Description
Book SynopsisDiophantine analysis, an area of number theory that helps to discover hidden treasures and truths within the numbers by exploring rational numbers, comprises two different but interconnected domains - diophantine approximation and diophantine equations. This book presents the fundamental ideas and theorems from diophantine approximation.
Trade ReviewThe author invites the reader right from the beginning, through his engaging and motivating style, to develop ideas actively and to find proofs for himself Zentralblatt MATH
Table of ContentsOpening 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.
