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.
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 Badly approximable numbers and quadratics 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 Further reading Acknowledgments Index.