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Customer Reviews (7)

Very informative book but not enough examples worked out
Elementary Number Theory: Second Edition is a relatively short mathematics book that deals a lot with prime number theory. The topics covered in this book are senior college level topics, making the name "Elementary" in the title somewhat deceptive. While topics they talk about in an intro to proofs class focus on the basics, this book focuses on the applied basics. The book is short and has many examples for the student to work out, but it is limited in that there are few examples that the author worked out. Because of the level of this book, many steps are also removed, however, even for a math student, some of the steps skipped left me stumped, and many topics are still out of grasp through independent study. There is a nice section of the book that has some induction problems, which is geared towards those who have already done proofs and now are working on number theory. This book is solid for the $10 it cost, but there is a reason other math texts run from $100-200, and that's because of the extensive amount of problems worked out in those texts. (You get what you pay for)

Undergrad math and phys major review
I picked up this book after taking a course in set theory/math logic.It was my first experience with proof based math and I found it very challenging and rewarding.I decided over the summer I would teach myself number theory as well.So far I am about 3 or 4 weeks into the summer and I've chopped down sections 1-6.Keep in mind I am also doing undergrad physics research and a directed study astrophysics course as well, you could easily progress further.I really have enjoyed this book thus far. I often get so caught up in it that I am up until morning toying with problems.The author did a fantastic job of finding the fine line between too difficult and elementary.It seems just short of a graduate text but a little above a common undergraduate text. Thanks to this text I am becoming much more confident in my ability to set up and execute proofs.I don't want to spoil the methods- but it seems MOSTLY up until this point all proofs are found in the same manner.I have not been able to execute a variety of proof methods (e.g. deduction, mathematical induction, contrapositive, and contradiction.)I would have liked to be able to switch it up some. However, I am no expert and this may be my own doing.
If that is the best "CON" I can come up with for this book, that says a lot.I HIGHLY recommend ANY student who enjoys thinking to pick up this book and complete it in it's entirety. I feel deprived that I wasn't given this opportunity sooner.

I am a 3rd year math+physics major.This book changed me from physics and chemistry to math and physics student and possibly from a physics grad student to a math grad.

Excellent on Elementary Number Theory
Excellent text on number theory.Topic coverage is extensive: quadratic reciprocity, linear diophantine equations, Lagranges's 4-square theorem, Pell's equation and more.Proofs are very good and plenty of exercises.

Good overview to basic number theory
This is a great book for the mathematically-interested layman, non-math-major undergraduate, and people in other fields (such as computer science) who want a fairly quick read on the basics of this fascinating field.It would also serve well as a a precursor/warmup for more advanced treatments of the subject.

It is a book that has aged very well
Published in 1978, this book suffers very little from the illness of being dated. Of course, Fermat's last theorem has been proven and computers have grown much more powerful. However, those advances have little affect on the value of the book. Done well, basic number theory is timeless, and Dudley does it very well. The explanations of the fundamentals are sound and solved exercises are scattered throughout. Solutions to most of the odd numbered problems at the end of the chapters are also included.
The coverage is fairly typical, so the book can still be used as the text for a course in beginning number theory. Chapter 23 contains a set of 267 additional problems, a set for each of the other chapters in the book. Solutions to most of the odd problems in this set are also included. I commend Dudley for including so many solutions; I have little time for authors of books who do not provide solutions to at least some of the exercises. If I ever teach a course in basic number theory, this is the book that I would use.
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Customer Reviews (9)

Classic Number Theory Text
This is one of the classic texts on Number Theory.It is a challenging book for anyone.The problem sets range from easy to hard.There are some hints, but those are few and far between.Like most Dover math books, it's dry and concise.There are better books on the market, but not for this price.

An incredible text in elementary number theory
Despite the deceptively small size of the text compared to many of its type, be sure to carry at least twice as many sheets of paper to fully get all you can out of it. George Andrew's pedagogical style of using combinatorics (basic gambling probability) to explain advanced concepts in number theory is executed brilliantly, and leaves even first-year undergraduates like me without a doubt in the world.

It is essential to do the problems in this book! Do not skip them thinking writing down the definitions and theorems will be enough-- some of the problems will kill you if you go in only knowing the written theorems, without any proper thought into the subject. Like any mathematical subject, it requires rigorous thinking and hours of reading before even considering going on to more advanced topics, like algebraic number theory, abstract algebra, or residue theory.

Breaking down the book into parts, I find it slightly disconcerting that despite the small nature of the book, the concept of quadratic congruences are only introduced in a less-than-introductory fashion, in comparison to other number theory books. It may be true that the author's main research was based off partition theory (the largest section in the book), but quadratic congruences have large applied mathematical influences, and should be considered to be read on, after the book as been finished.

Despite that, this text is an incredible foray into elementary number theory, and is a recommended buy for all those interested in the mathematical world.

Essential Number Theory
This book is great for the price and if you can handle the terseness of a Dover book I would say it is great in general. The back of the book indicates it would be good for liberal arts majors. That is just crazy. However, you don't need much more than a solid foundation in mathematics through the Calculus of sequences and series. To get the most out of this book, you should do as many of the exercises as you can, even the ones without answers. Also, plan on supplementing the text with some online research. A general review of generating functions may be useful. Chapter 3 is a bit out of place and easy to lose patience with. Perhaps it can be read following Part 1. With that said, you can get a lot out of this book with regard to number theory (which arguably may not be generally useful).

best bang for the bucks in number theory books
Price is a factor and Hardy is the classic text.
This one by George E. Andrews is well written has good examples and exercises
and doesn't cost an arm and leg used.It has a good prime section and a good quadratic section. I found stuff I hadn't seen before here.
It is going to take me a while to get everything out of this one!

Good
The author tries to make things easy, and he succeeds in most parts. However, some proofs seem to be simple, but they actually involve complicated reasoning. My suggestion is that the author should not try to hide these difficult parts by reformulating abstract things into simple objects because it doesn't really help. I'd rather see difficulties in proofs than follow them with no idea why they have to be like that. ... Read more

Product Description
An Introduction to the Theory of Numbers by G. H. Hardy and E. M. Wright is found on the reading list of virtually all elementary number theory courses and is widely regarded as the primary and classic text in elementary number theory. Developed under the guidance of D. R. Heath-Brown, this Sixth Edition of An Introduction to the Theory of Numbers has been extensively revised and updated to guide today's students through the key milestones and developments in number theory.

Updates include a chapter by J. H. Silverman on one of the most important developments in number theory - modular elliptic curves and their role in the proof of Fermat's Last Theorem -- a foreword by A. Wiles, and comprehensively updated end-of-chapter notes detailing the key developments in number theory. Suggestions for further reading are also included for the more avid reader.

The text retains the style and clarity of previous editions making it highly suitable for undergraduates in mathematics from the first year upwards as well as an essential reference for all number theorists. ... Read more

Customer Reviews (16)

There is no such thing as 'number theory'!
I'm not as impressed as the other reviewers here with this book, despite it's being in some sense a 'classic'.

I take it some people - I have no idea what proportion of the population - now and then are struck by properties of numbers. For example, if a number AB is added to BA, the result always divides by 11. If the difference is worked out, this is always a multiple of 9.(E.g. 75 minus 57 is 18, a multiple of 9). Or (e.g.) Any number cubed, less the number, will always divide by 6 and is also the product of three consecutive numbers.(e.g. 5 cubed is 125; less 5 gives 120, which is 4x5x6).This sort of thing is the basis of 'number theory'.

There are at least two problems with this book. Firstly, there is in fact not yet such a thing as 'number theory'. This book is a ragbag of techniques and things which have been identified and passed down by lecturers. But it is NOT a coherent 'theory' in any sense. Perhaps I might compare it with a book on 'chess theory'. Chess books have accounts of such things as opening gambits, sacrifices, end games - including some with extremely precise techniques needed for victory. And there are things like 'zwischenzug' and assorted events which are rare, but have some interest. But does this make up a body of 'theory'? I'd say not. Anyone looking to this book for insight into the Pythagorean mystery of number will in my view be more or less disappointed.

Now, what follows from this is my second point, which perhaps is to do with human psychology, or the capacity of the human brain. What is it that makes some people fix on a certain type of problem? For example, this book, like most or maybe all on number theory, starts with prime numbers - probably discovered as a result of packaging and division of actual objects. This of course had practical applications, such as the Babylonian 360 degrees, and our 12 inches, 14 pounds, 1760 yards, and so on. A collection of techniques (e.g. Eratosthenes' sieve), formulas, limits and other results has accumulated. Looking at Euclid's proof of the infinity of primes, his method was to multiply all the primes, and add 1. This function in effect is designed to use the properties of primes to generate a new prime. However Hardy and Wright don't attempt to generalise this process. Maybe Fermat's Last Theorem could be proved elegantly by inventing some ingenious function which combines the properties of addition, multiplication, and powers - repeated multiplication by the same number? What is it that makes some problems (so far) insoluble - and many of them are very trivial to state?

So we have here a collection of results, embodied in symbolism which is far enough from the actualities to (perhaps) look more impressive than it really is. Integration, for example, is basically simple enough, but the long s and the notation removes the reader from the real world...

And there's a related problem, which is that the connective material, explaining why the next bit is there and what it is supposed to illustrate, is completely missing. The result is like a tour of museum exhibits, where the tourist is expected to infer the significance of all the specimens. Or like a concert, where one sample piece of music is played after another, from which the auditor is presumably left to infer a theory of music. In fact, I've just decided to demote the book to two stars!

Easy read
This book provides a gentle presentation to many subfields of number theory: including analytic, algebraic, and elementary. It discusses generating functions in everyday language. The book's section on the zeta function is incredible. I would recommend this gem to anyone who taken calculus.

awesome book on number theory
I am an undergrad student in computer engineering. I bought this book after I looked at the table of contents and found some topics which I interested in. This is by far the best book on number theory I ever came across. It is very readable, fairly free of errors (the ones that are there are easy to spot and do not cause confusion). In comparison to another number theory book I read before. This one has the charm of making previously confusing concept clear. Different proofs are often given on major theorems. I do not really have a good way to describe it, but this book really "flows". The logic is clear and easy to follow. If I read this one to start with, it would save me a lot of time and I would have a much better understanding of the subject by now. I know, this review is totally uninformative, you have to see it for yourself to be sure, but I totally recommend this one.

The only downside is the price dropped by like 20$ since I bought it.

Number Theory
The book was an excellent accumulation
of Number Theoretic ideas. However, it
failed to produce applications or clearcut
examples of the theorems.

A Mathematical Classic Reviewed
Even though I have only read a small portion of this book, I can already tell that it deserves its "classic" label. I have been accumulating some mathematical classics, mostly purchased through Amazon, and this one certainly fits in with the rest. Like Euclid's Elements, it has a timeless appeal even as mathematics advances. And, judging by the table of contents, it goes well beyond a mere "introduction" to number theory, like H. S. M. Coxeter's "Introduction to Geometry." ... Read more

Product Description
Elementary Number Theory, Sixth Edition, blends classical theory with modern applications and is notable for its outstanding exercise sets. A full range of exercises, from basic to challenging, helps readers explore key concepts and push their understanding to new heights. Computational exercises and computer projects are also available. Reflecting many years of professors' feedback, this edition offers new examples, exercises, and applications, while incorporating advancements and discoveries in number theory made in the past few years. ... Read more

Customer Reviews (12)

half truth
The book itself is a fine book. However, the book is not in good shape. It was marked as in good condition and it is falling apart. I am very disappointed and will think twice about buying books on here again.

So obvious, it's difficult
Great book! Rosen explains tough concepts well and the homework problems really drive it home. The computational homework for programmers is especially nice. However, some of the problems are worded in such a way that it obscures the intent of the question. Then again, this IS number theory.

Moderately Helpful
For a good number of the odd problems the solution manual is good at working out the problem and coming to an answer. However, some of the answers are identical to what is in the back of the Number Theory textbook.

ODD PROBLEMS ONLY
The description is misleading. The solutions manual only contains the solutions to odd problems. The same 'answers' that are available at the back of the book are explained in a little bit more detail in the solutions manual. If you are looking for solutions to even numbered problems as well then don't buy this.

Omits obvious visual and intuitive ways of looking at concepts.
This book is wordy, but less clear than I think it could be.It moves slowly, yet omits helpful ways of looking at concepts in the interest of being elementary.Also, it does not pave the way for future study of the material, nor does it pave the way for later study of connections to algebra, analysis, or combinatorics.

This book overlooks intuitive, visual ways of representing basic concepts.For example, lattice/Hasse diagrams helped me understand divisibility and GCD's, but this book does not even mention this way of looking at divisibility.Congruence relations can be visualized in a number of ways, but this book only treats them using basic algebra.A book that moves as slowly and is as elementary as this one should really explore these sorts of visual presentations of concepts.

Since this book is elementary, it does not explore any connections to groups, rings and ideals, fields, or lattices, and I think this is a shame because these structures make number theory make more sense.I think the book would do better to introduce a few of these structures in a very basic way.The book also passes up the opportunity to introduce generating functions, a critical and fairly elementary topic in number theory, which are only touched in one exercise.The first few chapters of Wilf's Generatingfunctionology and Newman's "Analytic Number Theory" show that generating functions can be presented at an elementary level.

There are blurbs of history interspersed throughout the text, and I like the idea, but the history focuses almost exclusively on biographical information, with a tiny bit of history of famous problems and conjectures.There is little discussion of how the core mathematical ideas in the book were discovered and evolved over time.This book would do well to cut out the biographies and replace them with richer discussion of the historical development of the subject itself.

I like the idea of a number theory book that focuses on applications, but this book does so at the expense of other things: its treatment of truly fascinating topics (such as continued fractions) is so weak that I do not think it's worth the trade-off.The book does nothing to pave the way towards the study of either analytic or algebraic number theory.The Zeta function only gets a token mention.

This book is usable as a textbook or for self-study, but it is not outstanding for either of these goals, nor is it useful as a reference foradvanced students.Studying from this book alone won't help one develop a good sense of intuition in number theory.Stillwell's thin book is about as easy to follow as this one, and yet it slowly introduces ideals and other algebraic concepts by the end of the book.My favorite book on number theory is Apostol's "Analytic Number Theory".It is considerably more advanced than this one, but I think that students with a strong background will actually find it easier to learn from.I have not yet found a truly elementary book on number theory that I liked; I think students would be better off to first acquire enough background and mathematical maturity to dive right into some of these more advanced texts, than to spend their time working through a book like this. ... Read more

Product Description
This book gives an elementary undergraduate-level introduction to Number Theory, with the emphasis on carefully explained proofs and worked examples. Exercises, with solutions, are integrated into the text as part of the learning process. The first few chapters, covering divisibility, prime numbers and modular arithmetic, assume only basic school algebra, and are therefore suitable for first or second year students as an introduction to the methods of pure mathematics. Elementary ideas about groups and rings (summarised in an appendix) are then used to study groups of units, quadratic residues and arithmetic functions with applications to enumeration and cryptography. The final part, suitable for third year students, uses ideas from algebra, analysis, calculus and geometry to study more advanced topics such as Dirichlet series and sums of squares. The last chapter gives a concise account of Fermat's Last Theorem, from its origins in the ancient Babylonian and Greek study of Pythagorean triples to its recent proof by Andrew Wiles. ... Read more

Customer Reviews (15)

The knowledge I had expected, but too out of place, and does not help in solving basic questions.
I originally bought this book to gain the knowledge and have an advantage in competing in Math Olympiads, but this book deters from that purpose.
The book skips many important subjects, (such as qualities of divisors, essential qualities of prime numbers). Also, it is not that straightforward. Although I have learned a few tricks from this book, the book skips many steps in proving theorems, which, although may be filled by a teacher or student, the skipped steps are beyond the level of an average person reading this book.
Also, it is not very clear in its approaches, since it assumes the reader has learned many mathematical notations (such as cyclotomic polynomials, derivatives from Calculus, etc, etc), which can be confusing for the average high schooler.
Also, the exercises in this book are not very consistent with the topics of the book, such as proving theorems beyond the scope of the already learned knowledge, as well as the lack of problems actually consistent with the topics of the book (such as find the gcd(21390, 123, or 1290x+9233y=1, find solutions to x and y, if any). This book only gives me 1 or 2 of those problems, and then moves on. Many times I have found myself to have the need to re-read the materials from the book because of its lack of review sections. I have even seen online notes from people about number theory more consistently put together than this book, which also have firmly emplaced their knowledge into my mind.
Another realization I have is that this book does not fit the skill level of a high schooler, or even an undergraduate. Many times the exercises and the proofs in the book give either very easy problems, or very hard problems. Such problems include (find gcd(234, 123), then skipping to proving if 2^m+1 is prime, then m is prime as well, as well as proving Eintstein's criterion for polynomials).
They do not provide many "in between" problems, and so, part of the learning becomes defunct, or people might even be discouraged by the book.
However, I do realize that this book may be useful for undergraduates taking cryptography, this is however, not useful for anyone who is studying for math competitions.

Probably the best maths book I've ever read
This is a great little book thats packed full of great number theory results. It is well written.

I'm a real fan of the SUMS books (I've bought 4 of the titles in the series), because all of the books I've bought are well written, they're jammed full of useful information and they're relatively cheap!

The book strikes a good balance between keeping focused on number theory (there are chapters requirng a knowledge of rings and groups, but these structures only support the numbers, not abstract them away) and not being trivial (I've read too many number theory books that are 'bitty', in the sense that there is too much breadth and not enough depth).

Nice book with some flaws
"Elementary Number Theory" by Jones and Jones is a nice book, easily accessible to the average undergraduate math major. I used this book as a text for the junior level number theory course at my university. The price is very attractive, especially for students on a budget. The text is for the most part very readible. There are a few flaws with the book that prevent it from truly being an excellent text:

1) Complete (or nearly complete) solutions are given for every exercise. While this is good if you are trying to learn the material yourself, it makes it difficult to assign problems from the text.

2) Non-intuitive steps are often left out of proofs. While the instructor or a mature student can usually fill these in, some of the omissions are (in my opinion) beyond that of the average undergraduate.

With its excellent price point, I would recommend the book as a supplement to another text.

A Satisfactory Text for Elementary Number Theory
I've recently received my copy of Elementary Number Theory by Jones and Jones, and I'm (thus far) satisfied with the textbook. Although I'm not a professional mathematician, I have worked toward a degree in math and still love to study it. For me, the current textbook for number theory is a challenge to master, but with all the solutions to problems provided, I find it quite palatable to work toward an understanding of number theory, using this text.
In view of my current experiences with this textbook, I would recommend it to a mathematical hobbyist like myself, or to a professional student of mathematics -- or anyone wishing to tackle number theory.

An almost perfect square
That the book's almost square is easily gathered from the photo. That it's almost perfect must be verified by reading it. And what an enjoyable verification indeed awaits those who take on the challenge! Not that reading it is a challenge - on the contrary, the Joneses take every effort to ensure your learning experience be as painless as possible. Every proof is complete, all exercises are solved. The proofs are always selected for their instructional merit, rather than for their mathematical "elegance" (read: brevity and algebraic gimmickry). As one Amazonian reviewer put it: you could read it through, lying in a bubble bath.

Another Amazonian reviewer commented that "Number theory is like the cement on your driveway. Real and Complex analysis are the Porsche and Ferrari you drive home every night." I disagree. In any case, in my opinion the book's weak spots are those sections where the discussion forays into the realm of real and complex analysis, namely 9.4-6 ("Random Integers", "Evaluating Zeta(2)", "Evaluating Zeta(2k)"), 9.9 ("Complex variables"), 10.2 ("The Gaussian Integers"), a part of 10.6 ("Minkowsky's Theorem") and 11.9 ("Lame and Kummer"). "Sums of two squares" (Section 10.1) could also use improvement, but this is compensated by the excellent, independent treatment this topic receives in the "Minkowsky's Theorem" chapter.

On several occasions, from the very beginning, the book assumes familiarity with single-variable polynomials (particularly the division algorithm and the x^n-y^n expansion). Be prepared.

If it weren't for the forays mentioned above, the book would have been a straight fiver. But even as it stands, it'sa tour-de-force of pedagogy and expository mathematical writing.

One last quibble. The book doesn't have a homepage, nor is there any indication of a way to contact the authors. Textbook publishers should learn from their colleagues in the applied computer science publishing industry (such as O'Reilly, Wrox, Apress, etc.) and always make a homepage available for every book, with, at the minimum, a link to an errata page, and a forum where readers of the book can discuss it, (preferably with the involvement of the author(s)). ... Read more

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Customer Reviews (4)

Crystal clear
Samuel's book is a classic. It is a bit "antique", certainly not the most modern introduction to algebraic number theory. The topics covered in the book are algebraic and integral extensions, Dedekind rings, ideal classes and Dirichlet's unit theorem, the splitting of primes in an extension field and some Galois theory for number fields. So the range of topics is quite small (and the book is short, ~100 pages).

And yet I love this book. It is crystal clear, well written and well structured; it is quite dense (especially the last chapter) and makes the beginning student of algebraic number theory think a lot, but without ever getting too heavy to digest. The list of problems is fantastic: there are many very concrete problems which sharpen your understanding of the material considerably. And last but not least, it is ridiculously cheap.

I recommend this to anyone who wants to learn the basic material about number fields. Without any hesitation.

Samuel Knows Numbers
It's a little dense, and some proofs are lacking detail, but otherwise a great book.Extensive subject covered in one semester's worth of reading.

Good service
Never got shipping confirmation email but customer service was very quick to respond to my question about the shipping status.

A gem of a book
This is a lovely, lovely book -- the first I ever read on algebraic number theory. It is spare and direct, and a great introduction to the field. ... Read more

Product Description
Number Theory: A Lively Introduction with Proofs, Applications, and Stories, is a new book that provides a rigorous yet accessible introduction to elementary number theory along with relevant applications.

Readable discussions motivate new concepts and theorems before their formal definitions and statements are presented. Many theorems are preceded by Numerical Proof Previews, which are numerical examples that will help give students a concrete understanding of both the statements of the theorems and the ideas behind their proofs, before the statement and proof are formalized in more abstract terms. In addition, many applications of number theory are explained in detail throughout the text, including some that have rarely (if ever) appeared in textbooks.

A unique feature of the book is that every chapter includes a math myth, a fictional story that introduces an important number theory topic in a friendly, inviting manner. Many of the exercise sets include in-depth Explorations, in which a series of exercises develop a topic that is related to the material in the section. ... Read more

Customer Reviews (5)

Excelente libro
Excelente libro, ampliamente lo recomiendo. Vale la pena leerlo completo e intentar todos los ejercicios.

Genuinely funny!(Not to mention clear and well-written.)
This is an extraordinary book.A model of clarity, painstaking attention to detail, and all of it in a light-hearted, engaging manner.With cartoons!That are actually funny!

I compared this book's TOC to the catalog description of the Number Theory course at my school.It contains every topic in our syllabus.So it passes my first test.

Every dozen pages or so, one comes across a real gem.For example, the book discusses the AKS primality test and notes the significant contributions of undergraduates to that breakthrough.The illustrations of pineapples exhibiting Fibonacci numbers are both beautiful and illuminating.

Do not skip the "Math Myths"!They are not mere add-ons.They introduce the topics well, via engaging and well-thought-through analogies, and they are genuinely entertaining.The puns are of Daily-Show-headline calibre.It is one thing to ask a class for their ideas on how to solve the linear Diophantine equation aX+bY=1.It is another thing altogether to introduce the topic by asking the class to help Diophantus the "farmer" weigh potatoes using only a balance scale and the bricks he has on hand.The context provided by the latter provides the traction needed for students to *think* about the question for themselves.

I think this book would be especially well-suited to schools that offer Number Theory as a sophomore-level course, where students have not yet had experience with writing abstract proofs.No other book on this subject will soften that shock nearly as well, helping students make the transition to advanced mathematics by spelling out in detail the habits of mathematical thinking that we practioners of math tend to simply perform without explanation as autonomously as tying our shoes.

I will certainly be using this book for my class the next time I teach Number Theory.

One of the most helpful guides I have read.
Though it starts out at a nearly absurdly fundamental level, the authors do a fantastic job at guiding the reader through the material, guaranteeing that almost no prior knowledge is necessary to get the full experience. The stories and historical anecdotes, true or not, are almost a reason to read the book by themselves. Like one of the other reviewers said, an advanced middle-schooler would be able to go through this book and understand it, as would any undergrad who has yet to take number theory. The examples and exercises are fun and engaging even to someone who has learned the subject.
Be warned, however, that the book is oriented at those who want to learn the subject, and who are willing to take the time for a deep understanding. The lack of answers is not an accidental omission, it is meant to force readers to think through the problems and come up with their own solution. The book teaches not only number theory, but mathematical reasoning and logic as well.

monumental work of patience and love
When I first started reading this book, I was almost brought to tears at the appallingly elementary level that the book starts at: is the average American College Student so ignorant, unprepared, and childish to the point that one must start by teaching them material that would be covered in primary school in Asia and Europe, in a tone appropriate for kindergarten?However, as I kept on reading, I was amazed at the care and thought that the authors put into this book to keep the reader entertained and engaged, while at the same time making sure that no intermediate step, however small, was ever overlooked so that even the most unprepared students could be brought up to speed (as long as they work through the book diligently).

Starting from the Pythagorean theorem and the irrationality of the square root of 2, by the end of the book the reader will be able to understand the proof of Fermat's Last Theorem for the case n=4. Proofs of theorems are always preceded by numerical examples to give the reader a feel of the logic being used.Detailed instructions on how to perform proofs (including some basic English grammar) are also provided so that the reader can write proofs themselves.Various real world applications of the material are provided showing the reader the relevance of it all.

The problems are also very entertaining. The lack of solutions is not a drawback since for most problems one can easily tell whether the solution is correct or not by simply checking to see if one's solution satisfies the condition provided in the problem.Also, the joy of math is in trying to figure out the solution by oneself.If you cannot figure out an answer, just keep on thinking. You will learn nothing by trying to peek at the answer and trying to memorize it.

The authors' patience and their love of the field emanates from every page. The book is so well written that I would say that a gifted 6th or 7th grader will be able to understand it.This is a monumental piece of work that all future textbooks in math should be judged against. If you are interested at all in number theory, this book is a must have.


Postscript: as is often the case with first-editions, there are a few unfortunate typos which may be immensely confusing to the reader:

page 25, second equation: the left-hand-side should be x
page 29, first equation: the left-hand-side should be 2
page 275, table: the numbers in the second row under 5 and 8 should be 2

Hope these will be corrected in the second printing.

No answers
I've only covered the first couple of chapters so I can't comment on the overall effectiveness of this text. However, I can say that I am disappointed that there are no answers to the exercises. Most texts provide answers for at least the odd numbered exercises. The author alludes to a Student Companion on a web site but gives no link. On the Wiley site there is a page associated with this text, but it is a pay site. How lame. ... Read more

Product Description

This challenging problem book by renowned US Olympiad coaches, mathematics teachers, and researchers develops a multitude of problem-solving skills needed to excel in mathematical contests and research in number theory. Offering inspiration and intellectual delight, the problems throughout the book encourage students to express their ideas, conjectures, and conclusions in writing. Applying specific techniques and strategies, readers will acquire a solid understanding of the fundamental concepts and ideas of number theory.

Key features:

* Contains problems developed for various mathematical contests, including the International Mathematical Olympiad (IMO)

* Builds a bridge between ordinary high school examples and exercises in number theory and more sophisticated, intricate and abstract concepts and problems

* Begins by familiarizing students with typical examples that illustrate central themes, followed by numerous carefully selected problems and extensive discussions of their solutions

* Combines unconventional and essay-type examples, exercises and problems, many presented in an original fashion

* Engages students in creative thinking and stimulates them to express their comprehension and mastery of the material beyond the classroom

104 Number Theory Problems is a valuable resource for advanced high school students, undergraduates, instructors, and mathematics coaches preparing to participate in mathematical contests and those contemplating future research in number theory and its related areas.

Customer Reviews (2)

Great book but some solution can be simpler
It is a nice book.Some solution can be solved in different ways that is little simpler.

Number Theoretic Gem
I think that the only way for description of this book is buying it.
This book is as usual another gem, and this time in Number Theory, from
great math problemists Titu Andreescu and his colleagues Dorin Andrica and Zuming Feng.
If you would like to have fun and exciting in number theory, I highly recommend this fabulous book to you.

Congratulations to Titu Andreescu and his colleagues for their excellent books and attempts!!! ... Read more

Product Description
Elementary Number Theory, Seventh Edition, is written for the one-semester undergraduate number theory course taken by math majors, secondary education majors, and computer science students. This contemporary text provides a simple account of classical number theory, set against a historical background that shows the subject's evolution from antiquity to recent research. Written in David Burton’s engaging style, Elementary Number Theory reveals the attraction that has drawn leading mathematicians and amateurs alike to number theory over the course of history. ... Read more

Customer Reviews (13)

Low price, good book
The book condition is still good although its bought used. There are a few writings on the pages but its fine. However, there are some missing pages probably from printing error of the book.

Very clear and interesting
Got the book for free in electronic format. The prices for these books are criminal. Intellectual property is the fraud of frauds.

The book itself is excellent. I had not had any formal study of proofs when I started reading, but this book guides you very gently and clearly through the process. Number theory is NOT boring, and going through this book will make you feel enlightened. It's not easy, but very doable for anyone of average intelligence.

All the starting stuff is here
Most of this stuff I found in different books,
but usually not as well presented as this.
This book is a perfect one for someone starting in on number theory.
It even has some neat tables at the end.
I wish I had had this book ten years ago!

an excellent introductory book
before finally selecting this book for reading i 've spent a few hours in the library browsing through some number theory books. Coming from a a different background electical & computer engineer, I had no notion at all of number theory. I like his way of writing with the embedded historical notes and furthermore the proofs of the theorem and their chronological order particularly in the second chapter. I have to admit that I comment on the previous version but new versions are supposed to improve. It is not time consuming to go through the proofs while you understand the theorems and the techniques used behind. The flow is very coherent and solidly written. Overall an excellent introductory book as cited in a previous review.

Rigorous and not too hard

This is a textbook about Elementary Number Theory, where "elementary" does not
mean "simple" or "beginning", but rather those portions of the mathematics of
integers that do not rely on analysis (infinitesmal calculus).

Number theory allows many different orderings of topics, without omitting
proofs.I found Burton's order to be easy to follow. Many results in number
theory follow easily from results in abstract algebra or linear algebra.
The author does not depend on results beyond elementary algebra, but some
degree of mathematical maturity is required.Readers with a math degree will
still have to work to absorb the material.

There are many problems. Those with numerical answers are answered in the back
of the book. About half of the others are answered in an answer guide,
available separately.Almost everything is proved. I only recall two cases
of "left to the reader" except for the problems, of course. None of the
problems are used for future developments in the main text.

The author has a separate text about the history of mathematics.Most of the
chapters in this book start with a section about the history of the material
in the chapter and about the people that developed it. This is interesting
extra material, or padding that makes the book even more expensive than it
should be, depending on you.

This is the 6th edition. The only error I encountered was a consistent misspelling
of one name in chapter 10.I could not find any reported errors on the WWW.

I've used several other number theory books over the years. This one seems the best
for me.Perhaps that is due to Burton's skill, or perhaps it is because I finally
worked through one from front to back, instead of searching for the information
I needed just then.

... Read more

Product Description

Customer Reviews (5)

From Ore with love: old but instructive introduction to number theory.
Pros:
1. The book could teach basic number theory to a wide range of readers, from mathematically inclined high-school students to much more advanced lovers of mathematics. 2. It is enlivened by nice historical allusions.
3.The author shows and shares his fascination with the subject in the writing.
4. On a less lofty side, the font is large enough to avoid eye strain.

Cons:
1. First published 59 years ago, the book has to be dated. For example, many beautiful applications of number theory had been unknown at the time of writing.
2. Not all exercises require creativity, many of them are routine drills.

Bottom line:
If number theory is not your fortress, the book could strike a balance between enjoyable reading and learning.

A book for practically anyone
Ore's book is an excellent introduction to the fascinating topic of number theory.He takes his time explaining the history of numbers and goes into Euclid's algorithm so smoothly you hardly realize what you've learned.He discusses prime numbers and I was particularly delighted to see diophantine equations explained with lots of examples and an easy to follow method.The book is filled with interesting concepts, lots of examples, and good problems to do on your own.

At the end, for example, Ore talks of how number theory relates to geometry and I wish there were more of that in it.

I took this book on a very long trip, worked through many of the problems and simply found it a wonderful companion.If you get it, enjoy.One caution: if you already know some number theory you may find this book too simplistic.Still, it's worth having.

A book for practically anyone
Ore's book is an excellent introduction to the fascinating topic of number theory.He takes his time explaining the history of numbers and goes into Euclid's algorithm so smoothly you hardly realize what you've learned.He discusses prime numbers and I was particularly delighted to see diophantine equations explained with lots of examples and an easy to follow method.The book is filled with interesting concepts, lots of examples, and good problems to do on your own.

At the end, for example, Ore talks of how number theory relates to geometry and I wish there were more of that in it.

I took this book on a very long trip, worked through many of the problems and simply found it a wonderful companion.If you get it, enjoy.One caution: if you already know some number theory you may find this book too simplistic.Still, it's worth having.

Hamony?
A noted conjecture of the author's on the harmonic mean of the divisors is tucked unobtrusively in this pleasant reader: "Every harmonic number is even." See problem B2 in Richard K. Guy's Unsolved Problem's in Number Theory.

A good book (but not a great book). Very basic. For the more advanced historical approach, Andre Weil's Number Theory: An approach through history" is to be recommended. Or even Guy's book mentioned above.

Excellent theory interspersed with history
This book goes into detail on number theory, but it is often hard to follow with the history mingled with the theory.More advanced material is referenced without proofs.Two readers will especially like this book: those who want an introduction to number theory and those who want a good introduction to the history of number theory. ... Read more

Product Description

Starting with nothing more than basic high school algebra, this volume leads readers gradually from basic algebra to the point of actively performing mathematical research while getting a glimpse of current mathematical frontiers. Features an informal writing style and includes many numerical examples. Emphasizes the methods used for proving theorems rather than specific results. Includes a new chapter on big-Oh notation and how it is used to describe the growth rate of number theoretic functions and to describe the complexity of algorithms. Provides a new chapter that introduces the theory of continued fractions. Includes a new chapter on “Continued Fractions, Square Roots and Pell’s Equation.” Contains additional historical material, including material on Pell’s equation and the Chinese Remainder Theorem. A useful reference for mathematics teachers.

Customer Reviews (14)

Great for mathematicians who have not studied number theory..!
I think its a brilliant introduction to someone like myself, who teaches high school maths to some very able students who may well go on to undergraduate maths courses.
My first degree was in engineering, so I havent had the priviledge of a second level number theory course.. Its ideal! I love the conversational style.. and there are recommendations of books with more rigour, but dont be fooled into thinking this is easy.. Its a demanding read, at least I think so.

Useless piece of garbage
Okay, so I used this book in a semester-long course in elementary number theory. It's totally useless. Aside from the fact that the writing style is too chatty and to some extent patronizing, here's my main problem with this book:

The text gives minimal explanations of things -- basically it states a theorem, gives a few practical examples of why the theorem is true, then gives a chatty proof of said theorem. That'd be fine, but when you get to the exercises, you're left thinking, "huh?" The problems are either mind-numbingly routine or they are insanely beyond the scope of the text, requiring proofs of things that are MAJOR THEOREMS IN ELEMENTARY NUMBER THEORY with absolutely no context, hints, or help. Granted, the instructor of the course gave hints/direction for a lot of these problems, but without a good professor's guidance, good luck trying to prove major theorems all on your own for homework!

Yeah, don't use this book. It's just not very helpful to the student and should only be used if the course requires it AND you have a knowledgeable instructor who can give good guidance.

Great for a casual exploration of the topic
I can understand much of the criticism that I read here from frustrated math majors.I just want to say that, as an engineer who took a number theory course for fun, this was a great introduction to the subject.I found it very readable and easy to understand.It got me interested in number theory - enough so that I would consider reading a bit of it on my own time as I pursue further education in science.It seems that engaging non-mathematicians is the intent, so I have to consider the book a success.

P.S.Can you really complain about the style of the book after reading the title?You certainly can't claim false advertisement.

Pretty good!
I used this book for my Introduction to Number Theory class.I enjoy Silverman's writing style, but I wish there were some more examples and a little bit more theory involved.

It seemed to me as though there were a LOT of topics covered in a short about of time, but I would have liked to have seen some more of the actual meat behind it.

Not bad though!

For its intended audience, this is a gem....
I am a working oceanographer with a physics background who is interested in browsing through various areas of mathematics, particularly ones like number theory which are not a common part of a physicist's background. I picked up and read Dr. Silverman's "Friendly Introduction to Number Theory" and was thoroughly charmed. The book presented many of the basic results of number theory in a clear, concise fashion, and also gave a bit of context and background to the results. Basis computations for "non-experts" are stressed, and the reader for whom this book was intended goes away with a nice feeling of having picked up a bit of knowledge of a new topic. I would also add my voice to those who chided the math majors for panning this book. There are plenty of high level "theorem-proof" books out there for mathematicians, and to criticise a book that popularizes mathematics is both snobbish and counterproductive. We should heartily applaud and value good popularizations of science and technology. This book is a first rate popularization. ... Read more

Product Description
This undergraduate-level introduction describes those mathematical properties of prime numbers that can be deduced with the tools of calculus.Jeffrey Stopple pays special attention to the rich history of the subject and ancient questions on polygonal numbers, perfect numbers and amicable pairs, as well as to the important open problems.The culmination of the book is a brief presentation of the Riemann zeta function, which determines the distribution of prime numbers, and of the significance of the Riemann Hypothesis. ... Read more

Customer Reviews (4)

A Readable Delightful book
This is definitely one of the delightful math books to read.The material is so well organized that it flows very nicely.This book provides a very gentle introduction to such topics as Zeta function and Prime Number Theory.

One of my favorite math books
A little background on me. I have just finished my freshman year of high school, and this was my first book on number theory. However, I have read many other math texts. In the beginning of the book there are some new concepts introduced, but they are not too hard to understand. The middle is refreshing as it involves a lot of calculus, which the student is most likely familiar with. The latter part consists of a variety of new ideas, and the theorems can get quite lengthy. I do not fully understand all of them myself. The book is well written and also includes the history of many mathematical problems.

For the senior math undergraduate
A great book for senior undergraduates in mathematics or anyone with some background in calculus and complex numbers.Proofs are at a level where a careful reading makes them clear, and the author tells the reader when he is not being rigorous.Historical background and logical development of topics makes this a good read too.Most surprising to me was how the author tied in topics from prior chapters into later chapters--he didn't just jump from one topic to the next willy-nilly, but made the book flow as a whole.Problems given to the reader were helpful though sometimes too hard for me, a math major.

Do you like primers? ...and number theory? Well here you go!
There usually seems to be a pretty big gap between the math background needed to understand books on elementary number theory and what's needed to understand most books dealing with analytic number theory, and this book does a good job of making that gap seem smaller.The writing feels a bit like Silverman's "Friendly Introduction to Number Theory" and Derbyshire's "Prime Obsession."There are plenty of experiments for Mathematica and Maple.I could see this book being used in an undergraduate number theory class.The book doesn't assume any familiarity with complex variables. If you can integrate and aren't too afraid of series or logarithms, this book should be no problem.

The book goes over multiplicative functions, Mobius inversion, the Prime Number Theorem, Bernoulli numbers, the Riemann zeta function (and its value at 2n, its analytic continuation, its functional equation, and the Riemann Hypothesis), the Gamma function, Pell's equation, quadratic reciprocity, Dirichlet L-functions, elliptic curves (including their L-functions and the Birch and Swinnerton-Dyer conjecture), binary quadratic forms, and an analytic class number formula for imaginary quadratic fields.

I recommend this book to anyone who can read; and for those who can't read, this book is good motivation to become literate. ... Read more

Product Description
Superb introduction for readers with limited formal mathematical training. Topics include Euclidean algorithm and its consequences, congruences, powers of an integer modulo m, continued fractions, Gaussian integers, Diophantine equations, more. Carefully selected problems included throughout, with answers. Only high school math needed. Bibliography.
... Read more

Customer Reviews (5)

A good and brief introduction
The book covers main topics of elementary number theory. The book is very short (120 text pages) but not at cost of clarity: almost all theorems are proven in the text and many examples are given.

Not many problems have answer in the back, which is not good thing for self-studying.

The text does not require much mathematical background (I believe highschool is enough), and I can recommend the book to anyone interested in number theory. The book is very well worth its price. Buy this and if you still like number theory, buy one of those heavy books over $100 :-).

Readable, clear, but needs an errata page
As others have said, this is a fairly easy read.For me it's actually fun and I'm working through it for that reason.But:

- I don't normally use a highlighter, but found it necessary to highlight symbols where they were defined, because some of them come up only once in a while and it's easy to forget where the definition is.Symbols are not indexed.I have started my own symbol index in the back of the book.

- There are some annoying errors.The theorem to be proven in section 1-3, problem 2 is false for n=1.The decimal expansions in the chapter on continued fractions (page 75) are wrong (for example 1.273820... should actually be 1.273239...).It seems to me if you're going to give 7 digits they should be the right 7 digits.

On the other hand, these errors don't affect the overall flow of the text, and I'm having a great time working through this book on my own.I've read through the whole thing over the summer, and I'm going back through doing problems and writing programs.I was a math major 40 years ago, and haven't done much with it since, to give a context for that remark.

Very good

Very good book.First to comment on the fact that LeVeque has 2 dover books that cover basically the same topics (this one, and Fundamentals of Number Theory).I have looked at both, and this one is the better of the two.The other one uses slightly different definitions that have an Abstract Algebra twist to it.But the other book still doesn't use the power of abstract algebra so the different/akward definitions and explanations just make it hard to read.

An elementary number theory book should use elementary definitions and concepts (abstract algebra is meant for ALGEBRAIC number theory books).So avoid his other book, which is good, but not as easy to read as this one.

This book is very easy to read and concepts are introdced very clearly.Things come in small chunks which are easily digested.The thing about this book is, you can go through it faster than normal textbooks but you still end up learning everything you would by going slowing through hard-to-read texts (not like The Higher Arithmetic by Davenport, that book can lull you into reading it like a story book, but you end up learning nothing).

Good Introduction to Key Topics and Proofs of Number Theory
William J. LeVeque's short book (120 pages), Elementary Theory of Numbers, is quite satisfactory as a self-tutorial text. It should appeal to math majors new to number theory as well as others that enjoy studying mathematics.

Chapter 1 introduces proofs by induction (in various forms), proofs by contradiction, and the radix representation of integers that often proves more useful than the familiar decimal system for theoretical purposes.

Chapter 2 derives the Euclidian algorithm, the cornerstone of multiplicative number theory, as well as the unique factorization theorem and the theorem of the least common multiple. Speaking from experience, I recommend that you take the time necessary to master Chapter 2, not just because these basic proofs are important, but more critically to reinforce the skills and discipline necessary for the subsequent chapters.

Two integers a and b are congruent for the modulus m when their difference a-b is divisible by the integer m. In chapter 3 this seemingly simple concept, introduced by Gauss, leads to topics like residue classes and arithmetic (mod m), linear congruences, polynomial congruences, and quadratic congruences with prime modulus. The short chapter 4 was devoted to the powers of an integer, modulo m.

Continued fractions, the subject of chapter 5, was not unfamiliar and yet, as with congruences, I quickly found myself enmeshed in complexity, wrestling with basic identities, the continued fraction expansion of a rational number, the expansion of an irrational number, the expansion of quadratic identities, and approximation theorems.

I have yet to tackle the last two chapters, the Gaussian integers and Diophantine equations, but my expectation is that both topics will also require substantial effort and time. LeVeque's Elementary Theory of Numbers is not an elementary text, nor a basic introduction to number theory. Nonetheless, it is not out of reach of non-mathematics majors, but it will require a degree of dedication and persistence.

For a reader new to number theory, LeVeque may be too much too soon. I suggest first reading Excursions in Number Theory by C. Stanley Ogilvy and John T. Anderson, another Dover reprint. It is quite good.

Some caution: LeVeque emphasizes that many theorems are easy to understand, and yet this very simplicity is a two-edged sword. Simple theorems often provide no clues, no hints, on how to proceed. Discovering a short and elegant proof is often far from easy.LeVeque also stresses that a technique ceases to be a trick and becomes a method only when it has been encountered enough times to seem natural. A reader new to number theory may initially be overwhelmed by the variety of techniques used.

A nit: The Dover edition of LeVeque's Elementary Theory of Numbers would benefit from a larger font size. I occasionally found myself squinting to read tiny subscripts and superscripts.

Contents of this Book
1. Introduction 2. The Euclidean Algorithm and Its Consequences 3. Congruences 4. The Powers of an Integer Modulo "m" 5. Continued fractions 6. The Gaussian Integers 7. Diophantine Equations

Plus the sample problems and solutions in the above area. ... Read more

Product Description
Who were the five strangest mathematicians in history?What are the ten most interesting numbers? Jam-packed with thought-provoking mathematical mysteries, puzzles, and games, Wonders of Numbers will enchant even the most left-brained of readers. Hosted by the quirky Dr. Googol--who resides on a remote island and occasionally collaborates with Clifford Pickover--Wonders of Numbers focuses on creativity and the delight of discovery. Here is a potpourri of common and unusual number theory problems of varying difficulty--each presented in brief chapters that convey to readers the essence of the problem rather than its extraneous history. Peppered throughout with illustrations that clarify the problems, Wonders of Numbers also includes fascinating "math gossip." How would we use numbers to communicate with aliens?Check out Chapter 30. Did you know that there is a Numerical Obsessive-Compulsive Disorder? You'll find it in Chapter 45. From the beautiful formula of India's most famous mathematician to the Leviathan number so big it makes a trillion look small, Dr. Googol's witty and straightforward approach to numbers will entice students, educators, and scientists alike to pick up a pencil and work a problem. ... Read more

Customer Reviews (18)

Another Great Book on Recreational Mathematics
If you enjoy Recreational Mathematics and have enjoyed other books by the author, then you will like this book as well. A few things to keep in mind about this specific book:

1. Chapters are VERY short, some less than 1 page. Author covers a lot of ground. It may not be detailed enough for some, but for the most part, I though there was enough background history and examples to sufficiently introduce each topic. It is better to know something exists than to never be exposed to it, and this book will expose you to a lot!
2. Answers and followup discussions are not at the end of the chapter where I think they should be, but in a separate section toward the end of the book. I don't like this style as it requires you to use 2 bookmarks and constantly go back and forth. One star penalty for this.
3. Many of the great problems posed in the book are answered, but some are specifically left out. I think it should be up to the reader to decide how much they want to spoil the fun.

Summed up, a good book well worth getting and reading.

I love Mathematics
I am still reading the book.
It is elementar, but very interesting.

A delightful collection of mathematical puzzles
This book contains a delightful collection of mathematical puzzles in the tradition of Martin Gardner. There are Klingon Paths, Hexagonal Cats, Messages from the Stars, and Doughnut Loops. If you liked the puzzles in Pickover's "Alien IQ Test", you will like the puzzles in this book.

The book is not all numbers. There are historical anecdotes and stories about mathematicians told by the author's alter-ego, Dr. Googol. Are all mathematicians insane? The answer is not clear. However, the author describes the five strangest. Did you know that Pythagoras believed that it was sinful to eat beans?

There are a number of interesting top ten lists. As one who thinks that the proper role of mathematics is to solve the problems of the physical world, I was happy to note that Dr. Googol chose equations of physics for six of the ten most important mathematical expressions, e.g. Gauss' law and Newton's law of gravitation. Dr. Googol must have some physicist friends.

This is just one in a series of wonderful books that Dr. Pickover has written. I also recommend "The Science of Aliens, or Time: A Traveler's Guide", and his new book "A Passion for Mathematics".

Mind blowing !!
The book provides very valuable information about mathematics.The language is simple and any leyman can understand it well.The book also provides brain teasers to refresh your mind.And DEFINATELY this book will generate your interest in Mathematics.Thanks Clifford Pickover.
KB.

More unusual mathematics from a master
Narrated by the outstanding and eccentric mathematician Dr. Francis Google, this book is a collection of unusual mathematics problems, from those involving very large numbers to those defined by applying operations. For example, the Leviathan number (10^666)! is used to demonstrate that it is not necessary to compute a number to learn some of the properties that it has. Sets of numbers such as apocalyptic numbers, those that involve 666, the number of the beast, appear several times. One of my favorites are the Schizophrenic numbers, defined by the formula f(n) = 10 * f(n-1)+n, f(0) = 0, which is a set of integers demonstrating a simple pattern. However, the action starts when the square roots of the numbers are taken. These roots exhibit an unusual, repeated pattern in their digits.
Some incidents of mathematical history that are interesting trivia are also used. The number 365, 365, 365, 365, 365, 365 is supposedly the largest number that was ever squared in the head of a human. Other segments were based on surveys, where people answered questions such as, "Which would have had the greatest impact on the world as we know it today: `If Albert Einstein had lived another twenty years with a clear mind?', `If mathematician Srinivasa Ramanujan had lived another twenty years with a clear mind?", If Steven Hawking was not afflicted with Lou Gehrig's disease?'."A ranking of the top eight female mathematicians of all time, a listing of the five greatest scandals in mathematics history, the ten most important unsolved mathematical problems, the ten most influential mathematicians of all time, the ten most influential mathematicians alive today and the ten most difficult areas of mathematics to understand provide additional intellectual fodder.
Every time I read a Pickover book, the number of ideas used as the seeds to generate the text astounds me. He always seems able to come up with new twists on old problems and sometimes new problems that set your brain moving in circular motions as you try to comprehend the consequences of the statements and attempt to follow the logical consequences of the transformations. While some of the best books keep you reading from page to page without stopping, others cause you to read a little, process a lot and then read some more. That is what this book did to me, and I am sure that it will do the same to you.

Published in the recreational mathematics e-mail newsletter, reprinted with permission. ... Read more

Product Description
This is a second edition of Lang's well-known textbook. It covers all of the basic material of classical algebraic number theory, giving the student the background necessary for the study of further topics in algebraic number theory, such as cyclotomic fields, or modular forms. Part I introduces some of the basic ideas of the theory: number fields, ideal classes, ideles and adeles, and zeta functions. It also contains a version of a Riemann-Roch theorem in number fields, proved by Lang in the very first version of the book in the sixties. This version can now be seen as a precursor of Arakelov theory. Part II covers class field theory, and Part III is devoted to analytic methods, including an exposition of Tate's thesis, the Brauer-Siegel theorem, and Weil's explicit formulas. This new edition contains corrections, as well as several additions to the previous edition, and the last chapter on explicit formulas has been rewritten. ... Read more

Customer Reviews (1)

Classic
Awesome text.For the more well-versed reader in Algebraic Number Theory.Great resource for a variety of topics. ... Read more

Product Description
Traditionally, elementary number theory is a branch of number theory dealing with the integers, but without use of techniques from other mathematical fields. In this concise and elegant book, the author has sought to pare away all material that might be considered extraneous to a three-hour-per-week, twelve-week semester course in elementary number theory. The author presents in natural sequence the basic ideas and results of elementary number theory, laying a strong foundation for later studies in algebraic number theory and analytic number theory. The only background knowledge required of the reader is of some simple properties of the system of integers. Elementary Number Theory begins with a few preliminaries on induction principles, followed by a quick review of division algorithms. Then in the second chapter, the author touches upon the usage of divisors, the greatest (or least) common divisor (multiple), the Euclidean algorithm, and linear indeterminate equations. This foundation supports discussions in the subsequent chapters concerning: prime numbers; congruences; congruent equations; and, finally, three additional topics (comprising cryptography, Diophantine equations and Gaussian integers). Each chapter concludes with exercises that both illustrate the theory and provide practice in the techniques. Answers to even-numbered problems are given at the end of the book. ... Read more

Product Description

The central theme of this book is the solution of Diophantine equations, i.e., equations or systems of polynomial equations which must be solved in integers, rational numbers or more generally in algebraic numbers. This theme, in particular, is the central motivation for the modern theory of arithmetic algebraic geometry. In this text, this is considered through three of its most basic aspects. The book contains more than 350 exercises and the text is largely self-contained. Much more sophisticated techniques have been brought to bear on the subject of Diophantine equations, and for this reason, the author has included five appendices on these techniques.

Product 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. ... Read more

Customer Reviews (4)

good book
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
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!
I have started my studies in Number Theory reading this book from thepreface to the last word. It is amazing! I think it is a betterintroduction to the subject than the classical Hardy and Wright...it is"more objective" and almost 100% elementary...a good high schoolreader could do well with it. The chapter of diophantine equations has somedivine proofs, very clever and very beautiful. And there is an easy proofof the irracionality of Pi. The only negative point is the existence ofsome points where the authors could be less concise and a bit clearer,stating the theorems before giving the demonstrations, instead of saying atthe 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...
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. ... Read more

Product Description

"This book is the first volume of a two-volume textbook for undergraduates and is indeed the crystallization of a course offered by the author at the California Institute of Technology to undergraduates without any previous knowledge of number theory. For this reason, the book starts with the most elementary properties of the natural integers. Nevertheless, the text succeeds in presenting an enormous amount of material in little more than 300 pages."-—MATHEMATICAL REVIEWS

Customer Reviews (6)

Introduction to Analytic Number Theory
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
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
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 isquite 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
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
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 0The 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'. ... Read more

Product Description
A description of 148 algorithms fundamental to number-theoretic computations, in particular for computations related to algebraic number theory, elliptic curves, primality testing and factoring. The first seven chapters guide readers to the heart of current research in computational algebraic number theory, including recent algorithms for computing class groups and units, as well as elliptic curve computations, while the last three chapters survey factoring and primality testing methods, including a detailed description of the number field sieve algorithm. The whole is rounded off with a description of available computer packages and some useful tables, backed by numerous exercises. Written by an authority in the field, and one with great practical and teaching experience, this is certain to become the standard and indispensable reference on the subject. ... Read more

Customer Reviews (3)

Great book for computational aspects
I bought this book for the math course I had taken having the same title. This is an excellent book, but only if you are really interested in its content. It's not a casual read, since it's graduate text. Also, a background in number theory will be of great help - being a CS major, I had a little tough time in the beginning, but things turned out just fine. As for content, it has excellent coverage of the subject, and I would highly recommend this as a reference in this subject. Remember, though, that this book deals COMPUTATIONAL aspects, for only number theory, look for other books like Ireland-Rosen.

Definitely belongs on the shelf of all number theory lovers
This book is an excellent compilation of both the theory and pseudo-code for number theoretic algorithms. The author also takes the time to prove some of the major results as background to the algorithms, in addition to sets of exercises at the end of the book. The book is too large to do a chapter by chapter review, so instead I will list the algorithms in the book that I thought were particularly useful:

1. Most of the algorithms on elliptic curves. The author reminds the reader that number-theoretical experiments resulted in the famous Swinnerton-Dyer Conjecture and the Birch Conjecture. (a) the reduction algorithm, which for a given point in the upper half plane, gives the unique point in the half plane equivalent to this point under the action of the special linear group along with the matrix that maps these two points to each other. (b) The computation of the coefficient g2 and g3 of the Weierstrass equation of an elliptic curve. (c) The computation of the Weierstrass function and its derivative. (d) Determination of the periods of an elliptic curve over the real numbers. (e) The determination of the elliptic logarithm. (f) The reduction of a general cubic (f) The Shanks-Mestre algorithm for computing the order of an elliptic curve over a finite field F(p), where p is prime and greater than 457. (g) The reduction of an elliptic curve modulo p for p > 3. (h) The reduction of an elliptic curve modulo 2 or 3. (i) Reduction of an elliptic curve over the rational numbers. (j) Determination of the rational torsion points of an elliptic curve. (k) Computation of the Hilbert class polynomials and thus a determination of the j-function of an elliptic curve.

2. A few of the algorithms on factoring. (a) The Pollard algorithm for finding non-trivial factors of composites. (The author does not give the improved algorithm due to P. Montgomery, but does give references) (b) Shanks Square Form Factorization algorithm for finding a non-trivial factor of an odd integer. (c) Lenstra's Elliptic Curve test for compositeness.

3. Primality tests (a) The Jacobi Sum Primality Test for a positive integer. (b) Goldwasser-Killian elliptic curve test for a positive integer not equal to 1 and coprime to 6.

The author gives an overview of the computer packages used for number theory, including Pari, which was written by him and his collaborators. I have not used this package, but instead use Lydia and Mathematica for most of the number theoretic computations I need to do.

Excellent!
Cohen (the world renowned expert) starts with the most basic of algorithms(i.e. Euclid & Shanks). He moves seamlessly into Linear Algebra &Polynomials (bedrocks of most CAS). Although meant to be concise, heproves, or sketches a proof of the important results. Finally, the meat ofthe book, C.A.N.T. One important problem is finding the "classnumber" (has to do with unique factorization, which we are allaccustomed to in Z). A detailed description of the continued fractionalgorithm (for finding the fundamental unit), and others made it veryenlightening. He then deals with primality testing and factoring, two veryimportant problems, the latter because of RSA. First, a description of thealgorithm, then the theory behind it. He covered everything, from TrialDivision (Dark Ages) to Pollard Rho to NFS (cutting-edge). Also includedare some useful tables.

Of course, CAS information from 1993, won't bethat helpful (look in his newest, Advanced Topics inC.A.N.T.).

Excellent. Also try Knuth's "Semi-numericalAlgorithms" for a more computer oriented approach. ... Read more