Product Description
Covers addition, multiplication and exponentiation of cardinal numbers, smallest transfinite cardinal numbers, ordinal types of simple ordered aggregates and operations on ordinal types. Develops theory of well-ordered aggregates; investigates ordinal numbers of well-ordered aggregates and more.
... Read more

Customer Reviews (2)

Detailed Axiomatic Development of Transfinite Numbers - Not Suitable as Introduction
Georg Cantor's final and logically purified memoir on transfinite numbers was published in the late 1890s.This Dover reprint is the 1915 English translation by the mathematician Philip E. B. Jourdain; it also includes a lengthy, technically diffcult introduction by Jourdain.

Contributions to the Founding of the Theory of Transfinite Numbers is not suitable as an introduction. I unwisely disregarded caution from an earlier reviewer that Cantor's work would not be appropriate for a beginner in set theory. (I thought that I was reasonably acquainted with set theory, but I do admit that I was not a math major.)

The 82-page introduction by Jourdain assumes that the reader is reasonably familiar with the work of key nineteenth century mathematicians.While it is possible to skip the introduction, Jourdain's context setting is quite helpful. Cantor's transfinite numbers are so innovative and so unexpected that it almost seems as though they spring forth in a vacuum, but Jourdain shows that the earlier work of Dirichlet, Cauchy, Riemann, and Weierstrass helped point the way for Cantor.

Cantor's memoir (that is, his two-part discussion of transfinite number theory) comprise the remaining 125 pages. The difficulty with Cantor's axiomatic presentation is two-fold. First, the material itself is not easy - despite Cantor's careful approach. I even bogged down for awhile on his early discussion of the exponentiation of powers and how this leads to aleph-zero. And second, much of his terminology is outdated and unfamiliar. For example, there is no mention of sets, just aggregates and parts. Another example is that Cantor speaks of reciprocal and univocal correspondence. I have yet to complete Cantor's work, but I am continuing to plod along.

A recommendation: A much better starting point for readers new to transfinite numbers is a fascinating book by Mary Tiles, titled The Philosophy of Set Theory - An Historical Introduction to Cantor's Paradise. This work targets mathematics and philosophy majors, but is accessible to others.

An interesting book
There's nothing like reading the original.Here is the abstract theory of transfinite ordinals described by its originator, Georg Cantor.

It's probably not the best introduction to set theory for a beginner.The bookfocuses more on ordinal numbers than on cardinals or general sets.It'snot a great reference, either, since so many important results in settheory have been proven in the 100 years since Cantor.But I like thisbook a lot nonetheless.The exposition is beautiful -- concise, clear, andlogical.It's one of the most nicely presented math books I've read. ... Read more

Product Description
The purpose of this book is to describe the classical problems in additive number theory, and to introduce the circle method and the sieve method, which are the basic analytical and combinatorial tools to attack these problems. This book is intended for students who want to learn additive number theory, not for experts who already know it. The prerequisites for this book are undergraduate courses in number theory and real analysis. ... Read more

Customer Reviews (2)

Advanced graduate level text in additive number theory
Advanced graduate level text in additive number theory, covers the classical bases. This book is the first comprehensive treatment of the subject in 40 years. If the topic of additive number theory interests you, then this is the book to get as there is no comparable (single) book available.Requires a solid understanding of complex analysis.Note, a nice introduction to additive number theory can be found in Hardy and Wright's Introduction to Number Theory.

Some highlights: 1) Chen's theorem that every sufficiently large even integer is the sum of a prime and a number that is either prime or the product of two primes. 2) Brun's sieve for upper bound on the number of twin primes. 3) Vinogradov's simplification of the Hardy, Littlewood, and Ramanujan's circle method.

Clear coverage of Waring's Problem and Goldbach Conjecture
This book gives clear and complete proofs of Waring's Problem (that every positive integer is the sum of a bounded number of nth powers) and of all current results in the Goldbach Conjecture (Brun's theorem that the sum of the reciprocals of the twin primes converges, Vinogradov's theorem that every large odd integer is the sum of three primes, and Chen's theorem that every large even integer is the sum of a prime and and number that is either prime or the product of two primes). The focus of the book is these specific problems; it develops many general methods while attacking these problems, but does not develop the general methods for their own sake. The book assumes a little prior knowledge of analysis and number theory, and it quotes a few advanced results (for example, the Bombieri-Vinogradov theorem on primes in arithmetic progressions) that are proved in Davenport's book "Multiplicative Number Theory", but otherwise it is a complete exposition of these two problems.

The best feature of the book, apart from its complete coverage, is that all the calculations are written out in full; there's no need to keep your pencil and paper handy to check the steps. This is especially valuable in the sieve sections where the combinatorial explosion can be overwhelming.

The exercises are the weak point of the book; most of them are either routine or are omitted steps from proofs, and they don't present much challenge to someone who has already worked through the body of the book. The book's layout and production quality are good. There are only a small number of typographical errors, none confusing. ... Read more

Product Description
Elementary Methods in Number Theory begins with "a first course in number theory" for students with no previous knowledge of the subject. The main topics are divisibility, prime numbers, and congruences. There is also an introduction to Fourier analysis on finite abelian groups, and a discussion on the abc conjecture and its consequences in elementary number theory.

In the second and third parts of the book, deep results in number theory are proved using only elementary methods. Part II is about multiplicative number theory, and includes two of the most famous results in mathematics: the Erdös-Selberg elementary proof of the prime number theorem, and Dirichlet's theorem on primes in arithmetic progressions. Part III is an introduction to three classical topics in additive number theory: Waring's problems for polynomials, Liouville's method to determine the number of representations of an integer as the sum of an even number of squares, and the asymptotics of partition functions.

Melvyn B. Nathanson is Professor of Mathematics at the City University of New York (Lehman College and the Graduate Center). He is the author of the two other graduate texts: Additive Number Theory: The Classical Bases and Additive Number Theory: Inverse Problems and the Geometry of Sumsets. ... Read more

Product Description
This introduction to number theory has been written specifically for mathematics and computing undergraduates. Computer programs in BASIC are accompanied by basic text which explains the subject and demonstrates how computers have opened up new horizons for number theorists. ... Read more

Customer Reviews (1)

A feast for the maths amateur who likes to experiment with a computer
This is an excellent book for the interested amateur mathematician who likes experimenting with numbers using a computer, as well as a useful extra book for the maths student. Everyone reading it can join in the fun of numbers straight away without much maths background.
It is readable, fun, gives lots of BASIC programs which are easily translated into your favourite programming language too, has many exercises which - and this is rare in maths books in my experience - are fun too!It takes things at a nice pace, never going too fast but giving lots of examples and applications and notes.There are many extra sections with details of the lives of the famous mathematicians whose results are presented.
The 'starter' in this feast is divisibility and the main course is the the primes, looking at decimal numbers and repeating fractions along the way. There are side-orders of Pythagoras's theorem and finding right-angled triangles with integer sides, writing a number as a sum of two square numbers, quadratic reciprocity, complex (Gaussian) numbers; the continued fractions section is especially interesting and accessible and ends with a little cryptography (codes).An appendix gives programs for adding, subtracting, multiplying and dividing numbers which can be as large as you like (multiprecision arithmetic).
Altogether a wonderful well-written book to both dip in to as well as to study from cover to cover.
The authors have been (UK) university maths lecturers for many years and their experience shows.Although it was written in 1989, the maths is classic and timeless and the programs still work! ... Read more

Product Description

This edition has been called ‘startlingly up-to-date’, and in this corrected second printing you can be sure that it’s even more contemporaneous. It surveys from a unified point of view both the modern state and the trends of continuing development in various branches of number theory. Illuminated by elementary problems, the central ideas of modern theories are laid bare. Some topics covered include non-Abelian generalizations of class field theory, recursive computability and Diophantine equations, zeta- and L-functions. This substantially revised and expanded new edition contains several new sections, such as Wiles' proof of Fermat's Last Theorem, and relevant techniques coming from a synthesis of various theories.

Customer Reviews (2)

very nice reference book
This is an excellent reference for the researcher in number theory. The audience for this book would consist of professional mathematicians and university libraries.

Wonderful Arithmetic Geometry Book
This Book is a cornerstone in Arithmetic Geometry.

It is the first time in a single Book so different
arguments find a common place.

Let me say that the idea of dividing the work into
three parts,depending on the approach, is entirely
new. In fact,

Part 1 starts with elementary theory & applications(primes,diophantine equations& approx)

Part 2 gives an account of recent ideas and theory
(ch.3:Logic & Recursion, with a sketch of proof of
Matiyasevic's Theorem;ch.4:Algebraic NumberTheory;
ch.5:Arithmetic of Algebraic Varieties;ch.6: deals
with Zeta functions and modular forms;ch.7:gives a
picture, complete indeed, of Wiles'proof of Fermat
Last Theorem)

Part 3 gives "Analogies and Visions",i.e. the link
between numbers fields and function fields(usually
this analogy is only admitted, but never explained
in other books) and other analogies involving many
recent arguments in Arithmetic Geometry (such as :
Schottky uniformization, Arakelov Geometry, Zetas,
Dynamics and Cohomology). ... Read more

Product Description
This is the English translation of the original Japanese book. In this volume, "Fermat's Dream", core theories in modern number theory are introduced. Developments are given in elliptic curves, $p$-adic numbers, the $\zeta$-function, and the number fields. This work presents an elegant perspective on the wonder of numbers. Number Theory 2 on class field theory, and Number Theory 3 on Iwasawa theory and the theory of modular forms, are forthcoming in the series. ... Read more

Customer Reviews (2)

An accessible path from numbers to the sophisticated methods
This brilliant précis of algebraic number theory goes from the simplest basics up to suggestive outline proofs of the most important theorems. It is probably not a good first book on the subject although it is pretty self-contained in principle. You should already be comfortable with groups, rings, and fields.

Its great strength is connecting easily stated questions about Diophantine equations to more sophisticated methods. Chapter 0 quickly relates Pell's equations x^2+Ny^2=1 to square and triangular numbers, and both to the groups of units of algebraic number rings. Chapter 1 on elliptic curves is illustrated throughout by a handful of typical examples, and it proves many steps, but not all, in the Mordell theorem: the rational points on an elliptic curve form a finitely generated Abelian group.

There are many insightful comments on how to think about the ideas overall. These rise to poetry when we learn that the p-adic numbers are like the night sky, beauty obscured when the "sun" of the real numbers blots them out.

The high point, typifying the book's style, is the last chapter explaining how arithmetic in an algebraic number field K is captured by the kernel and cokernel of one group homomorphism (the group map that takes each unit of the field to its principal fractional ideal-kernel is the unit group and cokernel the class group). The remark is lightly made. Examples are given. But in fact the stage is being set for cohomological methods.

All this in 140 pages. Plus there are exercises, and answers to them.

An accessible path from numbers to the sophisticated methods
This brilliant précis of algebraic number theory goes from the simplest basics up to suggestive outline proofs of the most important theorems.It is probably not a good first book on the subject although it is pretty self-contained in principle.You should already be comfortable with groups, rings, and fields.

Its great strength is connecting easily stated questions about Diophantine equations to more sophisticated methods.Chapter 0 quickly relates Pell's equations x^2+Ny^2=1 to square and triangular numbers, and both to the groups of units of algebraic number rings. Chapter 1 on elliptic curves is illustrated throughout by a handful of typical examples, and it proves many steps, but not all, in the Mordell theorem: the rational points on an elliptic curve form a finitely generated Abelian group.

There are many insightful comments on how to think about the ideas overall.These rise to poetry when we learn that the p-adic numbers are like the night sky, beauty obscured when the "sun" of the rational numbers blots them out.

The high point, typifying the book's style, is the last chapter explaining how arithmetic in an algebraic number field K is captured by the kernel and cokernel of one group homomorphism (the group map that takes each unit of the field to its principal fractional ideal-kernel is the unit group and cokernel the class group).The remark is lightly made.Examples are given.But in fact the stage is being set for cohomological methods.

All this in 140 pages.Plus there are exercises, and answers to them.
... Read more

Product Description

Numbers have fascinated people for centuries. They are familiar to everyone, forming a central pillar of our understanding of the world, yet the number system was not presented to us "gift-wrapped" but, rather, was developed over millennia. Today, despite all this development, it remains true that a child may ask a question about numbers that no one can answer. Many unsolved problems surrounding number matters appear as quirky oddities of little account while others are holding up fundamental progress in mainstream mathematics.

Peter Higgins distills centuries of work into one delightful narrative that celebrates the mystery of numbers and explains how different kinds of numbers arose and why they are useful. Full of historical snippets and interesting examples, the book ranges from simple number puzzles and magic tricks, to showing how ideas about numbers relate to real-world problems, such as: How are our bank account details kept secure when shopping over the internet? What are the chances of winning at Russian roulette; or of being dealt a flush in a poker hand?

This fascinating book will inspire and entertain readers across a range of abilities. Easy material is blended with more challenging ideas about infinity and complex numbers, and a final chapter "For Connoisseurs" works through some of the particular claims and examples in the book in mathematical language for those who appreciate a complete explanation.

As our understanding of numbers continues to evolve, this book invites us to rediscover the mystery and beauty of numbers and reminds us that the story of numbers is a tale with a long way to run...

Customer Reviews (2)

Number Story is Wonder Story
This book has nothing wrong with it except a few typos.It's a well-written entry into higher math.Every chapter could be the summary of a PhD thesis.Yet, it's for you and me.I never got further than college statistics, yet, with a willingness to focus, I can roughly follow what Mr. Higgins is talking about.On the other hand, if I really knew my math, I'd be soaring into all sorts of mathematical heights and details.I personally loved the chapter on cryptography.Mr. Higgins tell the story of this fascinating field, which originated from war-time necessity.His simple non-mathematical explanations seduced me into actually delving with him into deeper math.So if you're like me, this chapter alone was worth the read.If you are a real mathematician, oh man, you'll follow all the math, and be so glad you read this book.

Classic
The book is a classic.It is well written.It is a joy to read.Do you know the story of numbers?Read the book, and find out.The author, in a delightful style of writing, explains the story of numbers.Peter Higgins, I feel, can get his information across to the reader so they'll understand it.You'll say to yourself, over and over again, that makes sense, how come in the past no one could explain it to me in this manner?There are professors, then there are Professors, Peter Higgins deserves the title of Professor of Mathematics.

These are my feelings, and I read the book with an open mind.Every now and then I'll re-read a chapter, or I'll dive into the chapter "For Connoisseurs."In the end, it's a classic. ... Read more

Product Description
Careful organization and clear, detailed proofs make this book ideal either for classroom use or as a stimulating series of exercises for mathematically minded individuals. Modern abstract techniques focus on introducing elementary valuation theory, extension of valuations, local and ordinary arithmetic fields, and global, quadratic, and cyclotomic fields.
... Read more

Customer Reviews (2)

Uncompromising and difficult
Make no mistae about it, this is a very difficult book. It is pitched at a highly advanced level and assumes a good deal of ring and field theory and makes no concessions by way of pausing to briefly summarise what is assumed at any point. Parts of the book also progress very quickly and consist of streams of definitions with next to no examples followed by a chunk of theorems that depend on the terms introduced to such an extent that the material has to be re-read many times to glean any understanding.Although in terms of the amount of material covered this is a comprehensive text, it is far too concise for student use. It might have some limited appeal as an advanced postgraduate reference book, but for anyone not already well up to speed in algebraic number theory this will be heavy going indeed.

A high level book
This book seems to be a good description of Algibraic valuations. Potential buyers should know that it assumes what seems to be a knowlage of graduate-level mathematics, particularly a thurogh knowlage of mathematicalfields. ... Read more

Product Description
Now into its Eighth edition, The Higher Arithmetic introduces the classic concepts and theorems of number theory in a way that does not require the reader to have an in-depth knowledge of the theory of numbers The theory of numbers is considered to be the purest branch of pure mathematics and is also one of the most highly active and engaging areas of mathematics today. Since earlier editions, additional material written by J. H. Davenport has been added, on topics such as Wiles' proof of Fermat's Last Theorem, computers & number theory, and primality testing. Written to be accessible to the general reader, this classic book is also ideal for undergraduate courses on number theory, and covers all the necessary material clearly and succinctly. ... Read more

Customer Reviews (5)

some basic material explained
The title leads one to higher expectations than the text
actually delivers?
The book has some basic topics in number theory,
but isn't so new or so well written as one might expect.
That the book has gone past seven printing seems to show that it has been a popular seller?
As for me I don't find myself desperate to buy it.

Disappointing
I've purchased this book based on the rave reviews it's received on Amazon.com, both on this page and elsewhere. I've been greatly disappointed.

This is the eighth edition, and, as such, is low on error count, so if all you're looking for in a math textbook is that it be error-free, this may be the book for you.

If you are looking for a little more than that: say, an interesting, well-motivated and pedagogically sound lecture, you'd be better off looking for it elsewhere, for instance in Jones & Jones' superb "Elementary Number Theory".

"The Higher Arithmetic"'s style of writing is unstructured prose (as opposed to the Definition-Theorem-Proof structure), supposedly rendering the text less rigid and more "friendly", when, in fact, it accomplishes the exact opposite effect: You're never sure where a proof begins and where it ends. This compounds unnecessary intellectual and psychological strains on top of those already naturally present whenever one learns new material.

The unstructured-ness also makes this book quite useless as a work of reference.

The proofs aren't particularly elegant or insightful (in fact, they are quite difficult to follow in some cases, for no good reason).

There's very little in terms of historical background and in terms of interesting applications and recreations.

Finally, the book is uncannily devoid of that geeky sense of humor that embellishes the best of math textbooks (e.g. "in this sense, at least, the prime 2 is very odd!", Jones & Jones, 1998, p. 106).

This book can best be recommended to those who have already studied number theory, and would like a refresher of the main topics an introductory course is likely to include.

P.S.
This review is based on my impressions of the first three chapters (which constitute roughly one third of the book in terms of number of pages). I simply couldn't bear reading any further. I can't preclude the possibility that it gets better down the road.

Dated due to lack of material on modern encryption, still suitable for learning number theory
The higher arithmetic is more commonly known as number theory and is one of the most enjoyable and complex areas of mathematics. Simultaneously simple and hard, the problems are generally easy to understand yet can be horrendously difficult to solve. Furthermore, the initial areas of number theory are easy to comprehend; in general it only takes a basic knowledge of algebra to manage the main points.
In this book, Davenport takes you through the basics of number theory, starting with prime factorization and going through some simple Diophantine equations. The chapter titles are:

*) Factorization and the primes
*) Congruences
*) Quadratic residues
*) Continued fractions
*) Sums of squares
*) Quadratic forms
*) Some Diophantine equations

This book is a solid introduction to number theory and can be understood by the advanced high school student. The primary drawback for the modern reader is that there is no coverage of the use of number theory in modern encryption techniques.

Good book, but if you have the money, there are better
Well, this is definitely a very good introduction to number theory.The author provides clear, readable proofs of all the most basic theorems on topics such as congruences, sums of squares, etc.He explains things quite well.However, despite costing almost 2.5 times as much, I would recommend Hardy and Wright's book An Introduction to the Theory of Numbers more highly than Davenport's book.Seriously, although it may seem good that Davenport doesn't require a knowledge of calculus as a prerequisite for his book (which Hardy DOES require), one probably shouldn't learn number theory until one has a good backrground on topics ranging from improper integrals to infinite series.Because Davenport does not require calculus as a prerequisite, he neglects HUGE aspects of what could actually be considered BASIC number theory: namely, the basic analytic aspects (such as Tchebycheff's results on the Prime Number Theorem) and the additive theory (i.e. partitions and such, as well as the basics of the generalized theory surrounding Waring's problem for high powers of integers).So, my recommendation is, wait until you know integral calculus and the theory of infinite series BEFORE buying a book on number theory, and then buy Hardy and Wright's book rather than this one.

This is a MUST BUY if you want to learn Number Theory!
This book is an AMAZING introduction to the Theory of Numbers. It assumes no previous exposure to the subject, or any technical mathematical knowledge for that matter. Its prose is lucid and the style appealing.Davenport chose NOT to write a lemma-theorem-proof kind of book, and theresult is a marvelous, eminently readable introduction to the subject. Itswonderful to read a book where good prose is used to appropiatelysubstitute a massive collection of uninviting symbols. I've also beenreading other books on Number Theory, such as Hardy & Wright, but noneare as clear as this one.

I found the chapter on quadratic residues(which includes the reciprocity law) to be especially well written. Thesection on computers and number theory is excelent as well. A concise andcoherent discussion of crytography and the RSA system is included here.The organization of the book's chapters is fantastic. Each chapter buildsup on results proven in the previous ones, showing well the connectionsbetween the different aspects of Number Theory. The exercises of the bookrange from simple to challenging, but are all accesible to someone willingto put effort into them.

This would be an excelent source for learningnumber theory for mathematical competition purposes, such as the ASHME,AIME, USAMO, and even for the International Mathematical Olympiad. The bookcontains much more than what is needed for these competitions, but theolympiad/contest reader will benefit greatly from a study of Davenport'swork.

The book can certainly be used for an undergraduate course inNumber Theory, though it might need supplementary materials, to cover asemester's worth of work. I know the book has been used in the past inprevious editions as the main text for Math 124: Number Theory at HarvardUniversity.

I would also recommend this book to anyone interested inacquanting themselves with Number Theory.

Awesome!There is simply noother word that describes The Higher Arithmetic. ... Read more

Product Description
The purpose of this book is to introduce the reader to arithmetic topics, both ancient and modern, that have been at the center of interest in applications of number theory, particularly in cryptography. No background in algebra or number theory is assumed, and the book begins with a discussion of the basic number theory that is needed. The approach taken is algorithmic, emphasizing estimates of the efficiency of the techniques that arise from the theory.A special feature is the inclusion of recent application of the theory of elliptic curves. Extensive exercises and careful answers have been included in all of the chapters. Because number theory and cryptography are fast-moving fields, this new edition contains substantial revisions and updated references. ... Read more

Customer Reviews (14)

Koblitz'sCourse in Number Theory and Cryptography
This book is a real gem - very clearly written and covering the subject matter concisely but comprehensively.Particularly welcome are the exercises which are ingenious and extend the subject matter rather than just test knowledge of the chapter.It is extremely helpful too (and rare in a graduate text book) that solutions to all of the problems are provided at the back of the book.Exceptionally, and again very helpfully, there seem to be zero errors/typos in the text.

Strongly recommended as the best introduction to this fascinating and important field

Essential for your secrets
How Neal Koblitz manages to squeeze the amount of material he presents into this slim volume is a miracle of nature. It even includes what most authors of graduate works leave out as a matter of course: answers to exercises. More amazing still is that far from being terse and unreadable the text is a delight.

My advice to anyone interested in this field is to have this book by their side at all times. Then if the need arises to find out what makes an algorithm tick or to refresh one's mind about a well known concept it's just the flick of a page away.

Pleasant introduction to cryptography
Chapters 1 and 2 give some elementary background material on number theory and finite fields. Chapter 3 discusses some old and naive cryptosystems. Chapter 4 discusses public key cryptosystems. In the RSA system, the receiver chooses two large primes p,q and makes public their product pq=n and some integer e relatively prime to phi(n). The sender then sends his message to the power e reduced mod n. To invert this operation one must know phi(n), i.e. one must know the factorisation n=pq. Since factoring big numbers is hard, only the intended receiver will be able to decipher the message instantly. RSA thus uses the fact that multiplying is easy but inverting it is hard; similarly, one can employ other such "trapdoor functions", such as exponentiation in Z/nZ, to create other public key cryptosystems. In chapter 5 we look at various algorithms and tricks for factorisation and primality testing. As for the cryptosystems, classical number theory that is hundreds of years old still provides the best tools (modulo arithmetic, quadratic residues, continued fractions, etc.), and in chapter 6 we see how another classical theory--elliptic curves--also proves to be fruitful in cryptography. The points of an elliptic curve over a finite field form a finite group, which we can use as the basis for new cryptosystems, analogous to how we made cryptosystems out of Z/nZ for instance. And starting with an integer and constructing corresponding finite field elliptic curves we can employ these groups and elliptic curve techniques to give improved algorithms for primality testing and factorisation.

Outstanding presentations
This book is an outstanding introduction to cryptographic techniques and algorithms Although it's labelled as a "graduate text in mathematics", most of it should be accessible to anyone who knows a little linear algebra. For readers just interested in the how-to of the algorithms, not even that is needed. Koblitz does a thorough job of leading up to each algorithm and proving its formal properties. He also presents the algorithms themselves, unencumbered by denser material of interest to mathematicians.

The book covers a variety of topics - public-key encryption, primality testing, factoring, and cryptographic protocols. It introduces zero-knowledge proofs and blind transfer, techniques that offer real hope of personal privacy in a world where data transfer is mandatory. I was a little disappointed by the chapters on elliptic cryptography, however. I hoped that Koblitz would bring is explanatory powers to bear on the algorithms. Somehow, I never quite connected with his descriptions of elliptic curves - perhaps I'm just thick, or perhaps a bit more introductory material would have helped.

The rest of the book is a very fine example of clear, readable math writing. Its clarity its range of topics earn it a place with anyone interested in cryptography, factoring, and prime numbers.

Excellent book for self study
This is an excellent book fot those, who are interested in the theoretical background of cryptography. It was also my first book in number theory, and I had no trouble following most of the text ( except the chapter on Elliptic curves, which -as I realize now- IS difficult)

Highly recommendable! A pleasant surprise is, that there are virtually no typos. ... Read more

Product Description
This book contains lectures presented by Hugh L. Montgomery at the NSF-CBMS Regional Conference held at Kansas State University in May 1990. The book focuses on important topics in analytic number theory that involve ideas from harmonic analysis. One valuable aspect of the book is that it collects material that was either unpublished or that had appeared only in the research literature. This book would be an excellent resource for harmonic analysts interested in moving into research in analytic number theory. In addition, it is suitable as a textbook in an advanced graduate topics course in number theory. ... Read more

Customer Reviews (3)

old-fashioned material but still very attractive
I bought this book in 1986 and used it since then on a number of occasions to set up elementary problems on number theory for my students; although you will find nothing on RSA algorithms (for example...) inside (book was written in 1968), this book covers a lot of ground in (not so) elementary number theory; it is well organised and I found it very nice to read and work with. The last parts cover Dirichlet's theorem and the prime-number theorem among others (uniform distribution modulo 1 and Kronecker theorem on irrational numbers).

this must be chandrasekharan's book
covers classical material in analytic number theory; begins at an elementary level which means that there is much to learn in it...

Please do not buy this book. It is too expensive!!!!
This book consists of only about 150 more pages, and the contents
it present just ordinary and can be easily found in other textbooks which is much cheaper!!! I only recommend for those people who think they are rich!! There are many other choices!!!! ... Read more

Product Description
This accessible, highly regarded volume teaches thetheory of numbers. It incorporates especially complete and detailedarguments, illustrating definitions, theorems, and subtleties of proofwith explicit numerical examples whenever possible. The author hasorganized the results and constructed the arguments in such a way asto reveal the essential structure of the subject and to impart anunderstanding of the various methods of proof as methods rather thantricks. Hundreds of exercises, including computer-oriented problems,are included. ... Read more

Customer Reviews (1)

Good Book for Self Study
This is an intentionally small book that starts at a level somewhat below the usual introductory text. It presumes no background other than the equivalent of a course in college algebra. (Two brief sections in Chapter 3 do assume a knowledge of the Calculus but can be omitted if necessary). Since learning mathematics involves doing mathematics there are over 600 problems with detailed answers to selected exercises, including many proofs.

This book includes fewer topics than the typical introductiory text but the selection of material is excellent. Chapter 1 starts with a detailed treatment of the postulates of mathematical induction and well ordering. Congruences and the quadratic reciprocity law of Gauss are nicely covered. The Chinese Remainder Theorem gets a few pages of its own. Chapter 9 provides a brief look at simple continued fractions. Plus many others: The Euclidian Algorithm, the Fundamental Theorem of Arithmetic, the Prime Number Theorem, etc.

After you finish this book you will understand the essential methods of proofs and will be ready for introductory texts which take a more advanced approach to Number Theory. It's also a fun book to study. ... Read more

Product Description

Customer Reviews (4)

A Math book accessible to all users!
The name of the book might look pure mathematics but if one looks throughout the book he will easily notice mystic joyfulness of mathematics blended with the deep understanding of authors on the subject makes the content available to users.

Based on my impression from the book (see below for detail), I found this book readable and enjoyable separated from reader background in mathematics. Mainly what you get from the book does not depend on what you know but depends on what you want.

I divide my review into two parts, first the language of this book and second the content of the book.

There are not too many books in mathematics useable for other students, the main barrier in reading Mathematical books for non-mathematicians is strange notations and definitions but I, as a physics student, found this book very readable far from mathematical obscurity. Furthermore, very good details in each part make it smoothly understandable through the book.

Despite quite broad range of the content from elementary number theory to advanced topics in statistics, very interesting and intuitive examples and applications make all steps clear for all type of readers. Specifically, the forth part of the book is quite applicable for all science and engineering students and researchers.

A Serious Math Book for Serious Students
Reading a math book is usually not easy.There can be two reasons for this.The first is poor writing or lack of motivation.This book does not suffer from this defect; the prose is polished and minimalist, and keeps the reader focused on the math.This style of exposition, which is called by another reviewer "machine gun" is called by practicing mathematicians as "tight"; the mathematics is allowed to speak for itself, and the viewpoint, the voice, of the expositors is expressed clearly through the mathematics.The second reason is that the material is inherently deep and difficult and cannot be merely "learned" from a book; it must be experienced first-hand, through problem solving and deep reflection.This book is full of deep material, presented in great detail so that the reader can appreciate the nuances of the methods presented and apply them as a practicing mathematician.The topics are specially chosen, as the book says, to invite the reader to pursue some threads in modern analytic number theory.There are also copious, well-thought-out exercises to help the reader gain the experience with the material which cannot be achieved simply through reading.One must only read the beautiful proof of Roth's theorem (Chapter 6) to see the quality of exposition, excellent choice of topics and the attention to detail that makes this book an excellent place to start for a serious student who wants to understand modern analytic number theory, and a jumping point to advanced books and research monographs on diophantine geometry and analytic number theory.

Few books engage the reader to the point of experiencing sheer delight, and this is one of them.
Most of the books on analytic number theory are very good, and so is this one. Yet there is more to it than being a good book.
The exposition is brilliant, rigorous, well paced, absolutely non-flippant and elegant (it feels like I am reading a latexed version of a G.H. Hardy book).

It is highly innovative since it has material that you normally do not expect to find in one single book. There are books about probabilistic number theory, but those books are devoted wholly to that subject. Same thing goes for random matrix theory. But the most surprising case is that of continued fractions. Books on continued fractions are *generally* elementary and not very long.

No single book out there combines introductions to the interactions of probability and random matrix theory with number theory in addition to treating more standard subjects (cryptography, group theory, continued fractions, circle method, L-functions, ...) exquisitely. This has been wonderfully achieved by Miller and Takloo-Bighash.

All in all, the flavour of this book is best summarized with the word: modern.

This book is not a popular math book. Yet not quite a textbook either, it is, as the title suggests, an invitation. And a serious invitation, for that matter. A little effort will be needed, however I have found out that whatever amount of effort you invest in it will be rewarded with interests!
I would say that courses in group theory, elementary number theory and complex analysis would constitute an adequate background.

It need not be read linearly, which is also a bonus. Within reason, you could move on to whichever subjects you find more interesting.
(By the way, the introductions to Fourier analysis and probability theory put certain 'methods' books to shame!)

My favourite parts are the summary of the status of the Riemann Hypothesis. It made me interested in finding out more about the Xian-Jin Li's criterion, Hardy's Theorem, the whole function-theoretic aspect of the book, and the connection between number theory and physics.

Finally, financially speaking, it is *very* well priced for a hardcover.

dense, hurried, sketchy and most unsatisfying
here is yet another dense math book unreadable to everybody
except those already familiar with the material-- why the
authors opted for this broad and sketchy treatment is unclear ---
introducing more topics than a russian novel has characters,
the book is more likely to lead the reader to a state of exhaustion
rather than enlightenment -- one has to ask who could possibly benefit

from this machine gun style exposition as there are more comprehensive
and gentle surveys available of each of the many subjects they touch on --
answer = no one -- ps it is an abomination to compare this unwieldy mess
to a pedagogical masterpiece like Hardy and Wright -- don't believe their
lies ... Read more

Product Description
This handbook covers a wealth of topics from number theory, special attention being given to estimates and inequalities. As a rule, the most important results are presented, together with their refinements, extensions or generalisations. These may be applied to other aspects of number theory, or to a wide range of mathematical disciplines. Cross-references provide new insight into fundamental research.

Audience: This is an indispensable reference work for specialists in number theory and other mathematicians who need access to some of these results in their own fields of research. ... Read more

Product Description
Number theory is concerned with the properties of the natural numbers: 1,2,3,.... During the seventeenth and eighteenth centuries, number theory became established through the work of Fermat, Euler and Gauss. With the hand calculators and computers of today, the results of extensive numerical work are instantly available and mathematicians may traverse the road leading to their discoveries with comparative ease. Now in its second edition, this book consists of a sequence of exercises that will lead readers from quite simple number work to the point where they can prove algebraically the classical results of elementary number theory for themselves. A modern high school course in mathematics is sufficient background for the whole book which, as a whole, is designed to be used as an undergraduate course in number theory to be pursued by independent study without supporting lectures. ... Read more

Customer Reviews (1)

Burn uses problems to introduce number theory ideas.
This book is a carefully sequenced set of problems along with answers and a few comments. Burn uses those problems to introduce important number theory ideas. I enjoyed working through the problems to learn more aboutnumber theory. Most problems are accessible to those with a good highschool mathematics background. ... Read more

Product Description
This book is an introduction to algebraic number theory via the famous problem of "Fermat's Last Theorem." The exposition follows the historical development of the problem, beginning with the work of Fermat and ending with Kummer's theory of "ideal" factorization, by means of which the theorem is proved for all prime exponents less than 37. The more elementary topics, such as Euler's proof of the impossibilty of x+y=z, are treated in an elementary 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 Kummer's ideal theory to quadratic integers and relates this theory to Gauss' theory of binary quadratic forms, an interesting and important connection that is not explored in any other book. ... Read more

Customer Reviews (3)

Old school algebraic number theory with heavy Kummer bias
Algebraic number theory eventually metamorphosed into a sub-discipline of modern algebra, which makes a genetic approach both pointless and very interesting at the same time. Edwards makes the bold choice of dealing almost exclusively with Kummer and stopping before Dedekind. Kummer's theory is introduced by focusing on Fermat's Last Theorem. As Edwards confirms, this cross-section of history is on the whole artificial--Fermat's Last Theorem was never the main driving force; not for Kummer, nor for anyone else--but it fits its purpose quite well, and besides, Edwards only adheres to it for about half the book. Kummer-Edwards's style has a heavily computational emphasis. Edwards defends this aspect fiercely. Perhaps feeling that the authority of Kummer is not enough to convince us of the virtues of excessive computations, Edwards trumps us with a Gauss quotation (p. 81) and we must throw up our hands.

Chapter 1 surveys Fermat's number theory. Chapter 2 deals with Euler's proof of the n=3 case of Fermat's Last Theorem, which is (erroneously) based on unique factorisation in Z[sqrt(-3)] and thus contains the fundamental idea of algebraic number theory. Still, progress towards Fermat's Last Theorem during the next ninety years is quite pitiful (chapter 3). The stage is set for our hero: Kummer, who developed a theory of factorisation for cyclotomic integers. One may of course not trust unique factorisation to hold here, but Kummer has a marvellous idea: the concept of "ideal" prime factors--curious ghost entities that save unique factorisation in many cases (chapter 4); enough to prove Fermat's Last Theorem for "regular" prime exponents (chapter 5). Telling whether a given prime is regular involves computing the corresponding class number, which is done analytically by means of an appropriate analog of the zeta function (chapter 6). Now, for all of this there is an analogous theory with quadratic integers in place of cyclotomic integers (cf. Euler above). Since it was not important for Fermat's Last Theorem, Edwards skipped past it before, but now we plunge into this theory and the allied theory of quadratic forms (chapters 7-9) to see how Kummer's theory helps elucidate some aspects of it, especially Gauss's notoriously complicated theory of quadratic forms.

great book
This is a great book.If you want to learn algebraic number theory from a very example/computational oriented book, then this is the book you want.it really has a lot of stuff in it.all other graduate books are theory without examples or motivation.this book is the exact opposite.the only drawback is that it doesn't use any modern algebra, but you can figure out how to shorten the arguments with algebra if you wanted to.

Read this if you're seriously interested in math.
There was a great burst of excitement, and several popular books, when Andrew Wiles proved "Fermat's last theorem". The popular books are fine, but they don't address the deepest issue: among all the many long-standing unsolved problems in number theory that are easy to state but resistant to solution, why did "Fermat's last theorem" attract the efforts of so many top-flight mathematicians: Euler, Sophie Germain, Kummer, and many others? The problem itself has no useful application or extension, and as stated seems like just another piece of obstinate trivia. So why is it mathematically interesting?

The answer, of course, is that attacks on the problem revealed deep and important connections between elementary number theory and various other branches of mathematics, such as the theory of rings. Thus, as so often in mathematics, the importance of the problem lies in where it leads the mind, rather than in the problem itself. Harold M. Edwards' book

is a minor classic of exposition, showing how the instincts of top-flight research mathematicians lead them to fruitful work from a seemingly unimportant starting point. I'm only sorry that Professor Edwards seems never to have completed the second volume he had hoped to write.

Thus book deserves to be read by a much larger audience than it has gotten; in particular, I believe every graduate student in math who hopes to do good research, regardless of specialty, would benefit from reading it. Beyond that, any mathematically inclined reader with a modicum of training in math, is likely to find this a fascinating book. ... Read more

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