Diophantine analysis

While its roots reach back to the third century, diophantine analysis continues to be an extremely active and powerful area of number theory. Many diophantine problems have simple formulations, they can be extremely difficult to attack, and many open problems and conjectures remain. Diophantine Analysis examines the theory of diophantine approximations and the theory of diophantine equations, with emphasis on interactions between these subjects. Beginning with the basic principles, the author develops his treatment around the theory of continued fractions and examines the classic theory, inclu. Read more... Abstract: Diophantine analysis is an extremely active field in number theory because of its many open problems and conjectures. Requiring only a basic understanding of number theory, this work is built around the detailed theory of continued fractions and features many applications and examples. Read more... While its roots reach back to the third century, diophantine analysis continues to be an extremely active and powerful area of number theory. Many diophantine problems have simple formulations, they can be extremely difficult to attack, and many open problems and conjectures remain. Diophantine Analysis examines the theory of diophantine approximations and the theory of diophantine equations, with emphasis on interactions between these subjects. Beginning with the basic principles, the author develops his treatment around the theory of continued fractions and examines the classic theory, including some of its applications. He also explores modern topics rarely addressed in other texts, including the abc conjecture, the polynomial Pell equation, and the irrationality of the zeta function and touches on topics and applications related to discrete mathematics, such as factoring methods for large integers. Setting the stage for tackling the field's many open problems and conjectures, Diophantine Analysis is an ideal introduction to the fundamentals of this venerable but still dynamic field. A detailed appendix supplies the necessary background material, more than 200 exercises reinforce the concepts, and engaging historical notes bring the subject to life Content: Chapter 1. Introduction : basic principles -- chapter 2. Classical approximation theorems -- chapter 3. Continued fractions -- chapter 4. The irrationality of [zeta](3) -- chapter 5. Quadratic irrationals -- chapter 6. The Pell equation -- chapter 7. Factoring with continued fractions -- chapter 8. Geometry of numbers -- chapter 9. Transcendental numbers -- chapter 10. The theorem of Roth -- chapter 11. The abc-conjecture -- chapter 12. p-adic numbers -- chapter 13. Hensel's lemma and applications -- chapter 14. The local-global principle.