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Exploring the Number Jungle: A Journey into Diophantine Analysis
Edward B. Burger, Williams College, Williamstown, MA
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Student Mathematical Library
2000; 151 pp; softcover
Volume: 8
ISBN-10: 0-8218-2640-9
ISBN-13: 978-0-8218-2640-9
List Price: US$25 Member Price: US$20
Order Code: STML/8

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.

Reviews

"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

• Opening thoughts: Welcome to the jungle
• A bit of foreshadowing and some rational rationale
• Building the rationals via Farey sequences
• Discoveries of Dirichlet and Hurwitz
• The theory of continued fractions
• Enforcing the law of best approximates
• Markoff's spectrum and numbers
• Solving the alleged "Pell" equation
• Liouville's work on numbers algebraic and not
• Roth's stunning result and its consequences
• Pythagorean triples through diophantine geometry
• A quick tour through elliptic curves
• The geometry of numbers
• Simultaneous diophantine approximation
• Using geometry to sum some squares
• Spinning around irrationally and uniformly
• A whole new world of $$p$$-adic numbers
• A glimpse into $$p$$-adic analysis
• A new twist on Newton's method
• The power of acting locally while thinking globally
• Selected big picture question commentaries
• Hints and remarks