classical algebraic geometry and number theory. A C.I.M.E. Summer School devoted to arithmetic geometry was held in Cetraro, Italy in September 2007, and presented some of the most interesting new developments in arithmetic geometry. This book collects the lecture notes which were written up by the speakers. The main topics concern diophantine equations, local-global principles, diophantine approximation and its relations to Nevanlinna theory, and rationally connected varieties. The book is divided into three parts, corresponding to the courses given by J-L Colliot-Thelene, Peter Swinnerton Dyer and Paul Vojta. ... Read more

Product Description
This digital document is an article from Science and Its Times, brought to you by Gale®, a part of Cengage Learning, a world leader in e-research and educational publishing for libraries, schools and businesses.The length of the article is 1521 words.The article is delivered in HTML format and is available in your Amazon.com Digital Locker immediately after purchase.You can view it with any web browser.The histories of science, technology, and mathematics merge with the study of humanities and social science in this interdisciplinary reference work. Essays on people, theories, discoveries, and concepts are combined with overviews, bibliographies of primary documents, and chronological elements to offer students a fascinating way to understand the impact of science on the course of human history and how science affects everyday life. Entries represent people and developments throughout the world, from about 2000 B.C. through the end of the twentieth century. ... Read more

Product Description
This digital document is an article from Science and Its Times, brought to you by Gale®, a part of Cengage Learning, a world leader in e-research and educational publishing for libraries, schools and businesses.The length of the article is 1445 words.The article is delivered in HTML format and is available in your Amazon.com Digital Locker immediately after purchase.You can view it with any web browser.The histories of science, technology, and mathematics merge with the study of humanities and social science in this interdisciplinary reference work. Essays on people, theories, discoveries, and concepts are combined with overviews, bibliographies of primary documents, and chronological elements to offer students a fascinating way to understand the impact of science on the course of human history and how science affects everyday life. Entries represent people and developments throughout the world, from about 2000 B.C. through the end of the twentieth century. ... Read more

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This monograph is concerned with counting rational points of bounded height on projective algebraic varieties. This is a relatively young topic, whose exploration has already uncovered a rich seam of mathematics situated at the interface of analytic number theory and Diophantine geometry. The goal of the book is to give a systematic account of the field with an emphasis on the role played by analytic number theory in its development. Among the themes discussed in detail are

* the Manin conjecture for del Pezzo surfaces;

* Heath-Brown's dimension growth conjecture; and

* the Hardy-Littlewood circle method.

Readers of this monograph will be rapidly brought into contact with a spectrum of problems and conjectures that are central to this fertile subject area.

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This book constitutes the refereed proceedings of the 9th International Algorithmic Number Theory Symposium, ANTS 2010, held in Nancy, France, in July 2010. The 25 revised full papers presented together with 5 invited papers were carefully reviewed and selected for inclusion in the book. The papers are devoted to algorithmic aspects of number theory, including elementary number theory, algebraic number theory, analytic number theory, geometry of numbers, algebraic geometry, finite fields, and cryptography. ... Read more

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This text offers a special account of Indian work in diophantine equations during the 6th through 12th centuries and Italian work on solutions of cubic and biquadratic equations from the 11th through 16th centuries. The volume traces the historical development of algebra and the theory of equations from ancient times to the beginning of modern algebra, outlining some modern themes such as the fundamental theorem of algebra, Clifford algebras, and quarternions. It is geared toward undergraduates who have no background in calculus. ... Read more

Customer Reviews (2)

Algebra in Ancient and Modern Times:A Teacher's Perspectiv
This book by V.S. Varadarajan gives a broad historical overview of the evolution of algeba as an independent field of mathematics.Varadarajan does an outstanding job of developing complex mathematical concepts in astep-by-step format.

The text begins, naturally, with the early Greekmathematicians and the first few chapters are about Euclid, Archimedes,counting principles of the ancient Egyptians, etc. Although the author doesnot devote much space to this early history, enough detail is provided touse this book as a reference in a mathematical classroom.

Section onethen proceeds to the period of Cardano, Tartalegia, Ferrari and themathematical competitions that have become so famous.Not only does theauthor give a standard account of the conflicts that arose over thepublication of Cardano's Ars Magna, he also writes tremendous detail on thesupporting equations and terms.Terms such as quadratic, multiple roots,polynomial and others are defined for the high school scholar, yet the textdoes not seem depreciating to students studying higher level collegemathematics.

The third and final section of Algebra in Ancient andModern Times, is devoted to Guass, whose work the author is obviouslyenthralled with, and other period four mathematicians. This sectiondevelops modern group theory, vector analysis and matrix algebra.Thedetail is again well written and enough historical content is given that,again, this book becomes a historical reference as well.

Overall, I wouldreccommend this to any teacher.It would be an excellent addition to aclassroom bookshelf for students from high school to graduate students incollege.There are additional problems and short notes that could beaccessed and referenced as well as using the book for a student material.

I reviewed the book originally for a History of Mathematics course atthe University of Lousiville and was quite pleased with my selection.

Algebra in Ancient and Modern Times:A Teacher's Perspectiv
This book by V.S. Varadarajan gives a broad historical overview of the evolution of algeba as an independent field of mathematics.Varadarajan does an outstanding job of developing complex mathematical concepts in astep-by-step format.

The text begins, naturally, with the early Greekmathematicians and the first few chapters are about Euclid, Archimedes,counting principles of the ancient Egyptians, etc. Although the author doesnot devote much space to this early history, enough detail is provided touse this book as a reference in a mathematical classroom.

Section onethen proceeds to the period of Cardano, Tartalegia, Ferrari and themathematical competitions that have become so famous.Not only does theauthor give a standard account of the conflicts that arose over thepublication of Cardano's Ars Magna, he also writes tremendous detail on thesupporting equations and terms.Terms such as quadratic, multiple roots,polynomial and others are defined for the high school scholar, yet the textdoes not seem depreciating to students studying higher level collegemathematics.

The third and final section of Algebra in Ancient andModern Times, is devoted to Guass, whose work the author is obviouslyenthralled with, and other period four mathematicians. This sectiondevelops modern group theory, vector analysis and matrix algebra.Thedetail is again well written and enough historical content is given that,again, this book becomes a historical reference as well.

Overall, I wouldreccommend this to any teacher.It would be an excellent addition to aclassroom bookshelf for students from high school to graduate students incollege.There are additional problems and short notes that could beaccessed and referenced as well as using the book for a student material.

I reviewed the book originally for a History of Mathematics course atthe University of Lousiville and was quite pleased with my selection. ... Read more

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Product Description
Author S.A. Stepanov thoroughly investigates the current stateof the theory of Diophantine equations and its related methods.Discussionsfocus on arithmetic, algebraic-geometric, and logicalaspects of theproblem. Designed for students as well as researchers,the bookincludes over 250 excercises accompanied by hints,instructions, and references. Written in a clear manner, this textdoes not require readers to have special knowledge of modern methodsof algebraic geometry. ... Read more

Customer Reviews (2)

A source for new, interesting problems
This book starts with an examination of some of the problems posed by Florentin Smarandache, one of the foremost mathematicians in the world today. The problems are from many different areas, such as sequences, primes and other aspects of number theory. Some of the problems are solved in the book, although in many cases the author raises additional questions.
The second part of the book deals with a function created by the author and given the name the Pseudo Smarandache function. It is defined as follows:

For any positive integer n, the value of the Pseudo Smarandache function is the smallest number m suchthat m evenly divides (1+2+3+ . . . +n).

There are many obvious consequences to this definition, but there are also many problems that are posed. Some are very easy to resolve, requiring only some of the basic notions of number theory. Others are very hard and may never be solved. However, that is part of the spirit of the chase, making the function all that more interesting.
If you are searching for new areas of mathematics to explore, I urge you to consider this book. An interest in number theory appears to be the common glue that binds mathematicians together and there are some new ideas about numbers to be found here.

Disclaimer: I served as the editor of the book, although I did not write any of the material.

Smarandache Sequences
Many Smarandache sequences are studied in this book by the Japanese author Kashihara. Also, the author makes more conjectures on them as in the Smarandache functions as well.

M.Marimutha ... Read more