Book Description
iAn Introduction to the Theory of Numbers/i 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 iAn Introduction to the Theory of Numbers/i 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 readerThe 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 (8)

Nice intro to number theory
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
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
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. Thebook 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!!!!
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.
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 textbookfor 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 atall 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. ... Read more

Amazon.com
When Andrew Wiles of Princeton University announced a solution of Fermat's last theorem in 1993 it electrified the world of mathematics. After a flaw was discovered in the proof, Wiles had to work for another year--he had already labored in solitude for seven years--to establish that he had solved the 350-year-old problem. Simon Singh's book is a lively, comprehensible explanation of Wiles's work and of the star-, trauma-, and wacko-studded history of Fermat's last theorem. Fermat's Enigma contains some problems that offer a taste for the math, but it also includes limericks to give a feeling for the goofy side of mathematicians.Book Description
xn + yn = zn, where n represents 3, 4, 5, ...no solution

"I have discovered a truly marvelous demonstration of this proposition which this margin is too narrow to contain."

With these words, the seventeenth-century French mathematician Pierre de Fermat threw down the gauntlet to future generations.  What came to be known as Fermat's Last Theorem looked simple; proving it, however, became the Holy Grail of mathematics, baffling its finest minds for more than 350 years.  In Fermat's Enigma--based on the author's award-winning documentary film, which aired on PBS's "Nova"--Simon Singh tells the astonishingly entertaining story of the pursuit of that grail, and the lives that were devoted to, sacrificed for, and saved by it.  Here is a mesmerizing tale of heartbreak and mastery that will forever change your feelings about mathematics. ... Read more

Customer Reviews (238)

Satisfied Customer
The book I ordered for a Christmas gift was received on time and was in perfect shape.

Epic Tale of Mathematical Mt. Everest
If you don't think math can be sexy and exciting, then you ought to give Singh's book a read.

By the time Pierre de Fermat - sometime civil servant and occasional brilliant mathematician - left this earth, he'd left an indelible mark on the 17th century.His work with Pascal on "laws of chance" (considered by some an oxymoron) was groundbreaking and enduring, as was his contribution to proto-calculus.

Fermat, though, was very often remembered for something he wrote in contrast with Pythaoreas' famous theorum (x squared plus y squared equals z squared).Fermat posited that x to the n power plus y to the n power equals z to the n power where n is any whole number greater than two has no solutions.He then wrote in his book "I have a truly marvelous proposition which this margin it too narrow to contain."

This entry - a margin note in Fermat's copy of a venerable math text - served as a nagging problem for the next several centuries.

One of the beauties of Dr. Singh's book is the accessible and fascinating mathematical history provided as a context for the narrative of this striking journey up a mathematical Mt. Everest.From the secrets of the ancient Greek Pythagorean Brotherhood through the tragic tale of 20th century Japanese number-cruncher Yutaka Taniyama, the story unfolds as a marked human tale complete with duels, contests, clashes of personality and numerous painful mathematical dead-ends.

The backdrop for the entire story is the modern account of Andrew Wiles whose lifelong goal was to solve Fermat's riddle.This quiet mathematician was thrust into the spotlight through his wrestling with the venerable problem, and this is in a great way his story.Dr. Singh also produced the award winning documentary "The Proof" for PBS.

Great book
The book presents the history of number theory, starting from Pythagoras and culminating with Andrew Wiles, the mathematician who solved Fermat's Last Theorem. The author explains, for example, how and why negative and imaginary numbers were introduced, and sometimes throws in a riddle for the reader to solve (the answer is in the appendix). You don't need to know advanced mathematics in order to understand this book, since the author does a good job at explaining things in a way that everybody understands. Along the way, he gives an insight on the life of the great mathematicians, some of which are tragic. Good read, I highly recommend it!!!

One of the deep pleasures of my reading life...
Once upon a time, two roads diverged into a yellow wood and I elected to pursue the path of an English and literature major. Along the neglected wayside lay a path toward mathematics, a discipline in which I was more academically proficient (not that I was bad at the other) but the purpose and pleasure of which had somehow escaped me. Though math had always been relatively easy for me, I believed -- perhaps through the way it had been taught and presented -- that it had never been "fun." Somehow its beauty had lain hidden beneath the drudgery of pedagogy. It was only years later that I discovered G. H. Hardy's "A Mathematician's Apology" and E. T. Bell's "Men of Mathematics" and learned to appreciate the joy of mathematics and its symmetry (ironically, the same appreciation that fires my love for Jane Austen). Though I will never become a professional mathematician, it is through books like "Fermat's Enigma" that I can share in the mathematician's joy of discovery that is the holy grail of every intellectual pursuit.

To ignore or to shy from this book because of a fear of math is to place unnecessary limits on your understanding of the world. Simon Singh does a masterful job of making accessible the story of an impenetrable math problem and telling it through the history of the mathematics that surround it and through the personalities of the mathematicians who pitted themselves against it and who ultimately led to its solution. There are no threatening equations here that are not solved for you -- only the challenge to get your mind around ideas that perhaps are new.

There is tremendous human drama along the way -- from the struggles of Sophie Germaine, who disguised herself as a man to study under the great mathematician Carl Gauss, to Evariste Galois, the brilliant young mathematician who died at the age of twenty in a dual he felt powerless to avoid, to Yutaka Tanyama, the brilliant Japanese mathematician whose conjecture pointed the way to Andrew Wiles' ultimate solution but who tragically took his own life before he was to ever see the proof, to the elation and doubts of Wiles himself when his much-heralded proof was found eventually to contain a fatal error. The story is not just a thumbnail history of the purest of sciences, but a story of human redemption as we see Wiles' struggles to correct his proof beneath the uncomfortable gaze of an anxious academic community.

Give yourself a chance on this one. I have read it time and again (and again) and have never failed to be moved by Wiles' ultimate triumph -- or in how articulately he uses metaphor to state his problem or the honesty with which he addresses his failures and shares his joy.

Informative, but not casual reading
Simon Singh makes difficult topics readable, although not necessarily totally understandable. This comment applies to Fermat's Enigma. The bulk of the book concerns the history of number theory and here it is truly outstanding. The description of the final solution was Jabberwocky to me. ... Read more