Diophantine analysis

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 Anal...

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Bibliographic Details
Main Author: Steuding, Jörn
Format: Book
Language:English
Published: Boca Raton : Chapman & Hall/CRC, 2005.
Series:Discrete mathematics and its applications
Subjects:
Diophantine analysis.
Analyse diophantienne.
Diophantische Analysis.
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's theorem
  • Padé 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
  • Apéry'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 Morko® spectrum
  • Badly approximable numbers
  • Notes on the metric theory
  • Exercises
  • The Pell equation. The cattle problem
  • Lattice p 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
  • CRFAC
  • 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 ErdÄos-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.

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