**Student Mathematical Library**

Volume: 8;
2000;
151 pp;
Softcover

MSC: Primary 11;

Print ISBN: 978-0-8218-2640-9

Product Code: STML/8

List Price: $27.00

AMS Member Price: $21.60

MAA Member Price: $24.30

**Electronic ISBN: 978-1-4704-2126-7
Product Code: STML/8.E**

List Price: $25.00

AMS Member Price: $20.00

MAA Member Price: $22.50

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# Exploring the Number Jungle: A Journey into Diophantine Analysis

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*Edward B. Burger*

Welcome to diophantine analysis—an area of number theory in
which we attempt to discover hidden treasures and truths within the
jungle of numbers by exploring rational numbers. Diophantine
analysis comprises two different but interconnected
domains—diophantine approximation and diophantine equations.
This highly readable book brings to life the fundamental ideas and
theorems from diophantine approximation, geometry of numbers,
diophantine geometry and \(p\)-adic analysis. Through an engaging
style, readers participate in a journey through these areas of number theory.

Each mathematical theme is presented in a self-contained manner and is
motivated by very basic notions. The reader becomes an active participant
in the explorations, as each module includes a sequence of numbered
questions to be answered and statements to be verified. Many hints and
remarks are provided to be freely used and enjoyed. Each module then
closes with a Big Picture Question that invites the reader to step back
from all the technical details and take a panoramic view of how the ideas
at hand fit into the larger mathematical landscape. This book enlists the
reader to build intuition, develop ideas and prove results in a very
user-friendly and enjoyable environment.

Little background is required and a familiarity with number theory is not
expected. All that is needed for most of the material is an understanding
of calculus and basic linear algebra together with the desire and ability
to prove theorems. The minimal background requirement combined with the
author's fresh approach and engaging style make this book enjoyable and
accessible to second-year undergraduates, and even advanced high school
students. The author's refreshing new spin on more traditional
discovery approaches makes this book appealing to any mathematician and/or
fan of number theory.

#### Readership

Undergraduate and graduate students and mathematicians interested in number theory.

#### Reviews & Endorsements

A wealth of information … designed as a textbook at the undergraduate level, with lots of exercises. The choice of material is very nice: Diophantine approximation is the unifying theme, but the tour has side trips to elliptic curves, Riemann surfaces, and \(p\)-adic analysis. The writing style is relaxed and pleasant … For this trip, the guide has chosen an ascent to an accessible summit, but with the emphasis always on teaching and motivating important techniques so that the beginner can advance to a higher level.

-- MAA Monthly

The author invites the reader right from the beginning, through his engaging and motivating style, to develop ideas actively and to find proofs for himself … Remarks at the end of each of the 20 short sections, into which this readily readable introduction to diophantine analysis is divided, extend the material and stimulate the reader to deeper study and involvement with it.

-- Zentralblatt MATH

This short book presents a nice enjoyable introduction to Diophantine analysis, which invites the motivated reader to rediscover by himself or herself many of the fundamental results of the subject, with hints given in an appendix for the more difficult results.

-- Mathematical Reviews

Here … is something truly different … number theory is such a large subject and so much of it is initially accessible without too many prerequisites, that one would expect to see a different take on the subject every once in a while. And that, happily, is what we have here both, in content and in style of presentation … For professors with the requisite background, this may be just the right book to use in an upper-level undergraduate seminar. Students working through this book will learn some nice material and will probably also emerge from the course with a much greater confidence in their ability to do mathematics.

-- MAA Online

#### Table of Contents

# Table of Contents

## Exploring the Number Jungle: A Journey into Diophantine Analysis

- Cover Cover11 free
- Title i3 free
- Copyright ii4 free
- Table of Contents v5 free
- Opening thoughts: Welcome to the jungle 17 free
- Modules 713
- Module 1. A bit of foreshadowing and some rational rationale 713
- Module 2. Building the rationals via Farey Sequences 1117
- Module 3. Discoveries of Dirichlet and Hurwitz 1723
- Module 4. The theory of continued fractions 2329
- Module 5. Enforcing the law of best approximates 2935
- Module 6. Markoff's spectrum and numbers 3541
- Module 7. Badly approximabie numbers and quadratics 4349
- Module 8. Solving the alleged "Pell" equation 4955
- Module 9. Liouville's work on numbers algebraic and not 5561
- Module 10. Roth's stunning result and its consequences 6167
- Module 11. Pythagorean triples through diophantine geometry 6773
- Module 12. A quick tour through elliptic curves 7581
- Module 13. The geometry of numbers 8187
- Module 14. Simultaneous diophantine approximation 8995
- Module 15. Using geometry to sum some squares 95101
- Module 16. Spinning around irrationally and uniformly 99105
- Module 17. A whole new world of p-adic numbers 105111
- Module 18. A glimpse into p-adic analysis 113119
- Module 19. A new twist on Newton's method 117123
- Module 20. The power of acting locally while thinking globally 123129

- Appendices 129135
- Acknowledgments 147153
- Index 149155
- Back Cover Back Cover1160