Book Description
The Fifth Edition of one of the standard works on number theory, written by internationally-recognized mathematicians. Chapters are relatively self-contained for greater flexibility. New features include expanded treatment of the binomial theorem, techniques of numerical calculation and a section on public key cryptography. Contains an outstanding set of problems.
Customer Reviews:
good book.......2002-01-09
This book (5th edition) cover the topics of undergraduate number theory well. The chapters are -
(1)divisibility
(2)congruences
(3)quadratic reciprocity and quadratic forms
(4)some funtions of number theory
(5)some diophantine equations
(6)farey fractions and irrational numbers
(7)simple continued fractions
(8)prime estimates and multiplicative number theory
(9)algebraic numbers
(10)partition funtion
(11)density of sequences of integers.
It also contains basic cryptography, basic group theory and basic elliptical curves in some of the chapters. The authors give notes on the end of each chapter about some research results, which I enjoy reading.
However, the author give too much hints spoling the fun of solving the problems. Eg 32-36, 40-3, 59-53, 108-36, 136-17, 312-8, and most of the problems in chapter 8. The author should put these hints at the back of the book. I suggest you look up IMO (imo.math.ca) for problems suitable for chapter 1-7 because IMO is well-knowned for its excellent number theory problems (especially 1990-3).
Overall this is an excellent book. I give it a rating of 4.5/5, I don't give it 5 because of the author give too much hints to problems instead of putting hints at back of the book.
Comprehensive.......2000-12-23
This is a fantastic book on number theory. It covers far more ground than most introductory text (comparable to Hardy and Wright in depth with much less concern for the big O). It covers material usually only available in separate texts: Rational points on elliptic curves, the partition function, and Dirchlet series. Quite readable chapters, well motivated theoretically, although the historic motivation for the subject matter comes largely in the end-of-the-chapter notes. It's an excellent refresher and reference for non-specialist who find themselves using an algorithm or formula they've forgotten (number theory now playing a role in physics and CS, like never before). It is well cross-referenced with regards to methods of proofs the can be accomplished in different section by different methods - this again making it an excellent reference.
Alas, it is pre-FLT. So you'll have to look elsewhere for that.
The best intro to the subject!.......2000-09-08
I have started my studies in Number Theory reading this book from the preface to the last word. It is amazing! I think it is a better introduction to the subject than the classical Hardy and Wright...it is "more objective" and almost 100% elementary...a good high school reader could do well with it. The chapter of diophantine equations has some divine proofs, very clever and very beautiful. And there is an easy proof of the irracionality of Pi. The only negative point is the existence of some points where the authors could be less concise and a bit clearer, stating the theorems before giving the demonstrations, instead of saying at the end of the paragraph "we then have proved the theorem of..." Its a good book for self-study. It has many exercises.
I've found a marvellous proof..........1999-11-23
It's a excellent book. Guide you through the simplest proofs until the great ones. If you can follow the book since start until end you'll be prepared for beginning research in this incredible world.
Average customer rating:
- no complaints
- Intermediate Level
- Short and to the point
- Very well written book
- Now I know how beautiful proofs can be
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An Introduction to Mathematical Reasoning: Numbers, Sets and Functions
Peter J. Eccles
Manufacturer: Cambridge University Press
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ASIN: 0521597188 |
Book Description
This book eases students into the rigors of university mathematics. The emphasis is on understanding and constructing proofs and writing clear mathematics. The author achieves this by exploring set theory, combinatorics, and number theory, topics that include many fundamental ideas and may not be a part of a young mathematician's toolkit. This material illustrates how familiar ideas can be formulated rigorously, provides examples demonstrating a wide range of basic methods of proof, and includes some of the all-time-great classic proofs. The book presents mathematics as a continually developing subject. Material meeting the needs of readers from a wide range of backgrounds is included. The over 250 problems include questions to interest and challenge the most able student but also plenty of routine exercises to help familiarize the reader with the basic ideas.
Download Description
This book eases students into the rigors of university mathematics. The emphasis is on understanding and constructing proofs and writing clear mathematics. The author achieves this by exploring set theory, combinatorics, and number theory, topics that include many fundamental ideas and may not be a part of a young mathematician's toolkit. This material illustrates how familiar ideas can be formulated rigorously, provides examples demonstrating a wide range of basic methods of proof, and includes some of the all-time-great classic proofs. The book presents mathematics as a continually developing subject. Material meeting the needs of readers from a wide range of backgrounds is included. The over 250 problems include questions to interest and challenge the most able student but also plenty of routine exercises to help familiarize the reader with the basic ideas.
Customer Reviews:
no complaints.......2007-09-30
In less than a week I had the book and I'm happy with my purchase
Intermediate Level.......2007-08-21
I'm biased. This is the sort of thing U.S. schools should require. Can't go wrong with this. This has 2X as much as "How to Read @ Do Proofs" (Daniel Solow). Wich is better? No Answr. I Luv em both. Former is more expository the other more mechanical. Go For It! Recomended.
Short and to the point.......2006-04-11
This book is excellent! It chapters are broken down into short sections and the content in each section is to the point! I also bought the book Proofs and Fundamentals by Ethan D. Bloch but found it to be long and drawn out. If you liked The Nuts and Bolts of Proofs by Antonella Cupillari then this book is for you!
Very well written book.......2004-09-30
I have a mathematics degree. Like most math majors, I struggled with proofs all through college. This book really has help me understand the art of writing proofs. The book is very well written and easy to read. This is just an awesome book!!!
Now I know how beautiful proofs can be.......2002-12-13
This book provides a nice introduction to mathematical reasoning and proofs. My intention on purchasing this book was to learn how to perform mathematical proofs. I believe it has achieved that purpose. The text is easy to follow and the author presents the work clearly.
Book Description
Bridging the gap between elementary number theory and the systematic study of advanced topics,
A Classical
Introduction to Modern Number Theory is a well-developed and accessible text that requires only a familiarity with basic abstract algebra. Historical development is stressed throughout, along with wide-ranging coverage of significant results with comparatively elementary proofs, some of them new. An extensive bibliography and many challenging exercises are also included. This second edition has been corrected and contains two new chapters which provide a complete proof of the Mordell-Weil theorem for elliptic curves over the rational numbers, and an overview of recent progress on the arithmetic of elliptic curves.
Customer Reviews:
Best book on the subject.......2005-05-16
I am currently finishing my third year of undergraduate math at Brown University, and have just completed a course that used this particular book. I have to say it's the most WELL WRITTEN math book I've ever read, and I've read many, many math books by now (more than I'm willing to count as I'm typing this). Professor Rosen (and Ken Ireland, God rest his soul) have made a book that has both fun and interesting problems as well as clear explanations of proofs in the text. It does of course require that you know the basics of abstract algebra (in particular, one is expected to know that "1" is a unit and therefore cannot be prime, so of course when we discuss problems involving factorization into primes, one will of course ignore the number 1). One is also expected to know the basics of formal logic (i.e. understanding how a proof by induction works, how a proof by contradiction works, and knowing that any proper subset of the natural numbers will have a least element), and I choose to point this out simply because MrBigBeast's review makes it obvious that all these facts were not understood. Despite the fairly large amount of assumed knowledge (this is a book intended for advanced undergrads and first year grad students, afterall), this book takes one on an amazing adventure through the depths of elementary number theory, as well as introduces you to very advanced topics in both algebraic and analytic number theory (ever want to know about Zeta Functions? This book treats the topic quite nicely, making a fairly difficult concept accessible). Truly a gem of a book and worth buying even if you never use it for a course.
Great Book.......2005-05-15
I'm currently an undergrad math and phsyics major at Brown, and I loved this book. Rosen is a great teacher and a great writer. As per the post below mine, the submitter is being overly nitpicky. If a reader cannot realize that unique factorization of Z+ extends to Z or understand immediately the nature of "1", then perhaps the reader shouldn't be trying to learn advanced number thoery. As per using the conclusion in the proof, it's called proof by induction. It's easy and trivial enough that I'm sure they didn't want to waste the readers time going through the incredibly obviouse steps.
The book is great. The problems are fun and interesting, and the book gradually generalizes which makes the abstraction easier to conceptualize. If you need something with tons of really baisc excersizes and proofs that will walk you through every step of the way, no matter how small, then this book may not be for you. But if you are a seriouse student looking for an interesting and insightfull introduction to the subject, I highly recomend this book
Covers many important areas.......2003-12-21
I have devoted a good portion of my life to the study of mathematics in general, especially algebra and number theory. This book is an extraordinary reference to many areas of number theory and extremely approachable. The book can be studied on its own or as a companion piece to more specialized texts such as Marcus's Number Fields.
Simply Amazing.......2003-05-26
I picked up this book as a junior in college and was simply stunned. The flow of ideas is so natural that there are times when you can even read the book like a novel. The exposition is clean, and the proofs are elegant.
However, keep in mind that this book IS a GTM. Hence, it requires pre-requisites by way of approximately a year of abstract algebra. As the author says in the preface, it's possible to read a the first 11 chapters without it. However, to appreciate the beauty of the theory, I would sincerely recommend algebra as pre-req.
The first 12 chapters can be considered 'elementary' (not easy, just fundamental). The others are specialized algebraic topics. For instance, the chapter on elliptic curves is useful to get a flavor of the subject. However, it includes very few proofs.
A Modern Classic.......1999-12-06
If ever there was a textbook of which one could say that it was a thing of beauty, this has to be it. The book is very clearly written, and it is readily accessible even to those without a deep understanding of algebra or analysis; despite this, it manages to touch upon a great deal of relatively sophisticated material, and in a way that makes clear the links between the problems of the past and those of the present. I'd imagine that the book would constitute an essential item of reference for anyone with more than a passing interest in number theory.
Amazon.com
This book is a rigorous exposition of formal languages and models of computation, with an introduction to computational complexity. The authors present the theory in a concise and straightforward manner, with an eye out for the practical applications. Exercises at the end of each chapter, including some that have been solved, help readers confirm and enhance their understanding of the material. This book is appropriate for upper-level computer science undergraduates who are comfortable with mathematical arguments.
Customer Reviews:
A Butchered Classic.......2007-09-28
I've heard that the first edition of this book is a classic. Reading the second edition, I can kind of see that -- occasionally there will be a stretch of 5 pages or so that is wonderfully clear, concise, and informative.
But overall, this edition is a disappointment. The explanations tend to be mechanical and unhelpful, and are sometimes confused or just incorrect. New sections on mathematical foundations and applications have been added, but there isn't really adequate space devoted to covering either topic, and the results are so rushed and lacking in context that I can't see those sections being useful to anyone who would need them in the first place. Finally, this edition needs to be proofread for correctness! It contains numerous mistakes, some of them in the presentations of key proofs.
Updated Classic Text.......2007-08-29
The previous edition of this text was published in the late 70's (1979), and it was still in use today in many schools and Universities across the world. For good reason too, the authors of this text really nail down the concept of computability as we understand it today. It is very difficult to find an undergraduate curriculum that does not include a course in Computability or theory of computation, and that is certainly a change from a couple of decades ago where this type of study was left to the Graduate level curricula. What this means to the reader is that one can not be a Computer Scientist without understanding the concepts and theory behind what computability really means.
Things like Context Free languages and grammar are used readily in things like XML and its accompanying standards such as the DTD. So, it makes sense to update a classic text to include such topics and further illustrate to the reader that what once was a theory is now center stage of Computer Science and the IT industry as a whole.
The text starts with the classics such as an introduction to automata theory followed by languages. The authors have taken a more relaxed approach to the topics as the proofs are less formal and easier to follow. Plain text is usually used to informally proof the topic at hand, and the authors go into a more formal approach on selected proofs. This is definitely a better approach than the other texts in the same topic that proofs are center stage of the discussion and the reader gets lost early on in the process. The text is easy to read for students, and easy to explain for the instructors. I remember when I took theory of Computation for my graduate work proofs were so convoluted and difficult to read that I had to spend many of nights trying to understand what the instructor was talking about in the class.
As one would expect, the book then goes into Turning Theory and Machine with the concept to computability and complexity. Well, the good news is that the authors' approach to the topic does not change; lots of explaining of the basics followed by a more detailed formal approach to the topic. All I need to say is that I wish my text was this reader friendly! Chapter 8, Introduction to Turing Machines, sets the ground work for the rest of the text. It explains reducibility and more importantly how to reduce a problem, something I have never seen in any other text in such detail! Automata and its relation to Turing Machine is depicted in detail, so there is no gap between the topics. What is interesting is that the authors close the loop with actually talking about, for example the Halting problem, in the real world with a program.
As one would expect, different classes of problems are explored in detail with many examples (theory and real-world examples) that accompany the topic at hand. Each chapter ends with a summary of topics discussed followed by a set of exercises. There are also a number of exercises at the end of each section in a given chapter in order to reel-in the topic for the reader.
All and all, this is one great text on automata and computation theory. It is easy to read and follow for the students without the loss of content. The authors relate abstract concepts to real-world examples to further illustrate the importance of the topic at hand.
Good, but just it.......2007-06-27
A good book, but just it.
It's like a normal book. It's not bad but not excellent...
Automata theory. The heart of Computer Science.......2007-04-06
Excellent book. Nothing to say for this one.
Eh... Whatever..........2007-01-21
Uhm... I had to buy this book because it was a required text for a required course. Who would buy a book like this otherwise? Duh!
Book Description
From the reviews: "The 2nd (slightly enlarged) edition of the van Lint's book is a short, concise, mathematically rigorous introduction to the subject. Basic notions and ideas are clearly presented from the mathematician's point of view and illustrated on various special classes of codes...This nice book is a must for every mathematician wishing to introduce himself to the algebraic theory of coding." European Mathematical Society Newsletter, 1993 "Despite the existence of so many other books on coding theory, this present volume will continue to hold its place as one of the standard texts...." The Mathematical Gazette, 1993
Customer Reviews:
Excellent book from mathematical standpoint.......2005-02-20
Very good intro textbook. It gives short, detailed preps to various coding areas (linear, cyclic, convolutional). The biggest advantage this book has is that it does not throw at You tonnes of unnecessary info (like many other thick books do). That is, it assumes reader has some basic understanding of algebra and probability theory. Let's say, it gives good theoretical presentation such that the reader gets good theoretical understanding, it is not example-based.
Book Description
This accessible, highly regarded volume teaches the theory of numbers. It incorporates especially complete and detailed arguments, illustrating definitions, theorems, and subtleties of proof with explicit numerical examples whenever possible. The author has organized the results and constructed the arguments in such a way as to reveal the essential structure of the subject and to impart an understanding of the various methods of proof as methods rather than tricks. Hundreds of exercises, including computer-oriented problems, are included.
Customer Reviews:
Good Book for Self Study.......2001-03-05
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.
Book Description
This is the fifth edition of a work (first published in 1938) which has become the standard introduction to the subject. The book has grown out of lectures delivered by the authors at Oxford, Cambridge, Aberdeen, and other universities. It is neither a systematic treatise on the theory of numbers nor a 'popular' book for non-mathematical readers. It contains short accounts of the elements of many different sides of the theory, not usually combined in a single volume; and, although it is written for mathematicians, the range of mathematical knowledge presupposed is not greater than that of an intelligent first-year student. In this edition the main changes are in the notes at the end of each chapter; Sir Edward Wright seeks to provide up-to-date references for the reader who wishes to pursue a particular topic further and to present, both in the notes and in the text, a reasonably accurate account of the present state of knowledge.
Customer Reviews:
Nice intro to number theory.......2007-03-13
This is an unusual number theory book in that it covers topics of interest to the authors which are not often found in the "standard" introductory treatment. My only mild complaints are: no subject index and some ambiguous and unusual notation here and there.
I agree that this book should be in the library of anyone serious about the topic, however, if you are beginning your study of number theory from scratch there are other books that may provide a better start. I would recommend Joe Roberts "Elementary Number Theory: A Problem Oriented Approach" and/or "An Introduction to the Theory of Numbers" by Niven, Zuckerman, and Montgomery.
Roberts offers a wide spectrum of problems, with detailed solutions, written along the lines of Polya & Szego's "Problems and Theorems in Analysis I & II". Nivens book is a solid traditional introduction.
It is fun to read Hardy and Wright though, it exhibits a style that is sadly missing today.
I have to say in closing that it would be good to ignore some of the previous reviews, specifically ones making reference to "idiots". They're unproductive, miss the point of reviewing, and exhibit a level of ignorance which Mark Twain identified years ago: "It is better to keep your mouth shut and appear stupid than to open it and remove all doubt."
Superb Introduction for the Mathematical Sophisticate.......2006-08-08
This classic deserves its reputation but be warned that it is not an introduction for mathematical neophytes. The authors take detours (which sometimes are looks ahead) from the main path of development that the sophisticate will enjoy but the novice may not be able to recognize as detours. Examples are the geometry of numbers (introduced in chapter 3), the Farey dissection of the continuum, and trigonometric sums.
The authors also present deeper material than is usually considered an introduction. Their presentations are excellent but require sophistication for the following topics among others: quadratic fields, generating functions of arithmetical functions, Selberg's proof of the Prime Number Theorem, and Kronecker's theorem.
This is a book to buy and keep provided you have the necessary mathematical sophistication.
Final note: this book nicely complements Apostol's Introduction to Analytic Number Theory.
One of the greatest.......2005-01-10
First of all, let me say this about the one star review. Do not let yourself be infuenced by lesser mathematicians. Idiots in my opinion. To give this book one star, you must posses some special kind of mediocracy. Keep your stupidity to yourself Lucas.
No one writes like this anymore. Mathematicians like Hardy have passed. The subject has ballooned, and now you have to specialize within Number Theory. There are fewer and fewer that can posses knowledge of the entire subject of Number Theory. Remember what Harold M. Edwards said. You have to read the classics, and beware of secondary sources. Authors give their own spin on ideas. And who is to say they have a greater or lesser understanding of the subject. Furthermore, who can determine how well can they express themselves. How many mathematicians our days bother to study grammar and literature? The best example is Gauss' Disquisitiones Arithmeticae. Would you rather read a book written by Gauss himself, the man that established the subject? Or by some one who learned what some one learned what some one learned over a period of 200 years? Also know what Axler, author of Linear Algebra Done Right, said about reading mathematics books. For a mathematics book, if you spend less than half an hour per page you are going too fast. The last thing i will say is again attributed to Edwards. In his book on Advanced Calculus he encourages the reader to jump chapters. A book does not have to, and sometimes it should not, be read in order. It may take some practice to see how you need to jump around, but you will find that you can maximize your reading by doing so.
There are several point in which this book excels. First, in the writing style. Second, in how many ideas it introduces. Or how good an understanding the reader obtains of Number Theory. It is invaluable to have the big picture. Third, the author has in mind the future material the reader will encounter. He knows you will go beyond this book, and prepares you for what is to come. You do not enter higher courses blind.
The writting style is representative of that of Wiles and Loiville. It will show you how your mathematical writting should be. It takes a lot of practice to learn mathematical formalism and how to write proofs. This is the book to learn from. The author is not afraid to connect the ideas you are learning to other advanced ideas and to mathematical history, unlike present day authors. If you plan to be a mathematician, you must know its history. The writting is in a mathematical sense superfluos. It does not assume you are a genius, but strikes balance between what you should know and what you should be told.
The book is successful in providing you with the big picture, and how ideas you are learning reflect one ideas you will learn or have already learned. Having a big picture of the subject, which he describes in the second chapter, lets you know what you are learning now and puts the entire material in context. Gives you great perspective of the subject. Because a great deal of branches of number theory are discussed, you are not only better equiped to choose which branch might interest you, but it eases the transition to more advanced courses, such as Analytical Number Theory.
The author from the start discusses unanswered questions in Number Theory. I know alot of professors which think that the student should not be exposed to questions that surpass his mathematical knowledge. They are the weak mathematicians. Mathematics is about exploring and breaking limits. You should know what is beyond your reach, and the reach of every one else. The questions that still stand might be answered by some one that was intrigued by the challenge of answering them when they are helpless to do so. Fermat's Last Thorem is such an example. The guy learned it at the age of 10.
The last thing i will say about the book is this. Number theory has one scope. Namely, prime numbers. This book make it clear that the purpose of number theory is to determine the properties of numbers. It discusses the limitations of mathematics in attaining answers to Riemann Hypothesis, Fundamental theorem, trancedental and irrational and algebraic numbers, and so on. The book is, in my opinion, an expansion of the section on unanswered questions. And in doing so many more questions are asked and analyzed. There are prime numbers, and nothing else.
THE BOOK on number theory---BUY IT!!!!.......2004-07-03
It was always claimed that of all the mathematicians who ever lived, Hardy was one of the greatest writers. This book certainly confirms that view. From the very beginning, one thinks, "Wow, this guy REALLY knows what he's talking about." Hardy was, in fact, one of the greatest number theorists of the twentieth century. Hardy gives actual intuitive motivation for almost all of the theorems in the book (intuition is often overlooked by mathematical authors who use the confusing traditional "theorem-proof" approach), and his proofs are elegant and easy to follow. Once, I spoke to the chair of the math department at a major University (Wash U. in St. Louis) and he told me that he reads Hardy and Wright at least once a year to refresh himself on the basics. I would recommend this book to anyone who is learning about number theory for the first time, and wishes to pursue the subject through self-study.
A classic introduction to a wide range of topics........2001-09-02
Every serious student of number theory should have this classic book on their shelf. Even though only "elementary" calculus and abstract algebra are used, a certain mathematical maturity is required. I feel the book is strongest in the area of elementary --not necessarily easy though -- analytic number theory (Hardy was a world class expert in analytic number theory). An elementary, but difficult proof of the Prime number Theorem using Selberg's Theorem is thoroughly covered in chapter 22.
While modern results in the area of algorithmic number theory are not presented nor is a systematic presentation of number theory given (it is not a textbook), it contains a flavor, inspiration and feel that is completely unique. It covers more disparate topics in number theory than any other n.t. book I know of. The fundamental results in classical, algebraic, additive, geometric, and analytic number theory are all covered. A beautifully written book.
Other recommended books on number theory in increasing order of difficulty:
1) Elementary Number Theory, By David Burton, Third Edition. Covers classical number theory. Suitable for an upper level undergraduate course. Primarily intended as a textbook for a one semester number theory course. No abstract algebra required for this book. Not a gem of a book like Davenport's The Higher Arithmetic, but a great book to seriously start learning number theory.
2) The Queen of Mathematics, by Jay Goldman. A historically motivated guide to number theory. A very clearly written book that covers number theory at a graduate or advanced undergraduate level. Covers much of the material in Gauss's Disquisitiones, but without all the detail. The book covers elementary number theory, binary quadratic forms, cyclotomy, Gaussian integers, quadratic fields, ideals, algebraic curves, rational points on elliptic curves, geometry of numbers, and introduces p-adic numbers. Only a slight bit of analytic number theory is covered. The best book in my opinion to start learning algebraic number theory. Wonderfully fills the otherwise troublesome gap between undergraduate and graduate level number theory.
Full of historical information hard to find elsewhere, very well researched. To cover all the material in this book would likely take two semesters, though most of the important material could be covered in one semester. Requires a background in abstract algebra (undergraduate level), and a little advanced calculus. Some complex analysis for sections 19.7 and 19.8 would be helpful, but not at all a requirement. The author recommends Harold Davenport's The Higher Arithmetic, as a companion volume for the first 12 chapters; according to Goldman it is a gem of a book.
3) Additive Number Theory, by Melvyn Nathanson. Graduate level text in additive number theory, covers the classical bases. This book is the first comprehensive treatment of the subject in 40 years. 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.
Average customer rating:
- Wonderful Arithmetic Geometry Book
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Introduction to Modern Number Theory: Fundamental Problems, Ideas and Theories (Encyclopaedia of Mathematical Sciences)
Yu.I. Manin , and
Alexei A. Panchishkin
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Introduction to Modern Number Theory: Fundamental Problems, Ideas and Theories (Encyclopaedia of Mathematical Sciences)
Customer Reviews:
Wonderful Arithmetic Geometry Book.......2006-08-20
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).
Book Description
This introductory textbook is designed to teach undergraduates the basic ideas and techniques of number theory, with special consideration to the principles of analytic number theory. The first five chapters treat elementary concepts such as divisibility, congruence and arithmetical functions. The topics in the next chapters include Dirichlet's theorem on primes in progressions, Gauss sums, quadratic residues, Dirichlet series, and Euler products with applications to the Riemann zeta function and Dirichlet L-functions. Also included is an introduction to partitions. Among the strong points of the book are its clarity of exposition and a collection of exercises at the end of each chapter. The first ten chapters, with the exception of one section, are accessible to anyone with knowledge of elementary calculus; the last four chapters require some knowledge of complex function theory including complex integration and residue calculus.
Customer Reviews:
Introduction to Analytic Number Theory.......2006-08-21
The reason I bought this book was to understand an elementary proof of the prime number theorem. Actually, it contains only an outline of an elementary proof. But the book introduces methods for the proof with awesome clarity. It must have been much greater if we could see the detailed elementary proof of the prime number theorem written by Apostol. He gives a reference to An Introduction to the Theory of Numbers by G.H. Hardy and E.M. Wright for the detailed proof, but the reader may be required to do unnecessary guessing (which is believed to be good in learning math, but seems to be nothing but trouble for me) to go through it.
Amazing.......2006-01-28
This book is absolutely incredible. The topics covered range from some very elementary topics on the theory of certain basic arithmetic functions, to much more advanced topics such as the theory of Dirichlet L-Functions. I have never seen a clearer explanation of the characters associated with finite Abelian groups, and the L-functions associated with Dirichlet Characters, than that provided by this book. Apostol makes even the most difficult concepts seem clear and simple. As an added bonus, the end-of-chapter exercises range from moderately difficult to almost excruciatingly so (but still very fun to work on) and give the reader excellent experience in solving problems in this field. With all this said, it should be pointed out that, as another reviewer stated, this book should not be read until the reader has already had a good deal of previous exposure to number theory. I myself would recommend the book of Hardy and Wright. As a second text on number theory, and an introduction to the aspects of number theory related to function theory and analysis, I believe that Apostol's book is the best that anyone could possibly hope for.
Exceptional readability.......2005-09-27
You normally dont talk so much about readability of a book on Math, but of all the other books on number theory that I've seen, this is quite a page turner. Strikes just the right ballance between theory, proofs and examples. As mentioned somewhere in the book, one of the aims of the author is to arouse reader's interest in number theory..which this book will certainly do..especially since its main emphasis is on prime numbers.
Unsurpassed SECOND text on number theory.......2004-06-29
The amazing positives of this book are accurately described in the other reviews so I will skip them. There is no negative, but the other reviewers assert that the reader needs no prior exposure to number theory. I completely disagree.
While this book does quickly cover elementary number theory, a reader new to this field will quickly feel lost. Without more exposure and a good prior feel for elementary number theory, the use of analytic techniques will seem ad hoc instead of following a logical pattern. By way of example, three areas covered in this book that are not part of analytic number theory and for which the reader would do better to learn from a less sophisticated text are the Fermat-Euler Theorem, Diophantine equations, and quadratic reciprocity.
Excellent texts for a first exposure to number theory are, from simpler to more difficult:
1. Elementary Number Theory by Underwood Dudley
2. An Introduction to the Theory of Numbers by Niven, Zuckerman and Montgomery
3. An Introduction to the Theory of Numbers by Hardy and Wright
Apostol's book on analytic number theory is a classic that may never be surpassed. It is a marvelous second book on number theory.
well presented, delightfully written.......2001-12-06
I think that there will be little harm if the title of the book is changed to 'Introduction to elementary number theory' instead. The author presumes that the reader has not any knowledge of number theory. As a result, materials like congruence equation, primitive roots, and quadratic reciprocity are included.
Of course as the title indicates, the book focusses more on the analytic aspect. The first 2 chapters are on arithmetic functions, asymptotic formulas for averaging sums, using elementary methods like Euler-Maclaurin formula .This lay down the foundation for further discussion in later chapters, where complex analysis is involved in the investigation. Then the author explain congruence in chapter 4 and 5. Chapter 6 introduce the important concept of character. Since the purpose of this chapter is to prepare for the proof of Dirichlet's theorem and introduction of Gauss sums, the character theory is developed just to the point which is all that's needed. ( i.e. the orthogonal relation). Chapter 7 culminates on the elementary proof on Dirichlet's theorem on primes in arithmetic progression. The proof still uses L-function of course, but the estimates, like the non-vanishing of L(1) , are completely elementary and is based only on the first 2 chapters.
The author then introduce primitve roots to further the theory of Dirichlet characters. Gauss sums can then be introduced. 2 proofs of quadratic reciprocity using Gauss sums are offered. The complete analytic proof, using contour integration to evaluate explicitly the quadratic Gauss sums, is a marvellous illustration of how truth about integers can be obtained by crossing into the complex domains.
The book then turns in to the analyic aspect. General Dirichlet series, followed by the Riemann zeta function, L function ,are introduced. It's shown that the L- functions have meremorphic continuation to the whole complex plane by establishing the functional equation L(s)= elementary factor * L(1-s). The reader should be familiar with residue calculus to read this part.
Chapter 13 may be a high point of this book, where the Prime Number Theorem is proved. Arguably, it's the Prime Number Theorem which stimulate much of the theory of complex analysis and analyic number theory. As Riemann first pointed out, the Prime Number Theorem can be proved by expressing the prime counting function as a contour integral of the Riemann zeta function, then estimate the various contours. The proof given in this book , although not exactly that envisaged by Riemann , is a variant that run quite smoothly. As is well known , a key point is that one can move the contour to the line Re(s)=1, and to do this one have to verify that zeta(s) does not vanish on
Re(s)=1.The proof , due to de la vale-Poussin, is a clever application of a trigonometric identity. Unfortunately, the method does not allow one penetrate into the region 0
The last Chapter is of quite differnt flavour, the so-called additive number theory. Here the author only focusses on the simplest partition function ---the unrestricted partition. However interesting phenomeon occur already at this level. The first result is Euler's pentagonal number theorem, which leads to a simple recursion formula for the partition function p(n). 3 proofs are given. The most beautiful one is no doubt a combinatorial proof due to Franklin. The third proof is through establishing the Jacobi triple product identity, which leads to lots of identites besides Euler's pentagonal number theorem. Jacobi's original proof uses his theory of theta functions, but it turns out that power series manipulaion is all that's needed.
The book ends with an indication of deeper aspect of partition theory--- Ramanujan's remarkable congrence and identities ( the simplest one being p(5m+4)=0(mod 5) ). To prove these mysterious identites, the "natural"way is to plow through the theory of modular functions, which Ramanujan had left lots more theorem ( unfortunately most without proof). However an elementary proof of one these identites is outlined in the exercises.
This book is well written, with enough exercises to balance the main text. Not bad for just an 'introduction'.
Book Description
Number theory and algebra play an increasingly significant role in computing and communications, as evidenced by the striking applications of these subjects to such fields as cryptography and coding theory. This introductory book emphasises algorithms and applications, such as cryptography and error correcting codes, and is accessible to a broad audience. The mathematical prerequisites are minimal: nothing beyond material in a typical undergraduate course in calculus is presumed, other than some experience in doing proofs - everything else is developed from scratch. Thus the book can serve several purposes. It can be used as a reference and for self-study by readers who want to learn the mathematical foundations of modern cryptography. It is also ideal as a textbook for introductory courses in number theory and algebra, especially those geared towards computer science students.
Download Description
Number theory and algebra play an increasingly significant role in computing and communications, as evidenced by the striking applications of these subjects to such fields as cryptography and coding theory. This introductory book emphasises algorithms and applications, such as cryptography and error correcting codes, and is accessible to a broad audience. The mathematical prerequisites are minimal: nothing beyond material in a typical undergraduate course in calculus is presumed, other than some experience in doing proofs - everything else is developed from scratch. Thus the book can serve several purposes. It can be used as a reference and for self-study by readers who want to learn the mathematical foundations of modern cryptography. It is also ideal as a textbook for introductory courses in number theory and algebra, especially those geared towards computer science students.
Customer Reviews:
Really a treasure.......2006-03-30
I'm a student digging into the cryptology for an year. The more article I read, the more confusion I encounter because of my poor mathematical background. However, when I get this, I could find answer to my puzzles, and make an more explicit way to settle down my own idea.
The background you really need, clear and sweet.......2005-11-06
This book is a marvel. It is clear and concise yet thorough. The author is obviously a bit of an obsessive compulsive, he has found the shortest paths from the clearest definitions to the most important results, each given with the cleanest, most insight-inducing proofs ... the results (and definitions) he gives are the ones any student (practitioner!) of modern computer science (especially cryptology) *needs* to know -- having this book on your shelves (and its contents in your head) should be a requirement for any degree, at any level, in computer science.
[Caveat: I know the author and have read his book in draft form. I also required my students to get it and read it, in a computer science course I taught.]
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