Diophantine Analysis

Diophantine Analysis

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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.show more

Product details

  • Hardback | 280 pages
  • 160 x 248.9 x 20.3mm | 521.64g
  • Taylor & Francis Inc
  • Chapman & Hall/CRC
  • Boca Raton, FL, United States
  • English
  • 11 black & white illustrations, 11 black & white halftones
  • 1584884827
  • 9781584884828
  • 2,430,763

Table of contents

INTRODUCTION: BASIC PRINCIPLES Who was Diophantus? Pythagorean triples Fermat's last theorem The method of infinite descent Cantor's paradise Irrationality of e Irrationality of pi Approximating with rationals Linear diophantine equations Exercises CLASSICAL APPROXIMATION THEOREMS Dirichlet's approximation theorem A first irrationality criterion The order of approximation Kronecker's approximation theorem Billiard Uniform distribution The Farey sequence Mediants and Ford circles Hurwitz' theorem Pade approximation Exercises CONTINUED FRACTIONS The Euclidean algorithm revisited and calendars Finite continued fractions Interlude: Egyptian fractions Infinite continued fractions Approximating with convergents The law of best approximations Consecutive convergents The continued fraction for e Exercises THE IRRATIONALITY OF z(3) The Riemann zeta-function Apery's theorem Approximating z(3) A recursion formula The speed of convergence Final steps in the proof An irrationality measure A non-simple continued fraction Beukers' proof Notes on recent results Exercises QUADRATIC IRRATIONALS Fibonacci numbers and paper folding Periodic continued fractions Galois' theorem Square roots Equivalent numbers Serret's theorem The Marko(R) spectrum Badly approximable numbers Notes on the metric theory Exercises THE PELL EQUATION The cattle problem Lattice points on hyperbolas An infinitude of solutions The minimal solution The group of solutions The minus equation The polynomial Pell equation Nathanson's theorem Notes for further reading Exercises FACTORING WITH CONTINUED FRACTIONS The RSA cryptosystem A diophantine attack on RSA An old idea of Fermat CFRAC Examples of failures Weighted mediants and a refinement Notes on primality testing Exercises GEOMETRY OF NUMBERS Minkowski's convex body theorem General lattices The lattice basis theorem Sums of squares Applications to linear and quadratic forms The shortest lattice vector problem Gram-Schmidt and consequences Lattice reduction in higher dimensions The LLL-algorithm The small integer problem Notes on sphere packings Exercises TRANSCENDENTAL NUMBERS Algebraic vs. transcendental Liouville's theorem Liouville numbers The transcendence of e The transcendence of pi Squaring the circle? Notes on transcendental numbers Exercises THE THEOREM OF ROTH Roth's theorem Thue equations Finite vs. infinite Differential operators and indices Outline of Roth's method Siegel's lemma The index theorem Wronskians and Roth's lemma Final steps in Roth's proof Notes for further reading Exercises THE ABC-CONJECTURE Hilbert's tenth problem The ABC-theorem for polynomials Fermat's last theorem for polynomials The polynomial Pell equation revisited The abc-conjecture LLL & abc The ErdAos-Woods conjecture Fermat, Catalan & co. Mordell's conjecture Notes on abc Exercises P-ADIC NUMBERS Non-Archimedean valuations Ultrametric topology Ostrowski's theorem Curious convergence Characterizing rationals Completions of the rationals p-adic numbers as power series Error-free computing Notes on the p-adic interpolation of the zeta-function Exercises HENSEL'S LEMMA AND APPLICATIONS p-adic integers Solving equations in p-adic numbers Hensel's lemma Units and squares Roots of unity Hensel's lemma revisited Hensel lifting: factoring polynomials Notes on p-adics: what we leave out Exercises THE LOCAL-GLOBAL PRINCIPLE One for all and all for one The theorem of Hasse-Minkowski Ternary quadratics The theorems of Chevalley and Warning Applications and limitations The local Fermat problem Exercises APPENDIX: ALGEBRA AND NUMBER THEORY Groups, rings, and fields Prime numbers Riemann's hypothesis Modular arithmetic Quadratic residues Polynomials Algebraic number fields Kummer's work on Fermat's last theorem BIBLIOGRAPHY INDEXshow more

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