Exploring the number jungle : A journey into diophantine analysis

Includes index.

hapter 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.

This book brings to life the fundamental ideas and theorems from diophantine approximation and equations, geometry of numbers, diophantine geometry, and p-adic analysis. Through his engaging style, the author creates an inviting atmosphere where readers actively build intuition, develop ideas, and prove results as they journey through some beautiful areas of number theory. The minimal background requirements and the author's fresh approach make this book enjoyable and accessible to a wide range of students, mathematicians, and fans of number theory.