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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.

Edward B. Burger: Williams College, Williamstown, MA

Cover 1
Title 3
Copyright 4
Table of Contents 5
Opening thoughts: Welcome to the jungle 7
Modules 13
Module 1. A bit of foreshadowing and some rational rationale 13
Module 2. Building the rationals via Farey Sequences 17
Module 3. Discoveries of Dirichlet and Hurwitz 23
Module 4. The theory of continued fractions 29
Module 5. Enforcing the law of best approximates 35
Module 6. Markoff's spectrum and numbers 41
Module 7. Badly approximabie numbers and quadratics 49
Module 8. Solving the alleged "Pell" equation 55
Module 9. Liouville's work on numbers algebraic and not 61
Module 10. Roth's stunning result and its consequences 67
Module 11. Pythagorean triples through diophantine geometry 73
Module 12. A quick tour through elliptic curves 81
Module 13. The geometry of numbers 87
Module 14. Simultaneous diophantine approximation 95
Module 15. Using geometry to sum some squares 101
Module 16. Spinning around irrationally and uniformly 105
Module 17. A whole new world of p-adic numbers 111
Module 18. A glimpse into p-adic analysis 119
Module 19. A new twist on Newton's method 123
Module 20. The power of acting locally while thinking globally 129
Appendices 135
1. Selected Big Picture Question commentaries 135
2. Hints and remarks 141
3. Further reading 151
Acknowledgments 153
Index 155
A 155
B 155
C 155
D 155
E 155
F 155
G 155
H 156
I 156
J 156
K 156
L 156
M 156
N 156
O 156
P 156
Q 156
R 156
S 156
T 156
U 157
Back Cover 160
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