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In the spirit of The Book of the One Thousand and One Nights, the authors offer 1001 problems in number theory in a way that entices the reader to immediately attack the next problem. Whether a novice or an experienced mathematician, anyone fascinated by numbers will find a great variety of problems--some simple, others more complex--that will provide them with a wonderful mathematical experience. ... Read more

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This volume is a translation of Dirichlet's Vorlesungen über Zahlentheorie which includes nine supplements by Dedekind and an introduction by John Stillwell, who translated the volume.

Lectures on Number Theory is the first of its kind on the subject matter. It covers most of the topics that are standard in a modern first course on number theory, but also includes Dirichlet's famous results on class numbers and primes in arithmetic progressions.

The book is suitable as a textbook, yet it also offers a fascinating historical perspective that links Gauss with modern number theory. The legendary story is told how Dirichlet kept a copy of Gauss's Disquisitiones Arithmeticae with him at all times and how Dirichlet strove to clarify and simplify Gauss's results. Dedekind's footnotes document what material Dirichlet took from Gauss, allowing insight into how Dirichlet transformed the ideas into essentially modern form.

Also shown is how Gauss built on a long tradition in number theory--going back to Diophantus--and how it set the agenda for Dirichlet's work. This important book combines historical perspective with transcendent mathematical insight. The material is still fresh and presented in a very readable fashion.

This volume is one of an informal sequence of works within the History of Mathematics series. Volumes in this subset, "Sources", are classical mathematical works that served as cornerstones for modern mathematical thought. (For another historical translation by Professor Stillwell, see Sources of Hyperbolic Geometry, Volume 10 in the History of Mathematics series.) ... Read more

Customer Reviews (1)

Gauss and then some
Dirichlet is all about quadratic forms. But first there are three preliminary chapters on the tools we will need: unique factorisation, modulo arithmetic, quadratic reciprocity. Then in chapter 4 we get to the quadratic forms, ax^2+2bxy+cy^2. "The whole theory originates in the problem of deciding whether a given number is representable by a given form" (p. 92). (Remember, for example, that Fermat solved the case a=1, b=0, c=1 -- which integers are sums of two squares?) "The number b^2-ac, on which the properties of the form mainly depend, is called the determinant of the form", and two forms are equivalent (represent the same numbers) when one results from the other by applying a variable transformation matrix of determinant 1. And now the problem above reduces to "the two main problems in the theory of equivalence: I. To decide whether two given forms of the same determinant are equivalent. II. To find all substitutions that send one of two equivalent form to the other." (p. 100). We spend the rest of the chapter solving there two problems for any determinant D, and we work out the applications in the cases D=-1,-2,-3,-5 (which includes the theorem of Fermat above). In the cases D=-3,-5 representations are not generally unique (which we secretly think of as the manifestation of the loss of unique factorisation in Z[sqrt(-3)] and Z[sqrt(-5)]) and this goes hand in hand with the fact that the number of equivalence classes (the "class number") of forms in those cases is 2, not 1. Such matters are the motivation for Dirichlet's great contribution: the determination of the class number for any D (chapter 5). Apart from this motivation of measuring "how far quadratic integers deviate from unique prime factorisation", as Stillwell puts it (p. xvii), Dirichlet also assigns his solution of the class number problem great intrinsic beauty: "This problem is the last and most important solved in this book, and is connected with the most beautiful algebraic and analytic investigations of this century" (p. 100).

This is a pleasant book. Dirichlet is a celebrated expositor and quite rightly so. There is also an excellent 10 page introduction by Stillwell and some 70 pages of supplements by Dedekind. The most interesting supplement is certainly Dirichlet's famous L-series proof that there are infinitely many primes in essentially any arithmetic progression. ... Read more

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"The present book has as its aim to resolve a discrepancy in the textbook literature and ... to provide a comprehensive introduction to algebraic number theory which is largely based on the modern, unifying conception of (one-dimensional) arithmetic algebraic geometry. ... Despite this exacting program, the book remains an introduction to algebraic number theory for the beginner... The author discusses the classical concepts from the viewpoint of Arakelov theory.... The treatment of class field theory is ... particularly rich in illustrating complements, hints for further study, and concrete examples.... The concluding chapter VII on zeta-functions and L-series is another outstanding advantage of the present textbook.... The book is, without any doubt, the most up-to-date, systematic, and theoretically comprehensive textbook on algebraic number field theory available." W. Kleinert in Z.blatt f. Math., 1992 "The author's enthusiasm for this topic is rarely as evide!nt for the reader as in this book. - A good book, a beautiful book." F. Lorenz in Jber. DMV 1995 "The present work is written in a very careful and masterly fashion. It does not show the pains that it must have caused even an expert like Neukirch. It undoubtedly is liable to become a classic; the more so as recent developments have been taken into account which will not be outdated quickly. Not only must it be missing from the library of no number theorist, but it can simply be recommended to every mathematician who wants to get an idea of modern arithmetic." J. Schoissengeier in Montatshefte Mathematik 1994 ... Read more

Customer Reviews (3)

Must have for AN
This book is great.It takes a different approach than some other texts I have recently read but is an excellent starting point for those interested in AN.

One of the most beautifully written math books
This book is basically all you need to learn modern algebraic number theory. You need to know algebra at a graduate level (Serge Lang's Algebra) and I would recommend first reading an elementary classical Algebraic number theory book like Ian Stewart's Algebraic Number Theory, or Murty and Esmonde's Problem's in Algebraic Number theory.

10 stars if I could.
After having no fun with Lang's text "Algebraic Number Theory" I began seking out something more complete and which was full of quality exposition.As a result of Amazon's approach to marketing towards members, I was recommended this book and decided quickly that I must have it.This book is marvelously well written, examples are kept to an un-overwhelming minimum, the problems are not trivial (at least to me) and in fact I feel this is the kind of book on par with, say, Paulo Ribenboim's "Classical Theory of Algebraic Numbers" since these are both the type of book you would want to take with you on a long trip or as Paulo says, "while stranded on a desert island".This book is by no means intended for those who are not fluent in both Number Theory as well as Algebra, both at the graduate level and obviously for those who are Mahematically gifted.I highly recommend this book to graduate students interested in Algebraic number theory as well as those needing a splendid reference. ... Read more

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Product Description
Publisher: The Open Court Pub. Co.Publication date: 1901Subjects: Number theoryNumbers, Theory ofNumbers, IrrationalIrrational numbersMathematics / GeneralMathematics / Algebra / GeneralMathematics / LogicMathematics / Number TheoryMathematics / Set TheoryNotes: This is an OCR reprint. There may be typos or missing text. There are no illustrations or indexes.When you buy the General Books edition of this book you get free trial access to Million-Books.com where you can select from more than a million books for free. You can also preview the book there. ... Read more

Customer Reviews (5)

Foundations of the reals, foundations of the integers
The first essay (27 pages) is Dedekind's excellent exposition of his "Dedekind cuts" definition of the real numbers. This constructs the reals from the rationals and proves that there are no gaps left. The only downside, Dedekind anticipates, is that the principles on which this proof is built are so "common-place" that "the majority of my readers will be very much disappointed" (p. 11) that there is nothing more to it.

The second essay develops basic set theory and uses it in particular to build a foundation for the integers in terms of the successor function and induction. It's a plain definition-theorem-proof account, just as may be found in any foundations of mathematics book of today. There is none of the enthusiasm of the first essay. Dedekind even says that working out this theory was a "wearisome labor" (p. 41, referring to building the theory of infinite sets from the definition that a set is infinite if it can be put in one-to-one correspondence with a subset of itself---of course, Dedekind doesn't make this remark out of dislike of mathematics, but for the sake of alerting others to the brilliance of his work, which incidentally seems to have been one of his key motivations throughout).

Will Appeal to Students of Mathematics and Philosophy
Richard Dedekind (1831-1916) is recognized as one of the great pioneers in the logical and philosophical analysis of the foundations of mathematics. Dedekind completed his doctoral studies under Gauss, was a friend of Cantor and Riemann, and worked under Dirichlet.

This inexpensive, 115-page book, Essays on the Theory of Numbers, contains two essays: his brief, famous essay Continuity and Irrational Numbers and his longer paper The Nature and Meaning of Numbers. ThisDover edition (1963) is an unabridged and unaltered copy of the 1901 authorized English translation by mathematician W. W. Beman.

I particularly enjoyed his famous essay on the Dedekind cut and irrational numbers. Dedekind writes clearly and carefully and this first paper should appeal to all students of mathematics.The intent of the longer essay was to provide a logical basis for finite and infinite numbers as well as demonstrating the logical validity of mathematical induction. I had some difficulty with The Nature and Meaning of Numbers as some of Dedekind's terminology is outdated and unfamiliar.

Some statements can be reformulated easily to modern terminology. For example, simply substitute set for system and proper set for proper system. Dedekind uses the term transformation for function (or mapping). Inverse transformations and identical transformations are the same as inverse functions and identical mappings.

A system may be compounded from other systems (same concept as union of sets).The community of systems A, B, and C is the same as intersection of sets A, B, and C. While admitting that a null system has some value, Dedekind deliberately avoided using the concept of a null set in these essays. I did not at first recognize that similar or distinct transformations were equivalent to one-to-one mappings. I had difficulty with the Dedekind's use of the term chain when discussing the transformation of a system S into itself.

Dedekind was not successful in imposing his terminology on later mathematicians. Nonetheless, Dedekind's essays had considerable influence on mathematics, not only for their content, but for their clarity of expression.

Minor points: This 1901 translation often employs an unusual positioning of the verb 'is': If R, S are similar systems, then is every part of S also similar to a part of R. Also, while I encountered a few typos, none were particularly troublesome.

An interesting pair of historical essays
Richard Dedekind is one of the fathers of modern mathematical proofs. Reading his work will give you a glimpse into the early stages of this development. Indeed, his essay on Continuity and Irrational Numbers was, in part, written because Dedekind was trying to provide some rigor to what was not yet a rigorous science. The first essay is a classic. It is his description of a means of defining a number in a given space, which has since been referred to as a "Dedekind cut." His descriptions and proofs are exceptionally clear and straightforward. The second essay is a discussion of how a number system is constructed and its characteristics. It, too, shows Dedekind to possess a excellent ability to explain the ideas very clearly and simply.

There are two difficulties with the book, which I found serious enough to warrant only four stars. First, the terminology is rather antiquated, so that the descriptions are clear only once you are able to translate Dedekind's phrases; for instance, "a system S is compounded from the systems A and B" would today be written "the set S is the union of sets A and B." Second, there are a fairly large number of typos in the book, given its importance and the rigorousness of the work; for example, in the proof in paragraph 42 of The Meaning of Numbers, the phrase (not in Dedekind's shorthand) "the transformation of A is contained in B" should read "the transformation of A is contained in A." Most typos are as minor as this, but annoying in the unnecessary effort needed to bull ones way through them. A couple errors are more significant. I blame the translator and proofreader, not Dedekind.

All in all, the book is well worth the price and the effort to understand it.

Accessible genius
This is not a book of "number theory" in the usual sense. It is a book combining two essays by Dedekind: "Continuity and irrational numbers" is Dedekind's way of defining the real numbers from rational numbers; and "The nature and meaning of numbers" where Dedekind offers a precise explication of the natural numbers (using what are now called the Peano axioms, since Peano made so much of them after reading Dedekind). They are essays in logic, or foundations of mathematics, or philosophy, as you like. And they are brilliant, readable, works of genius.

Probably the main value of the book is as an introduction to Dedekind's way of thinking about mathematics: his clarity, precision, and way of cutting to the bare core of a subject. You can find the same genius in Dedekind's THEORY OF ALGEBRAIC INTEGERS (available in a fine English translation by John Stillwell) but of course that is a more advanced text. The same style of thought works powerfully in all of Dedekind's mathematics. But most of it is very hard stuff. Here you see it in easily accessible form, suitable for even a smart high school student willing to think hard.

A very good Introduction to Number Theory
If you wan a text to introduce you at Number Theory this is your best Philosophical option ... Read more

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Global class field theory is a major achievement of algebraic number theory based on the functorial properties of the reciprocity map and the existence theorem. This book explores the consequences and the practical use of these results in detailed studies and illustrations of classical subjects. In the corrected second printing 2005, the author improves many details all through the book.

Product Description
A noted mathematician and teacher offers a witty, historically oriented introduction to number theory, dealing with properties of numbers and with numbers as abstract concepts. Written for readers with an understanding of arithmetic and beginning algebra, the book presents classical discoveries of number theory, including the work of Pythagoras, Euclid, and others. ... Read more

Customer Reviews (8)

An informal introduction to number theory with a historical perspective.
Friedberg's text, which is written in an inviting conversational tone, is an idiosyncratic introduction to number theory which stresses the subject's historical development.The material is introduced through problems that motivate the results that Friedberg discusses.These results include Euclid's theorem that there are infinitely many prime numbers, the use of the sieve of Eratosthenes to find prime numbers less than the square root of a positive integer n, Gauss' Fundamental Theorem of Arithmetic, perfect and amicable numbers, Pythagorean triples, modular arithmetic, factoring numbers of the form x^2 + ny^2, and the Law of Quadratic Reciprocity.Friedberg ably links these topics together and places them in historical perspective.However, there are better introductions to number theory.This text has no formal exercises, so you do not have an opportunity to reinforce what you are learning. It is also a poor reference because definitions, theorems, and proofs are stated within paragraphs, the whimsical chapter titles do not convey what topics are covered, and there is no subject index to help you find the definitions and theorems that are buried within the paragraphs.Also, the scope of coverage is less than that of other introductions to number theory.

In his introduction, Friedberg, a physicist, distinguishes between the common and scientific meanings of the word theory.He also discusses the difference between a scientific theory and a mathematical theorem.

Friedberg uses sequences to introduce proofs by mathematical induction.Friedberg shows how proofs of mathematical induction work and discusses why they are valid.In the text, however, he tends to use Fermat's method of infinite descent to prove assertions indirectly rather than using direct induction proofs.

While discussing these sequences, Friedberg refers to 1 as a prime number, contrary to the usual definition that a prime number is a positive integer larger than 1 whose only factors are 1 and itself.Defining 1 to be prime would violate the assertion of the Fundamental Theorem of Arithmetic that each positive integer has a unique prime factorization.For instance, if you allow 1 to be prime,

6 = 2 x 3 = 1 x 2 x 3 = 1 x 1 x 2 x 3 = ...

This is problematic, so Friedberg disregards his own assertion that 1 is prime when discussing the Fundamental Theorem of Arithmetic.

Friedberg then discusses some results from ancient Greece, including the fact that the square root of 2 is irrational, Euclid's theorem that there are infinitely many primes, the sieve of Eratosthenes, perfect numbers, and amicable numbers.He also proves the Fundamental Theorem of Arithmetic while covering these topics.

During a brief discussion of Diophantine equations, Friedberg discusses how to find and factor Pythagorean triples, that is, triples (x, y, z) of positive integers that satisfy the equation x^2 + y^2 = z^2.A better explanation of how to find Pythagorean triples is given by John Stillwell in his texts Mathematics and its History and Elements of Number Theory.After Friedberg's discussion of the problem, he tackles the more general problem of how to factor numbers of the form x^2 + ny^2, where n is a positive integer.The mathematics used to solve this problem, including modular arithmetic, is quite powerful, which is conveyed by the simple proofs Friedberg provides of results proved with more difficulty earlier in the book and by his proofs of Fermat's Last Theorem for the cases n = 3 and n = 4.

Friedberg concludes the book with a proof of Gauss' proof of the Law of Quadratic Reciprocity.The material on quadratic residues calls upon many of the previous results.However, while there is a table classifying the theorems in the text (albeit without their actual formal statements), the lack of a subject index makes finding the necessary definitions and theorems difficult.Consequently, Friedberg's arguments are more difficult to follow than they need to be.

If you are seeking a basic introduction to the subject, try working through Oyestein Ore's an Invitation to Number Theory (New Mathematical Library), which is accessible to a bright high school student.Ore is also the author of a slightly more advanced text, Number Theory and Its History (Dover Classics of Science and Mathematics), which, like Friedberg's text, introduces number theory through its historical development.There are numerous more advanced treatments of the subject, which serve as good introductions.They include, among others, The Higher Arithmetic: An Introduction to the Theory of Numbers by H. Davenport, Elementary Number Theory by Gareth A. Jones and J. Mary Jones, and An Introduction to the Theory of Numbers by Ivan Niven, Herbert S. Zuckerman, and Hugh L. Montgomery.

What a carefully written exploration!
I think this book is a masterpiece in mathematical exposition.All you need to know is how to add, subtract, multiply, and divide and maybe a vague memory of algebra.Mr. Friedberg will walk you through a lot of number theory after which (or maybe even during which) you may find a number theory textbook more approachable.If you read carefully you will really internalize what a proof by contradiction is and what an infinite descent is.You'll get a real appreciation for the logic of a proof and you'll see some ingenius tricks used by some great mathematicians ... and you'll understand them!
This book is approachable and doable by anyone with a motivation for what can be understood about numbers.And I can't stress how carefully, thoughtfully, and articulately it is written.

Be carefull
You must have a medium understanding of mathematics and algebra.

"Quirky" is exactly the right word.
A lot of us know that you can't double the square.You can't find two square whole numbers, one of which is twice the other.This, of course, is an ancient Greek problem.

If b squared were equal to two times a squared, the right side of the equation would contain an odd factor of two, which is obviously impossible by the fundamental theorem of arithmetic.This is the modern way of proving this assertion.

Richard Friedberg prefers the old way.He uses Fermat's method. On page 45 we read:

"At each point we can prove that the numbers we have reached are even.So we can go on dividing forever. But this is impossible.Eventually we must reach 1, or some other odd number.Since we have proved something that is impossible, we must have assumed something that isn't true.The only thing we assumed was that there are two numbers a and b, such that two times a squared equals b squared.So there can be no such numbers."

He continues in this way all the way to quadratic recipocity, and concludes with a Table of Theorems, all rigorously proved in his own quirky way.

I continue to be frustrated by Friedberg's approach to number theory.It is historically accurate but very difficult to assimilate or combine into present day orthodoxy.I'm not sure whether he is worth my time, but nevertheless I continue to study his book. I've read it on and off now for more than five years.There is no doubt in my mind that he is a genius . . . hence the five stars.

Whether one wants to embark on this slippery slope of classical geometry, historical number theory, the defects in Euler's reasoning and other incredibly obscure topics in number theory, the reader must decide for himself or herself.I don't think I'll ever know as much about the history of number theory as Richard Friedberg does, so I decided to put in my two cents mid-way through the course of my studies.

Not that adventurous
I felt like I suited up for space travel and got grounded by equipment malfunction. Perhaps I took the title too literally. Since there are so many books on number theory, surely one with such a title should cover the outer reaches. This is nothing but a basic introduction. More is covered in Albert Beiler's "Recreations in the Theory of Numbers" and it's much more adventurous. Still worth 3 stars, and worth owning - but not worth keeping under your pillow. ... Read more

Product Description
Suitable for senior undergraduates and beginning graduate students in mathematics, this book is an introduction to algebraic number theory at an elementary level. Prerequisites are kept to a minimum, and numerous examples illustrating the material occur throughout the text. References to suggested readings and to the biographies of mathematicians who have contributed to the development of algebraic number theory are provided at the end of each chapter.Other features include over 320 exercises, an extensive index, and helpful location guides to theorems in the text. ... Read more

Customer Reviews (1)

Grab bag of good and bad
Strengths:
1. Easy reading, detailed proofs
2. Covered required algebra background (modules, ideals, Dedekind domains, etc)
3. Many, many examples

Weaknesses:
1. Too detailed in some cases
2. Does not develop more advanced ideas that actually make the material easier
3. Poor index
4. Examples are often too simple

This book takes the reader through the required algebra background and moves them into the realm of using these abstract algebraic construction to study the theory of numbers. The book is aimed at upper-level undergraduates, so it's easy reading. Sometimes too easy reading, as proofs are often long-winded and contain many trivial details. In some instances, I wanted all those details, often it was simply annoying.

The real strength of this book lies in the many explicit examples. It was worth the price for these examples, as most higher-level books offer few examples.

The index is terrible, but the additional reading section at the end of each chapter is a nice addition.

Overall, I learned a lot from this book, but would have liked to have the authors approached the material at a little bit higher level. For instance, instead of using complex conjugates extensively, I would have preferred introducing a mapping to the complex conjugates (say sigma) for use in most proofs. ... Read more

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In Biscuits of Number Theory, the editors have chosen articles that are exceptionally well-written and that can be appreciated by anyone who has taken (or is taking) a first course in number theory. This book could be used as a textbook supplement for a number theory course, especially one that requires students to write papers or do outside reading. The editors give examples of some of the possibilities.

Product Description
This book provides a brisk, thorough treatment of the foundations of algebraic number theory on which it builds to introduce more advanced topics. Throughout, the authors emphasize the systematic development of techniques for the explicit calculation of the basic invariants such as rings of integers, class groups, and units, combining at each stage theory with explicit computations. ... Read more

Customer Reviews (1)

Technical, but concrete
It is an unfortunate feature of number theory that few of the books explain clearly the motivation for much of the technology introduced.Similarly, half of this book is spent proving properties of Dedekind domains before we see much motivation.

That said, there are quite a few examples, as well as some concrete and enlightening exercises (in the back of the book, separated by chapter).There is also a chapter, if the reader is patient enough for it, on Diophantine equations, which gives a good sense of what all this is good for.

The perspective of the book is global.Central themes are the calculation of the class number and unit group.The finiteness of the class number and Dirichlet's Unit Theorem are both proved.L-functions are also introduced in the final chapter.

While the instructor should add more motivation earlier, the book is appropriate for a graduate course in number theory, for students who already know, for instance, the classification of finitely generated modules over a PID.It may be better than others, but would be difficult to use for self-study without additional background. ... Read more

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Three diverting puzzles involve the proof of a basic law governing the world of numbers known to be correct in all tested cases—the problem is to prove that the law is always correct. The challenges concern van der Waerden’s theorem on arithmetic progressions, the Landau-Schnirelmann hypothesis and Mann’s theorem, and a solution to Waring’s problem. Proofs and explanations of the answers included.
... Read more

Customer Reviews (4)

Three pearls of number theory
A jewel. Highly recommended for students ofnumber theory, expecilally amateurs and teachers.

Truly a pearl.
It is truly a pearl, and pearls have permanence; --they retain their beauty, from one generation to the next.

So too is the case for this little book. Measured in mathematical generations, you must count back a few;--back to the last year of the Second World War, and in what was then The Soviet Union; now Russia. The author, A. Y. Khinchin was (and is) a mathematical physicist of World Renown. He has seminal contributions to number theory, to statistics, to information theory, and to statistical physics.

The book is unique in many ways; for one, I believe it is for everyone, -- even if you don't know math. But readers with math background will know that it is possible for writing in math to be both moving and beautiful. This is the case for this little classic. Both the historical background and the subject are unique.

The nature of the book (64 pages in all!) is almost like a personal letter written by a loving teacher to one of his students, but it is much more than that.

At the time, the War had devastated Russia, and almost everyone from the young generation, including students of the sciences was at the front. The casualties everywhere in The Soviet Union were staggering; many had lost parents and relatives during 4 long years of destruction.

Khinchin's student Seryozha was recovering (at the time of the letter) in an army hospital, and he had written his former teacher, asking for math problems to work on. We can't begin to imagine the terrible conditions of army hospitals on the front at this time. The care Khinchin took in responding is moving. In fact Seryozha had only taken one or two beginning classes at university, before being sent to war. And even though Khinchin had only a vague recollection of Seryozha from a class, he truly wanted to send him something he could use, -- something that would make him happy. Students at the front were giving their lives for the rest of the country, and we must remember that this was a war where the difference between good and evil was crystal clear. Khinchin's students were heroes. The book opens with a moving and personal letter, full of empathy, gratitude and love.

As for the mathematics, Khinchin had carefully selected problems of great beauty, problems that can be stated and appreciated with little specialized knowledge; -- in modern lingo, with very few prerequisites. And at the same time, they are problems Seryozha can work on in his hospital bed. They are profound, and they can be attacked with elementary means. Naturally, since 1945, there have been a lot of advances on all three. The problems are from arithmetic (or number theory), and they go under the names: (a) van der Waerden's theorem on arithmetic progressions, (b) Landau's hypothesis and Mann's theorem, and (c) an elementary solution of Waring's problem.

By now these three problems take a different form in modern math books, but none as beautiful, in my opinion as Khinchin's in his loving letter to his student written toward the end of the war. Review by Palle Jorgensen, May 2005.

Come on, professors, write more like this!
This book is actually a letter from a Russian professor to a student sent off to war.It's short, but won't be an easy read. These are "pearls" but getting the oyster open is going to be tough. It's also remarkable for it's candid revelation of the mathematical process of professional practitioners at various universities in different countries.The first pearl is about a young student name van der Waerden. Yep, the guy who went on to prove so many results in Abstract Algebra and wrote the classic text on the subject influencing Artin and Noether. It's interesting to note, van der Waerden used finite differences in his proof recounted in the first pearl, and he's the only author I know that included finite differences in his abstract algebra text book. Both the candid historical confessions and the conversational exposition make this a great book. It's style and methods should be widely imitated. Come on, professors, write more like this! Future archaeologist of the 20th century will be glad this document is available for it's revelation of the habits of homo professorus mathematicus.

If you like number theory you I think you will enjoy this bo
A Y Khinchin was one of the greatest mathematicians of the first half of the twentieth century.He was also famous as a teacher and communicator. Fortunately, several of the books he wrote are still in print in English translations, published by Dover. LikeWilliam Feller andRichard Feynman he combines a complete mastery of his subject with an ability to explain clearly without sacrificing mathematical rigour.

This is a short book of three chapters:Chapter 1. Van der Waerden's theorem on arithmetic progressions. Chapter 2. The Landau-Shnirelmann hypothesis and Mann's theorem. Chapter 3. An elementary solution of Waring's problem.

These are all difficult problems from the theory of numbers and I think that the elementary proofs that Khinchin describes here are original.This book is a challenging but enjoyable read.

I also recommend his other book on number theory: "Continued Fractions". ... Read more

Product Description
This text presents a logical development of number theory, focusing on the axiomatic development of number theory, showing how to prove theorems and understand the nature of number theory. Drawing applications from physics, statistics, computer science, mathematics, astronomy, cryptography and mechanics, this book features extensive worked examples which illustrate many number theory patterns. It treats applications in depth with substantive discussion of the context of each application. ... Read more

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Number Theory or arithmetic, as some prefer to call it, is the oldest, purest, liveliest, most elementary yet sophisticated field of mathematics. It is no coincidence that the fundamental science of numbers has come to be known as the "Queen of Mathematics." Indeed some of the most complex conventions of the mathematical mind have evolved from the study of basic problems of number theory.

André Weil, one of the outstanding contributors to number theory, has written an historical exposition of this subject; his study examines texts that span roughly thirty-six centuries of arithmetical work — from an Old Babylonian tablet, datable to the time of Hammurapi to Legendre’s Essai sur la Théorie des Nombres (1798). Motivated by a desire to present the substance of his field to the educated reader, Weil employs an historical approach in the analysis of problems and evolving methods of number theory and their significance within mathematics. In the course of his study Weil accompanies the reader into the workshops of four major authors of modern number theory (Fermat, Euler, Lagrange and Legendre) and there he conducts a detailed and critical examination of their work. Enriched by a broad coverage of intellectual history, Number Theory represents a major contribution to the understanding of our cultural heritage.

Customer Reviews (2)

Weil's Number Theory
This item is of very great interest to me. It covers thoroughly, in brilliant exegesis, the whole history and mathematics of number theory from Hammurapi to Legendre at a level suitable for the general reader with an interest in number theory. The author was one of the pioneers in this field.

Respectable and enjoyable guide to pre-Gaussian number theory
When someone like Weil sets out to write a history of number theory it is destined to be the standard reference for decades to come. But this is not only an authoritative reference everyone loves to cite--it is also delightfully readable. It is not a substitute for a textbook (although Weil hints at this possibility is the preface), but even for readers with only a modest background in number theory this book will be a source of insight and joy.

Chapter 1 "Protohistory" treats briefly some of the scattered pre-Fermat attempts, which helped form the Diophantine tradition of what would constitute the staple problems of number theory--Pythagorean triples, sums of squares, Pell's equation, such things. These seeds blossomed in the hands of Fermat (chapter 2), with whom we start to see the formation of a coherent theory of numbers with some basic tools: infinite descent, modulo arguments, a precursor of elliptic curve arithmetic, etc. Fermat rarely wrote things down properly, and Euler (chapter 3) had to work hard to prove his theorems and conjectures, in the process adding some ideas of his own (the group theoretic core of modulo arithmetic and Z/pZ, auxiliary functions such as the phi function, etc.). Euler's further investigations along these lines also left many valuable ideas for future mathematicians such as the crystallisation of the importance of quadratic forms (taken up by Lagrange, chaper 4; later perfected by Gauss) and the statement of the law of quadratic reciprocity (taken up by Legendre, chapter 4; later proved in full by Gauss). Also highly decisive for the future development of number theory was Euler's bringing in of analytic ideas into number theory, in particular elliptic integrals (whose deep importance was later revealed by Jacobi) and the zeta function and L-series (whose deep importance was later revealed by Dirichlet and Riemann). ... Read more

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In this volume one finds basic techniques from algebra and number theory (e.g. congruences, unique factorization domains, finite fields, quadratic residues, primality tests, continued fractions, etc.) which in recent years have proven to be extremely useful for applications to cryptography and coding theory. Both cryptography and codes have crucial applications in our daily lives, and they are described here, while the complexity problems that arise in implementing the related numerical algorithms are also taken into due account. Cryptography has been developed in great detail, both in its classical and more recent aspects. In particular public key cryptography is extensively discussed, the use of algebraic geometry, specifically of elliptic curves over finite fields, is illustrated, and a final chapter is devoted to quantum cryptography, which is the new frontier of the field. Coding theory is not discussed in full; however a chapter, sufficient for a good introduction to the subject, has been devoted to linear codes. Each chapter ends with several complements and with an extensive list of exercises, the solutions to most of which are included in the last chapter.

Though the book contains advanced material, such as cryptography on elliptic curves, Goppa codes using algebraic curves over finite fields, and the recent AKS polynomial primality test, the authors' objective has been to keep the exposition as self-contained and elementary as possible. Therefore the book will be useful to students and researchers, both in theoretical (e.g. mathematicians) and in applied sciences (e.g. physicists, engineers, computer scientists, etc.) seeking a friendly introduction to the important subjects treated here. The book will also be useful for teachers who intend to give courses on these topics.

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First published in 1975, this classic book gives a systematic account of transcendental number theory, that is those numbers which cannot be expressed as the roots of algebraic equations having rational coefficients. Their study has developed into a fertile and extensive theory enriching many branches of pure mathematics. Expositions are presented of theories relating to linear forms in the logarithms of algebraic numbers, of Schmidt's generalization of the Thue-Siegel-Roth theorem, of Shidlovsky's work on Siegel's E-functions and of Sprindzuk's solution to the Mahler conjecture.The volume was revised in 1979, however Professor Baker has taken this further opportunity to update the book including new advances in the theory and many new references. ... Read more

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This is the first book that makes the difficult and important subject of transcendental number theory accessible to undergraduate mathematics students.

Edward Burger is one of the authors of The Heart of Mathematics, winner of a 2001 Robert W. Hamilton Book Award. He will also be awarded the 2004 Chauvenet Prize, one of the most prestigious MAA prizes for outstanding exposition.

Customer Reviews (1)

Review of Making Transcendence Transparent
This is a good introduction to the study of transcendental numbers, and it is pretty new (2004).
The authors provide motivation for complex proofs by working up from simpler proofs for special cases. For example, they prove various properties of the exponential function, and these culminate in proof of the full Lindemann theorem. Likewise a series of special cases leads up to proof of the Gelfond-Schneider theorem.
There is a nice description of Mahler's classification of transcendental numbers.
Chapter 7 concerns elliptic functions and does a good job of introducing the concepts surrounding elliptic curves.
One weakness in this book is that it does not have a good bibliography. It could better fill its niche in the market if the authors had made one. Moreover, there are many proofs I would like to see laid out better, and for this reason I have decided to give just 4 stars.
... Read more

Product Description
This book is intended for an introductory course in number theory, primarily taken by mathematics majors.The author wrote the text with four main goals in mind.He wrote a book that is both easy to read and easy to teach from.The author also aims to ease students into using proofs, and to develop a self-confidence in mathematics surrounding the difficulty of mathematical proof.Although the main users of this text will be mathematics students, a large audience could easily use the book. ... Read more

Customer Reviews (3)

Good book on a standard subject
I thought this book was a very good book on elementary number theory. It covered all the standard topics for such a book. The author was clear in his presentation of topics.

Not a bad book, but horribly expensive
I had to get this book for an undergraduate class in number theory, so I had no choice but to shell out the cash. Price aside, it really is a decent textbook. It is easy enough to understand for even a non math person, and there is a nice range of easy, medium, and hard problems. Still, although I haven't looked at too many other textbooks, I bet you could find a book just as good for half the price.

Plenty of choices...
Another thin book on a standard subject with horrendous price. Another cheap effort hope to reap in good cash. Look elsewhere, plenty of choices. ... Read more