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An Invitation to Modern Number Theory Hardcover – March 26, 2006


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Product Details

  • Hardcover: 519 pages
  • Publisher: Princeton University Press (March 26, 2006)
  • Language: English
  • ISBN-10: 0691120609
  • ISBN-13: 978-0691120607
  • Product Dimensions: 9.4 x 6.3 x 1.6 inches
  • Shipping Weight: 1.8 pounds (View shipping rates and policies)
  • Average Customer Review: 4.2 out of 5 stars  See all reviews (5 customer reviews)
  • Amazon Best Sellers Rank: #891,478 in Books (See Top 100 in Books)

Editorial Reviews

Review

"This is a great book. . . . [I]t is a fine book for talented and mathematically mature undergraduates, for graduate students, and for anyone looking for information on modern number theory."--Henry Ricardo, MAA Reviews

"This is the first text to present Random Matrix Theory and the Circle Method for German primes. This well-written book supplements classic texts by showing connections between seemingly diverse topics, by making the subject accessible to beginning students and by whetting their appetite for continuing in mathematics"--Mathematical Reviews

"I would highly recommend this book to anybody interested in number theory, from an undergraduate student to an established expert, since everybody will be able to find in this book lots of new interesting material, tempting problems, and interesting computational challenges. It could also be used as a textbook for a graduate course in number theory. To promote and stimulate independent research, it contains many very interesting exercises and even suggestions for research projects."--Igor Shparlinski, SIAM Review

From the Inside Flap

"The book provides a much-needed introduction to modern number theory that emphasizes analytic number theory. It should serve remarkably well as an advanced undergraduate textbook and its latter parts would be suitable for a beginning graduate course. Some of the material covered, such as the circle method and random matrix theory, is not readily available elsewhere in book form. These topics provide terrific examples of areas in number theory of great current interest that can be penetrated by students. I would seriously consider using this book in my own classes and recommend it with enthusiasm for highly motivated students."--William Duke, University of California, Los Angeles

"Having this selection of material available in essentially self-contained form is fantastic. Reading the book (or taking a class based on it) might easily decide the future endeavors of many a neophyte mathematician. I have yet to discover a clearer exposition of the works of the circle method. The inclusion of exercises and, especially, of problems for further research and theoretical or numerical exploration is extremely valuable. I would dare to compare the book to Hardy and Wright's classic An Introduction to the Theory of Numbers in that Miller and Takloo-Bighash expose readers to the lively work of number theory, to its proofs, ideas, and methods, assuming only a very modest background."--Eduardo Dueñez, University of Texas, San Antonio


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18 of 19 people found the following review helpful By J. N. M. ROBLES on January 31, 2007
Format: Hardcover
Most of the books on analytic number theory are very good, and so is this one. Yet there is more to it than being a good book.

The exposition is brilliant, rigorous, well paced, absolutely non-flippant and elegant (it feels like I am reading a latexed version of a G.H. Hardy book).

It is highly innovative since it has material that you normally do not expect to find in one single book. There are books about probabilistic number theory, but those books are devoted wholly to that subject. Same thing goes for random matrix theory. But the most surprising case is that of continued fractions. Books on continued fractions are *generally* elementary and not very long.

No single book out there combines introductions to the interactions of probability and random matrix theory with number theory in addition to treating more standard subjects (cryptography, group theory, continued fractions, circle method, L-functions, ...) exquisitely. This has been wonderfully achieved by Miller and Takloo-Bighash.

All in all, the flavour of this book is best summarized with the word: modern.

This book is not a popular math book. Yet not quite a textbook either, it is, as the title suggests, an invitation. And a serious invitation, for that matter. A little effort will be needed, however I have found out that whatever amount of effort you invest in it will be rewarded with interests!

I would say that courses in group theory, elementary number theory and complex analysis would constitute an adequate background.

It need not be read linearly, which is also a bonus. Within reason, you could move on to whichever subjects you find more interesting.
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14 of 17 people found the following review helpful By Aaron M. Silberstein on February 2, 2007
Format: Hardcover
Reading a math book is usually not easy. There can be two reasons for this. The first is poor writing or lack of motivation. This book does not suffer from this defect; the prose is polished and minimalist, and keeps the reader focused on the math. This style of exposition, which is called by another reviewer "machine gun" is called by practicing mathematicians as "tight"; the mathematics is allowed to speak for itself, and the viewpoint, the voice, of the expositors is expressed clearly through the mathematics. The second reason is that the material is inherently deep and difficult and cannot be merely "learned" from a book; it must be experienced first-hand, through problem solving and deep reflection. This book is full of deep material, presented in great detail so that the reader can appreciate the nuances of the methods presented and apply them as a practicing mathematician. The topics are specially chosen, as the book says, to invite the reader to pursue some threads in modern analytic number theory. There are also copious, well-thought-out exercises to help the reader gain the experience with the material which cannot be achieved simply through reading. One must only read the beautiful proof of Roth's theorem (Chapter 6) to see the quality of exposition, excellent choice of topics and the attention to detail that makes this book an excellent place to start for a serious student who wants to understand modern analytic number theory, and a jumping point to advanced books and research monographs on diophantine geometry and analytic number theory.
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1 of 1 people found the following review helpful By robert m on July 15, 2011
Format: Hardcover Verified Purchase
I have about a dozen books on number theory, analytical, algebraic, GTMs, etc. Not to mention everything on crytography. So why buy another book? It is always good to keep your skills fresh and also to see what other people are teaching from. This book is used by the MIT OCW so if you want to know if you are learning well you can follow their class outline. What is nicer than having a book and having it be part of a free class the purpose of which is to just help educate the world?
As for the book, if you have read any of the Burtons(my actual favorite) or perhaps a silverman there is a general outline. This book doesn't follow those exact lines but I think it is a better book in some respects. In some ways a small step ahead of them as perhaps the writer intended to skip some of the stuff that is in every introductory number theory book. If you like this then don't overlook apostol as your next number theory book. enjoy
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6 of 11 people found the following review helpful By Farzad Qassemi on March 6, 2007
Format: Hardcover
The name of the book might look pure mathematics but if one looks throughout the book he will easily notice mystic joyfulness of mathematics blended with the deep understanding of authors on the subject makes the content available to users.

Based on my impression from the book (see below for detail), I found this book readable and enjoyable separated from reader background in mathematics. Mainly what you get from the book does not depend on what you know but depends on what you want.

I divide my review into two parts, first the language of this book and second the content of the book.

There are not too many books in mathematics useable for other students, the main barrier in reading Mathematical books for non-mathematicians is strange notations and definitions but I, as a physics student, found this book very readable far from mathematical obscurity. Furthermore, very good details in each part make it smoothly understandable through the book.

Despite quite broad range of the content from elementary number theory to advanced topics in statistics, very interesting and intuitive examples and applications make all steps clear for all type of readers. Specifically, the forth part of the book is quite applicable for all science and engineering students and researchers.
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