Diophantine analysis comprises two different but interconnected domains—diophantine approximation and diophantine equations. This 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 is President of Southwestern University, Georgetown, TX .
Opening thoughts: Welcome to the jungle Chapter 1. A bit of foreshadowing and some rational rationale Chapter 2. Building the rationals via Farey sequences Chapter 3. Discoveries of Dirichlet and Hurwitz Chapter 4. The theory of continued fractions Chapter 5. Enforcing the law of best approximates Chapter 6. Markoff’s spectrum and numbers Chapter 7. Badly approximable numbers and quadratics Chapter 8. Solving the alleged “Pell” equation Chapter 9. Liouville’s work on numbers algebraic and not Chapter 10. Roth’s stunning result and its consequences Chapter 11. Pythagorean triples through Diophantine geometry Chapter 12. A quick tour through elliptic curves Chapter 13. The geometry of numbers Chapter 14. Simultaneous diophantine approximation Chapter 15. Using geometry to sum some squares Chapter 16. Spinning around irrationally and uniformly Chapter 17. A whole new world of p-adic numbers Chapter 18. A glimpse into p-adic analysis Chapter 19. A new twist on Newton’s method Chapter 20. The power of acting locally while thinking globally Appendix 1. Selected big picture question commentaries Appendix 2. Hints and remarks Appendix 3. Further reading