Number Theory (Dover Books on Mathematics) Revised ed. Edition

by George E. Andrews (Author)

166 ratings

ISBN-13: 978-0486682525

ISBN-10: 9780486682525

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Although mathematics majors are usually conversant with number theory by the time they have completed a course in abstract algebra, other undergraduates, especially those in education and the liberal arts, often need a more basic introduction to the topic.
In this book the author solves the problem of maintaining the interest of students at both levels by offering a combinatorial approach to elementary number theory. In studying number theory from such a perspective, mathematics majors are spared repetition and provided with new insights, while other students benefit from the consequent simplicity of the proofs for many theorems.
Among the topics covered in this accessible, carefully designed introduction are multiplicativity-divisibility, including the fundamental theorem of arithmetic, combinatorial and computational number theory, congruences, arithmetic functions, primitive roots and prime numbers. Later chapters offer lucid treatments of quadratic congruences, additivity (including partition theory) and geometric number theory.
Of particular importance in this text is the author's emphasis on the value of numerical examples in number theory and the role of computers in obtaining such examples. Exercises provide opportunities for constructing numerical tables with or without a computer. Students can then derive conjectures from such numerical tables, after which relevant theorems will seem natural and well-motivated..

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Editorial Reviews

About the Author

The Holy Grail of Number Theory
George E. Andrews, Evan Pugh Professor of Mathematics at Pennsylvania State University, author of the well-established text Number Theory (first published by Saunders in 1971 and reprinted by Dover in 1994), has led an active career discovering fascinating phenomena in his chosen field — number theory. Perhaps his greatest discovery, however, was not solely one in the intellectual realm but in the physical world as well.

In 1975, on a visit to Trinity College in Cambridge to study the papers of the late mathematician George N. Watson, Andrews found what turned out to be one of the actual Holy Grails of number theory, the document that became known as the "Lost Notebook" of the great Indian mathematician Srinivasa Ramanujan. It happened that the previously unknown notebook thus discovered included an immense amount of Ramanujan's original work bearing on one of Andrews' main mathematical preoccupations — mock theta functions. Collaborating with colleague Bruce C. Berndt of the University of Illinois at Urbana-Champaign, Andrews has since published the first two of a planned three-volume sequence based on Ramanujan's Lost Notebook, and will see the project completed with the appearance of the third volume in the next few years.

In the Author's Own Words:
"It seems to me that there's this grand mathematical world out there, and I am wandering through it and discovering fascinating phenomena that often totally surprise me. I do not think of mathematics as invented but rather discovered." — George E. Andrews

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Product details

ASIN ‏ : ‎ 0486682528
Publisher ‏ : ‎ Dover Publications; Revised ed. edition (October 12, 1994)
Language ‏ : ‎ English
Paperback ‏ : ‎ 288 pages
ISBN-10 ‏ : ‎ 9780486682525
ISBN-13 ‏ : ‎ 978-0486682525
Item Weight ‏ : ‎ 10.6 ounces
Dimensions ‏ : ‎ 5.5 x 0.75 x 8.75 inches

Best Sellers Rank: #91,325 in Books (See Top 100 in Books)
- #13 in Number Theory (Books)
- #317 in Mathematics (Books)

Customer Reviews:
166 ratings

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Lutolf Markus Franz

Don't buy the Kindle version

Reviewed in the United States on October 7, 2018

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One of the reviews of this book notes that the equations are images and rather on the small side. I checked the preview and decided they were big enough, and went ahead and bought the Kindle version. Big mistake. The equations when seen through the Kindle reader are about half as big as in the preview and make reading the book without painstakingly enlarging each equation next to impossible.

30 people found this helpful

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Philip A.

Good Introductory Text for the Mathematically Inclined

Reviewed in the United States on February 28, 2015

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As a retired statistician and teacher, I never had the opportunity to formally study number theory. Over the years I was exposed to the topic and learned some of the basics -- sort of the tip of the iceberg. I learned enough to want to know more; hence, the acquisition of this book. I am quite familiar with Dover Publications and a big fan of their libary. Dover typically publishes comprehensive texts at reasonable prices. So when I was looking for a book on this subject and saw this one, I decided to buy it.

I am glad I did. I am working my way through it -- problems and all -- and have finished the first three chapters. I find the material well presented and satisfying my needs. As a statistician I appreciate the fact that Dr. Andrews elected to take a combinatorial approach to the topic. Being familiar with this type of reasoning makes certain topics easier for me to comprehend. The book is not for the layman and takes an individual with a solid mathematical background to get through it in its entirety. However, if you're interested in the topic and willing to put in the effort, the book will pay off.

26 people found this helpful

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book

Good book

Reviewed in the United States on October 13, 2018

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Good book. It may help to be familiar with proof by induction before reading this book. However the book does spend 1 chapter in the beginning covering proof by induction.

5 people found this helpful

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Tanis Coralee Leonhardi

Interesting inclusion of computers

Reviewed in the United States on January 7, 2020

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Number theory is both easy and difficult. This book does a good job of highlighting some of these aspects in a clear and straightforward way. It also does a good job of discussing the role technology is playing for some in the field today.

One person found this helpful

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Native of Neptune

Might as well be renamed 'Combinatorial Number Theory'

Reviewed in the United States on February 22, 2012

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A few years ago, I read this book by George Andrews of Penn State University into chapter 8 and this 1971 textbook by him already shows his long interest in both combinatorics and number theory. Where I stopped reading was when the author's proofs started being multiple pages long.

Here are the titles of the chapters with their starting pages:

// PART I Multiplicativity-Divisibility // 1. Basis Representation-3 / 2. The Fundamental Theorem of Arithmetic-12 / 3. Combinatorial and Computational Number Theory-30 / 4. Fundamentals of Congruences-49 / 5. Solving Congruences-58 / 6. Arithmetic Functions-75 / 7. Primitive Roots-93 / 8. Prime Numbers-100 // PART II Quadratic Congruences // 9. Quadratic Residues-115 / 10. Distribution of Quadratic Residues-128 // PART III Additivity // 11. Sums of Squares-141 / 12. Elementary Partition Theory-149 / 13. Partition Generating Functions-160 / 14. Partition Identities-175 // PART IV Geometric Number Theory // 15. Lattice Points-201 / There are four mathematical appendices and the full set of indices after the 15 chapters--213-259.

From the complicated table of contents above, one can see a broad sweep of combinatorial number theory. Part I is mostly pretty straight number theory, and that is what I did read. Part III on additivity is almost fully combinatorics more than number theory though. Still the price of this book is quite low to have access to all of this big range of mathematics to pick and choose what is most interesting to any given reader. Recommended.

39 people found this helpful

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D. Taylor

Excellent text by expert in the field

Reviewed in the United States on December 22, 2000

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George Andrews is the reigning expert on partitions in the mathematical community who has written many seminal papers on the subject over the past half-century! If you don't know what partitions are in the theoretical sense, don't worry, the text provides ample introduction. I don't think you can find a more elementary introduction to the difficult, but extraordinarily powerful and elegant theory of partitions. The book covers the basics of number theory well, but it is the chapters on partitions that make this text stand out. It covers the Rogers-Ramanujan identities as well as the Jacobi triple product identity. It is rare in the mathematical community that an expert in a subject also writes a ground-level introductory text - but that's what you have here. Thanks to the dover edition, it's now quite affordable.

142 people found this helpful

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Vincent Cina

Good value - poor image quality on Kindle

Reviewed in the United States on December 8, 2016

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You simply can not beat the content for the price!

One huge complaint though for the Kindle Edition. On the big iPad Pro, some formulas are just about unreadable. The formulas appear to be images and their size can not be adjusted with the text size adjustment. The resolution on these formula image is so poor that some of the small parts of the formula, like subscripts and superscripts are simply unreadable. I've taken to trying to use a magnifying glass to help read some formulas, but that is simply not an enjoyable reading experience.

11 people found this helpful

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Lappoon Rupert Tang

Great Services

Reviewed in the United States on January 17, 2020

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A very nice book arrived on time.

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Nabla

Interesting and rigorous

Reviewed in the United Kingdom on November 12, 2014

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Very interesting introduction to number theory. The book starts by introducing Peano's axioms, as well as groups and semigroups, but quickly moves onto more advanced topics. The book is rigorous, proofs are given for each theorem.

3 people found this helpful

Thomas Cullen

Five Stars

Reviewed in the United Kingdom on July 26, 2017

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very good introduction to number theory

JPW

Five Stars

Reviewed in the United Kingdom on November 30, 2014

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An excellent book.

Edoardo Angeloni

Una miniera di problemi antichi ma sempre interessanti.

Reviewed in Italy on April 1, 2020

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E' un libro poco pretenzioso che a prima vista sembrerebbe contenere solo risultati scontati. Ma in realtà è una miniera di vecchi problemi, però riproposti in maniera originale ed innovativa. Il geniaccio americano per i numeri sembra non aver ancora abbandonato le aule universitarie. Insomma tanti antichi spunti che costituiscono ancora una sfida per le nuove generazioni di studenti.

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Maikol

Incredible book

Reviewed in Spain on September 20, 2020

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If you want to get deeper in Number Theory this book is a must.

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