Product Description
This text presents the principal ideas of classical number theory emphasizing the historical development of these results and the important figures who worked on them. It is intended to introduce third or fourth-year undergraduates to mathematical proofs by presenting them in a clear and simple way and by providing complete, step-by-step solutions to the problems with as much detail as students would be expected to provide themselves. This is the only book in number theory that provides detailed solutions to 800 problems, with complete references to the results used so that the student can follow each step of the argument. ... Read more

Customer Reviews (2)

More of a study guide, insulting in scope in places
Any undergraduate textbook that says the proof of Bertrand's Postulate is beyond the scope of the text (p201) is remiss. Proofs easier than quadratic reciprocity exists. Not surprising, this was my text at a school known for engineering, not pure math.

In 400 pages you'll only go one-third the distance covered in Hardy and Wright, or Niven and Zuckerman. There is almost no emphasis on the more advanced areas of number theory, nor even a hint that such branches exist. Although many of the elementary, but unsolved conjectures are mentioned.

But wait! Thebook does have some merit. The book is broken into bite-size pieces - number theory forAttention Deficit Disorders.Everything is broken up into these pieces, notes, problems, further readings. Even the proofs that aren't "beyond its scope" are broken up into lemmas.This book has more "problems with solutions" than any I've seen, which would make it a good study guide for more concise texts that leave all the problems to the reader. It is these problems and solutions that fill out the bulk of the text and limit its depth of coverage.(Hardy and Wright has only 24 more pages but covers so much more).

Excellent summary of basic number theory.
Adler and Coury's text on elementary number theory is one of the best I've ever seen; certainly for the purpose of independentreading or study. Aside from an otherwisestandard flow of theorems, proofs, exercises, etc, there are approximately 800 problemsall with solutions. Numeroushistorical and incidental notes are included as well, making this arewarding book to read and use.

Anyone who works through the materialshouldhave no difficulty with the more challengingclassics, like thoseof Hardy and Wright orthe more recent one by Hua. ... Read more

Product Description
Ian Stewart's Galois Theory has been in print for 30 years. Resoundingly popular, it still serves its purpose exceedingly well. Yet mathematics education has changed considerably since 1973, when theory took precedence over examples, and the time has come to bring this presentation in line with more modern approaches.To this end, the story now begins with polynomials over the complex numbers, and the central quest is to understand when such polynomials have solutions that can be expressed by radicals. Reorganization of the material places the concrete before the abstract, thus motivating the general theory, but the substance of the book remains the same. ... Read more

Customer Reviews (16)

Great book on Galois Theory
This updated addition is a wonderfully written exposition of the ideas in Galois Theory. Especially helpful is the focus on the concrete example of the complex field, before moving on to general fields.

great for independent study undergrad
I learned Galois Theory from this book as an undergrad in an independent study. Stewart hits the insoluability of the quintic and compass-straight edge constructions. There are a lot of steps to getting to the end, and he puts them out there without robbing the reader of the chance to be a part of the proof.

Many mistakes spoil the book
If you buy this book, be sure to find the half a dozen(!) pages of errata. Then reserve a few hours to go through almost every page to correct the many mistakes. Be warned! This book could have been very nice if it weren't for the many mistakes.

Galois Theory is great!
I've received the book very quickly and the book (a classic) is great! I recommand Amazon and Galois Theory as well!

Noanswerstoexercises, itisinconvienient.
itisnotgoodfor self-
study.


... Read more

Product Description
Contents: Preface. Reader's Guide. Index of Notation. The Origins of Algebraic Number Theory. Part I: Numbers. Quadratic and Cyclotomic Fields. Geometric Methods. Lattices. Minkowski's Theorem. Part II: Geometric Representation of Algebraic Numbers. Class-Group and Class-Number. Part III: Number-Theoretic Applications. Computational Methods. Fermat's Last Theorem. Dirichlet's Units Theorem. Appendix. Quadratic Residues. References. Index. ... Read more

Product Description

This book details the classical part of the theory of algebraic number theory, excluding class-field theory and its consequences. Coverage includes: ideal theory in rings of algebraic integers, p-adic fields and their finite extensions, ideles and adeles, zeta-functions, distribution of prime ideals, Abelian fields, the class-number of quadratic fields, and factorization problems. The book also features exercises and a list of open problems.

Product Description
In response to concerns about teacher retention, especially among teachers in their first to fourth year in the classroom, we offer future teachers a series of brief guides full of practical advice that they can refer to in both their student teaching and in their first years on the job. Number Theory for Elementary School Teachers is designed for preservice candidates in early and/or elementary education. The text complements traditional Math Methods courses and provides deep content knowledge for prospective and first year teachers. ... Read more

Product Description
This book is based on the AMS Short Course, The Unreasonable Effectiveness of Number Theory, held in Orono, Maine, in August 1991. This Short Course provided some views into the great breadth of applications of number theory outside cryptology and highlighted the power and applicability of number-theoretic ideas. Because number theory is one of the most accessible areas of mathematics, this book will appeal to a general mathematical audience as well as to researchers in other areas of science and engineering who wish to learn how number theory is being applied outside of mathematics. All of the chapters are written by leading specialists in number theory and provides excellent introduction to various applications. ... Read more

Product Description
Basic treatment, incorporating language of abstract algebra and a history of the discipline. Topics include unique factorization and the GCD, quadratic residues, number-theoretic functions and the distribution of primes, sums of squares, quadratic equations and quadratic fields, diophantine approximation, more. Many problems. Bibliography. Advanced undergraduate-beginning graduate-level. 1977 edition.
... Read more

Customer Reviews (6)

Recommend this book
This book on Number Theory is pitched at the undergraduate level.It is well written and organized. A nice feature is historical notes on past number theorists.

Great Number Theory Book
This book covers all the basics in number theory. The greatest common divisor, the Euclidean algorithm, congruences, primitive roots, quadratic reciprocity and more. If one has taken abstract algebra, then this is a great introductory number theory book, if one hasn't taken abstract algebra, then a few chapters would be difficult to understand (such as chapters 3, 4 and 8), but the rest don't need abstract algebra.
I like the topics that are covered, in particular, I think Leveque does a very good job of explaining important concepts in elementary number theory in chapter 6. I really like Brun's theorem on twin primes and the order of magnitude of several famous number theory functions.
The last chapter has some interesting sections including the proof of the trascendence of e.

Another thing I like about the book, is that it has mini biographies of important number theorists throughout the history of mathematics. I've always enjoyed reading about great mathematicians.

If you actually like number theory...
...stay away from this book.This book was required for an upper-division number theory course I took, and the teacher used it for one of our eight assignments.I assume he did this because there are few worthwhile problems in the book.The author spends too much time explaining the history behind the theorems and too little time discussing applications.He proves the theorems but the proofs make no sense to someone learning just from the book.The teacher for my course explained all of the theorems in ways that actually made sense and were easily reproducable.When studying I never looked at this book and instead used Nathanson's GTM for number theory.If you only care about the history of number theory, this book has some redeeming quality.Otherwise, stay away.Dover math books tend to be inferior to texts like UTMs and GTMs, as evidenced by the latter costing literally ten times as much sometimes.

Good for self teaching
I am currently working through this book, and I really like the format, which is usually a small, digestible chapter followed by a set of exercises (usually 10 or so).Quite a wide variety of topics are covered including congruences, primitive roots, analysis of the number theoretic functions (e.g., the number of primes below x), and a little on diophantine approximations and continued fractions.Nothing post-calculus is used in the book except for some algebraic structures such as fields and rings, however, they are fully explained at the beginning of the book.(And some previous acquaintance with these would probably be good.)The exercises are especially good, being not too easy and not too hard.In response to the review below, to actually understand math like this you must be willing to do some work yourself.If you are looking to sit back in your easy chair and be entertained, then you should buy a book on the history of number theory, not a textbook.

Deceptive Title
No concept is beyond the reach of an intelligent mind, so long as that concept is brilliantly explained.If there were a race for explanatory brilliance, this book would fall somewhere short of the starting line.Forexample, an appendix lists 58 mathematical symbols, (most of which youwon't encounter in high school).These symbols are blithely usedthroughout the text, yet none is adequately explained.If you don't havecalculus, statistics, trigonometry and a few other disciplines alreadyfirmly under your belt, forget this text.Of interest to all readers maybe the occasional insets giving concise biographies of importantmathematicians. ... Read more

Product Description
Algebraic number theory is one of the most refined creations in mathematics. It has been developed by some of the leading mathematicians of this and previous centuries. The primary goal of this book is to present the essential elements of algebraic number theory, including the theory of normal extensions up through a glimpse of class field theory. Following the example set for us by Kronecker, Weber, Hilbert and Artin, algebraic functions are handled here on an equal footing with algebraic numbers. This is done on the one hand to demonstrate the analogy between number fields and function fields, which is especially clear in the case where the ground field is a finite field. On the other hand, in this way one obtains an introduction to the theory of "higher congruences" as an important element of "arithmetic geometry".Early chapters discuss topics in elementary number theory, such as Minkowski's geometry of numbers, public-key cryptography and a short proof of the Prime Number Theorem, following Newman and Zagier. Next, some of the tools of algebraic number theory are introduced, such as ideals, discriminants and valuations. These results are then applied to obtain results about function fields, including a proof of the Riemann-Roch Theorem and, as an application of cyclotomic fields, a proof of the first case of Fermat's Last Theorem. There are a detailed exposition of the theory of Hecke $L$-series, following Tate, and explicit applications to number theory, such as the Generalized Riemann Hypothesis. Chapter 9 brings together the earlier material through the study of quadratic number fields. Finally, Chapter 10 gives an introduction to class field theory. ... Read more

Product Description
This book is written for undergraduates who wish to learn some basic results in analytic number theory. It covers topics such as Bertrand's Postulate, the Prime Number Theorem and Dirichlet's Theorem of primes in arithmetic progression.

The materials in this book are based on A Hildebrand's 1991 lectures delivered at the University of Illinois at Urbana-Champaign and the author's course conducted at the National University of Singapore from 2001 to 2008.

Contents:

• Facts about Integers
• Arithmetical Functions
• Averages of Arithmetical Functions
• Elementary Results on the Distribution of Primes
• The Prime Number Theorem
• Dirichlet Series
• Primes in Arithmetic Progression

Product Description
In this student-friendly text, Strayer presents all of the topics necessary for a first course in number theory. Additionally, chapter on primitive roots, Diophantine equations, and continued fractions allow instructors the flexibility to tailor the material to meet their own classroom needs. Each chapter concludes with seven Student Projects, one of which always involves programming a calculator or computer. All of the projects not only engage students in solving number-theoretical problems but also help familiarize them with the relevant mathematical literature. ... Read more

Product Description
This valuable book focuses on a collection of powerful methods of analysis that yield deep number-theoretical estimates. Particular attention is given to counting functions of prime numbers and multiplicative arithmetic functions. Both real variable ("elementary") and complex variable ("analytic") methods are employed. The reader is assumed to have knowledge of elementary number theory (abstract algebra will also do) and real and complex analysis. Specialized analytic techniques, including transform and Tauberian methods, are developed as needed. ... Read more

Product Description

Number theory, an ongoing rich area of mathematical exploration, is noted for its theoretical depth, with connections and applications to other fields from representation theory, to physics, cryptography, and more. While the forefront of number theory is replete with sophisticated and famous open problems, at its foundation are basic, elementary ideas that can stimulate and challenge beginning students. This lively introductory text focuses on a problem-solving approach to the subject.

Key features of Number Theory: Structures, Examples, and Problems:

* A rigorous exposition starts with the natural numbers and the basics.

* Important concepts are presented with an example, which may also emphasize an application. The exposition moves systematically and intuitively to uncover deeper properties.

* Topics include divisibility, unique factorization, modular arithmetic and the Chinese Remainder Theorem, Diophantine equations, quadratic residues, binomial coefficients, Fermat and Mersenne primes and other special numbers, and special sequences. Sections on mathematical induction and the pigeonhole principle, as well as a discussion of other number systems are covered.

* Unique exercises reinforce and motivate the reader, with selected solutions to some of the problems.

* Glossary, bibliography, and comprehensive index round out the text.

Written by distinguished research mathematicians and renowned teachers, this text is a clear, accessible introduction to the subject and a source of fascinating problems and puzzles, from advanced high school students to undergraduates, their instructors, and general readers at all levels.

Customer Reviews (2)

Very good book
It's not a book for beginners in number theory, it's perfect for people who're interest in study for mathematical olympiads and those who're interest in study deeper number theory. It's a fantastic book, like most books of titu and Dorin Andrica, i recomend !

Number Theory: Structure, examples, and Problems
It's perfect to prefare math olympiad contest. It contains fundamental concepts
of elementary number theory and practical problems of math contests with original
solutions. ... Read more

Product Description
This book effectively integrates computing concepts into the number theory curriculum using a heuristic approach and strong emphasis on rigorous proofs.Its in-depth coverage of modern applications considers the latest trends and topics, such as elliptic curves—a subject that has seen a rise in popularity in the undergraduate curriculum. ... Read more

Customer Reviews (3)

Covers a lot, but lots of errors
This book covers a lot of great material, but it also has a lot of errors. Watch out! It can be frustrating to work and rework a problem or example, only to get the 'wrong' answer. I ended up writing MatLab scripts on a number of subjects, and proved to myself that I got the answer right... the book was wrong! Also, the writing style is confusing rather than illuminating. Still, I learned a lot -- including by proving the book wrong.

Riddled with errors
A good intro to Number Theory for non-mathematicians. (Specfically, excellent for anyone who wants to pick up cryptography.)
However, the numerous errors found throughout the book take away from it's impact. Not to mention some outdated algorithms in Chapter 5.
A lower price would make this a better buy.

Excellent
This is a wonderfull book for people looking to combine their math skills with technology.The book also is easy to follow and understand for those with no computer skills. ... Read more

Product Description
Analytic Number Theory presents some of the central topics in number theory in a simple and concise fashion. It covers an amazing amount of material, despite the leisurely pace and emphasis on readability. The author's heartfelt enthusiasm enables readers to see what is magical about the subject. Topics included are; The Partition Function, The Erd"s-Fuchs Theorem, Sequences without Arithmetic Progressions, The Waring Problem, A "Natural" Proof of the Non-vanishing of L-Series, and a Simple Anlaytic Proof of the Prime Number Theorem -- all presented in a surprisingly elegant and efficient manner with clever examples and interesting problems in each chapter. This text is suitable for a graduate course in analytic number theory. ... Read more

Customer Reviews (3)

One Gem in Number Theory
This is one of the nicest math books I have ever seen.
A Great Book in Number Theory by a Great Mathematician; D. J. Newman.

Excellent book, very clear.A few errors still.
I find this book exceptionally clear.It's the thinnest GTM text I've ever seen, but there is a lot of material there.

It's true that this book doesn't have a comprehensive treatment of analytic number theory, I like the other reviewer's analogy of an appendix to generatingfunctionology--this book does focus primarily on sequences and generating functions.However, it is a very clear, fun, and easy to read book.The book's thinness may be misleading; there is plenty of explanatory prose, motivation, discussion, and proofs are generally pretty easy to follow.

The revised printing does correct many errors, but I was still able to find some (and I suspect this means there are a lot more than I was able to find).This was a little bit annoying, but only a little bit.Overall I think this is a great book.

Second printing corrects most typos
This book is somewhat in the spirit of Aigner and Ziegler's "Proofs from the Book": short, clear proofs of important results in Analytic Number Theory. My favorite parts are (1) the "natural" proof of the non-vanishing of L-series, which really does make it look inevitable; (2) the Crazy Dice, a simple and surprising example of the power that generating functions provide when you switch your viewpoint between formal power series and the functions they represent.

To some extent the author keeps the proofs short by leaving out steps, so you'll need to read it with pencil and paper nearby to work out the missing steps. The first printing was loaded with typographical errors; most (not all) of these are corrected in the 2000 second printing. Unfortunately not all the remaining typos are obviously typos; this combined with the brevity can make the exposition hard to follow. The first printing was fascinating (for its content) and exasperating (for its typos); the second printing is still fascinating, and occasionally exasperating. ... Read more

Product Description
Various developments in physics have involved many questions related to number theory, in an increasingly direct way. This trend is especially visible in two broad families of problems, namely, field theories, and dynamical systems and chaos. The 14 chapters of this book are extended, self-contained versions of expository lecture courses given at a school on "Number Theory and Physics" held at Les Houches for mathematicians and physicists. Most go as far as recent developments in the field. Some adapt an original pedagogical viewpoint. ... Read more

Customer Reviews (1)

I'm no expert
Since no one else has reviewed this I'll say a bit about it. The book consists of extensive notes, written up after lecture series on each topic, and for the most part they are very clear and useful as introductions. Only the notes on Zeta functions seem to assume you were in the audience for the lecture!

Bos on Riemann surfaces, and Stark on algebraic numbers and Galois theory, are tremendously clear and informative, on the basic math and also on the farther reaches relating it to number theory and physics. Those happen to be the ones I've read most. The chapters on elliptic curves, modular forms, p-adic analysis, are all very nice. I have not looked at the more physical chapters.

The chapters assume basic real and complex analysis, but not very much. You need to be able to work with abstraction on the level of non-Archimedean metrics. If that is the limit of your background, then you'll have to be able to read about math that is not explained in full detail all the time, but is clearly layed out. ... Read more

Product Description
Number Theory Through Inquiry; is an innovative textbook that leads students on a carefully guided discovery of introductory number theory. The book has two equally significant goals. One goal is to help students develop mathematical thinking skills, particularly, theorem-proving skills. The other goal is to help students understand some of the wonderfully rich ideas in the mathematical study of numbers. This book is appropriate for a proof transitions course, for an independent study experience, or for a course designed as an introduction to abstract mathematics. Math or related majors, future teachers, and students or adults interested in exploring mathematical ideas on their own will enjoy ;Number Theory Through Inquiry.;Number theory is the perfect topic for an introduction-to-proofs course. Every college student is familiar with basic properties of numbers, and yet the exploration of those familiar numbers leads us to a rich landscape of ideas. Number Theory Through Inquiry contains a carefully arranged sequence of challenges that lead students to discover ideas about numbers and to discover methods of proof on their own. It is designed to be used with an instructional technique variously called guided discovery or Modified Moore Method or Inquiry Based Learning (IBL). Instructorsmaterials explain the instructional method.This style of instruction gives students a totally different experience compared to a standard lecture course. Here is the effect of this experience:Students learn to think independently:they learn to depend on their own reasoning to determine right from wrong; and theydevelop the central, important ideas of introductory number theory on their own. From that experience, they learn that they can personally create important ideas.They develop an attitude of personal reliance and a sense that they can think effectively about difficult problems. These goals are fundamental to the educational enterprise within and beyond mathematics. ... Read more

Customer Reviews (4)

Good information
The information in this book was clear and very helpful for my coursework. Expensive, but useful!

Great book
I used this book to teach a semester course to talented high school students all of whom had at least two semesters of calculus before entering the course, so they were approximately at the college sophomore level.The students really learned how to prove things in a way that a standard approach did not.Because the students had to prove all the theorems, we covered less material than a standard lecture approach.My purpose, however, was not a huge amount of material, but rather to get the students to be able to write proofs, read proofs, and be critical of what they wrote themselves and what others wrote.It was a great experience for me and for them.At the end of the semester the students spontaneously broke into applause for the course.There were a few times when I helped them do proofs or gave some of the proofs as extra credit (if they were not central to the development of the material).I collected homework for almost every class and read it sometimes grading sometimes commenting.The book is well organized and has standard topics for such a course.I added the explicit statement of the division algorithm at the beginning of the course which helped in several instances later.I am using this book and this approach again.

worthless "textbook"
I had a teacher use this book to teach (well she really didn't give lectures, everything was left to the students) for an introductory number theory course.

This textbook gives a cursory, ambiguous review of number theory.The instructor used the "Moore Method" a.k.a the "let the students give your lectures for you" method.Therefore, the students trying to learn the material cannot depend on the veracity of the information the other students give in the lecture (yes you can do proofs, but of course you can do proofs INCORRECTLY). Therefore the only information that the student can depend on is feedback (if any) given by the instructure, which using the "Moore Method" is done through returned assignments. So, if the corrections on the assignments are done INCORRECTLY, through error or MALICE, then the student has NO WAY to verify the results or the methods used to obtain them.This is where a proper textbook would help, and where THIS TEXTBOOK UTTERLY FAILS.In a proper text book, a principle is demonstrated through theorem, and mabye a couple of examples.This book has non of this, only a vague hint on how something MIGHT work, and then it leaves you to "intuit" the answer from thin air.Well if I could "intuit" complex mathematical relationships out of thin air, I sure wouldn't need this book or even to take a course of study at a University, because I'd be rainman, or good will hunting, or something like that.This textbook and the "Moore Method" are THE WORSE way I have ever encountered for teaching mathematics.Like I said, if the student does not have any metric by which to measure how he is being evaluated and graded in a course of study, then how on earth is he to know that he is being treated fairly, and not with personal bias?Since there is no standard by which the student can review grades on coursework, it cannot be determined.You might say "get another textbook" but if the student does not know anything about the material, that is not a reasonable solution, since he cannot evaluate material about which he does not know independently. So, what you end up with is a radical feminist professor, whose Vita has more things on it that have to do with feminism than mathematics, who uses the "Moore Method", with its lack of an objective measure from which to form a grade, who uses it as a weapon to sandbag men.The feminist "professor" doesn't even have one published work in mathematics that is entirely her own work, so its very likely that she has just ridden gender politics through her entire career and relied on others to carry her water academically.

Can be used as a text, would require significant instructor supplements
If you were to use this book as a text in a number theory course, you first must have made the decision to teach it in a nonstandard manner. The approach used in the presentation of number theory is not the traditional listing of the fundamental theorems with their proofs. Concepts are stated as theorems but in no case is a proof offered.
There are several groups of specific exercises such as

Illustrate the division algorithm m = nq + r for m = 25, n = 7; m = 277, n = 4; m = 33, n = 22; m = 33, n = 45.

Questions such as

Do every two integers have at least one common divisor?
What other numbers can you show to be irrational? Make and prove the most general conjecture you can.
Which natural numbers can be written as the sum of two squares of natural numbers? State and prove the most general theorem possible about which natural numbers can be written as the sum of two squares of natural numbers, and prove it.

At several points in the text, there are exercises called "Blank paper exercises" which have the following structure.

After not looking at the material in this chapter for a day or two, take a blank piece of paper and outline the development of that material in as much detail as you can without referring to the text or to notes. Places where you get stuck or can't remember highlight areas that may call for further study.

The coverage is generally what is found in an introductory course in number theory. However, the lack of proofs means that either the students must derive them on their own, look them up in another reference or have the instructor provide them. While this does not preclude the use of this book, it will require extra effort on the part of the student and/or the instructor.

Published in Journal of Recreational Mathematics, reprinted with permission
... Read more

Product Description

This book surveys the current state of the "small" sieve methods developed by Brun, Selberg and later workers. The book is suitable for university graduates making their first acquaintance with the subject, leading them towards the frontiers of modern research and unsolved problems in the subject area.

Product Description
The majority of students who take courses in number theory are mathematics majors who will not become number theorists. Many of them will, however, teach mathematics at the high school or junior college level, and this book is intended for those students learning to teach, In addition to a careful presentation of the standard material usually taught in a first course in elementary number theory, this book includes a chapter on quadratic fields which the author has designed to make students think about some of the "obvious" concepts they have taken for granted earlier. The book also includes a large number of exercises, many of which are nonstandard. ... Read more

Customer Reviews (2)

Perhaps biased
This book was the required text for an independent study class I enrolled in.The class has been more difficult than I thought it would be, as has the text.It is complex material and doesn't provide a lot of clear-cut examples - instead assuming that you make the connection yourself. However, I am learning all on my own and the material may be more understandable with the help of a live professor.

A wonderful insight into number theory
In general, this book gives a comprehensiveaccount on elementary number theory. The first few chapters include some fundamental concepts like divisibility and congruences (i.e. a simple kind of modular arithmetic), aswell as famous yet basic theorems like the fundamental theorem ofarithmetic. Important topics in number theory such as Diophantineequations, fractional approximations for irrational numbers and Quadraticfields are there, and if you're interested in magic squares, I'd like tosay that a whole chapter is devoted to it.

There're some goodpoints featuring this book. It assumes no prerequisite in number theory.Just a bit knowledge about numbers and operations on them are needed.Results and theorems are closely related, allowing you to observe howthings are connected. Although not many examples are available, some arereally instructive and helpful enough to avoid misconceptions.

However, it's a pity to say that the materials contained are not reallywell-organized, especially those in Chapter 7: the geometric arguments usedin the development of the continued fraction algorithm lack concision, anda few proofs are quite annoying because the author failed to justify someclaims that shuold not be treated as something "obvious". It canbe motivating just to provide readers guidelines about how to work outthose minor stuff, but such things shouldn't have been misleadingly called"proofs". Another problem is that the illustratons presented areoccasionally insufficient, and this is particularly the case in the chapterabout Diophantine equations. Novices in the subject can hardly rely on thetext to solve harder exercises contained without tracing out more techniquewhich is not emphasized.

Overall, the book deserves to be a finereading for the interested ones new to number theory. But if you're seriousabout the topic, find an even better book instead. ... Read more

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Customer Reviews (2)

Number Theory as a Research Seminar
I have used these materials to teach an undergraduate course in number theory,
and I think they are an excellent way to give independent and motivated
students an experience with research.This book is just for the students
to use, and so it is deliberately incomplete.Instructors should contact
the publisher to receive the supplements.

This book is NOT designed to be a reference text.

The basic idea is the following: Let students discover and prove the main
results, then extend the material to include topics they couldn't reasonably
do by themselves.

A typical chapter includes a prelab, lab, chapter summary (instructor supplement),
and problems.The prelab introduces the basic ideas, and then in the lab,
students conjecture and prove the main results of the chapter.When they
are finished, they receive a chapter summary (available in the instructor
supplements), including lab results.Obviously it would be foolish to include
lab answers in the student material.Each chapter also includes homework problems.

The instructor materials also include Going Father sections, which include
material that would be too difficult for students to develop on their own
(e.g., Rabin-Miller).Taken as a whole, the entire package, student book
and instructor supplements, is comparable to the typical undergraduate textbook.

The CD includes the labs (for Mathematica, Maple, and the web), so there wasn't
any practical need to print them in the book, in triplicate.Perhaps they could
have included the Going Farther sections instead.The prelabs are fine as
written, but I supplement a few.The homework is minimal, but adequate, as
students have a lot of work to do in the labs.

Not enough depth to be a primary text in number theory
In the preface, the authors state that this book is designed to be used in a one-semester upper division undergraduate course in number theory. I would dispute that claim, in my opinion the level of mathematics is not high enough for an advanced undergraduate course. There are not enough examples where the reader is required to work their way through a formal proof.
The chapters are:

*) Divisibility and factorization
*) The Euclidean algorithm and linear Diophantine equations
*) Congruences
*) Applications of congruences
*) Solving linear congruences
*) Primes of special forms
*) The Chinese remainder theorem
*) Multiplicative orders
*) The Euler phi-function
*) Primitive roots
*) Quadratic congruences
*) Representation problems
*) Continued fractions

which certainly gives the impression that there is depth to the coverage. However, that is not the case. Each chapter begins with a few pages of what is called prelab. There are a few statements about the topic and then the student is to proceed to working a set of lab exercises.
For each chapter there is a set of lab exercises in triplicate, they are identical, except one is in Maple, one in Mathematica and the last runs Java applets in a browser. This repetition is a bit puzzling, as everyone has access to a browser. There is a set of homework problems at the end of each chapter with no solutions included. With the labs in triplicate and there being only a few pages of explanation of the mathematical background for those labs, there is nowhere near the depth of coverage that I would expect in an advanced undergraduate class.
However, I will say that this book would be an excellent text for any supplemental lab that you might want to do with a number theory course.
... Read more