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Diophantine Analysis

Publisher: 
Champman&Hall/CRC
Number of Pages: 
261
Price: 
69.95
ISBN: 
1-58488-482-7

This very nice text offers an impressive and breathtaking overview of Diophantine approximation, Diophantine equations, and related classical topics. The book covers a wide range of material from very elementary results through the more serious standard topics.

The book opens by introducing material usually found in an undergraduate number theory book. The writing style here is, at times, lively and cute. The book then moves into the elementary aspects of Diophantine approximation (a la Dirichet, Kronecker, and Hurwitz), introduces Pade approximation, and develops the theory of continued fractions and the connection with the so-called "Pell's Equation". The text then treats some topics that are less common in such books: Apery's and Beukers' proofs of the irrationality of ζ(3) and the connection between factoring and continued fractions.

The book then takes on a much more serious tone as it moves through the geometry of numbers (including a discussion, but no proof of Minkowski's Successive Minima Theorem), elementary transcendence results, and Roth's Theorem (whose proof is included in detail). The text closes with a treatment of some more modern topics including the abc-conjecture and an introduction to p -adic analysis.

While the book has exercises throughout, the intended audience is not clear. The voice of the book varies dramatically — sometimes very basic notions are described in lively detail as if the reader might be beginning his or her journey into mathematics, but then on page 8 one of the standard proofs of the irrationality of π is offered devoid of any intuition whatsoever. Then after this technically complicated argument, on page 12 we are informed of the definition of the Division Algorithm. Certainly material on Roth's Theorem is not designed for the usual undergraduate mathematics student.

Despite this slight unevenness, for the professional mathematician this book provides a fine resource. I enjoyed it and am sure I will be a useful reference.


Edward B. Burger teaches at Williams College. He is the author of many books, including Exploring the Number Jungle: A Journey into Diophantine Analysis.

Date Received: 
Tuesday, August 16, 2005
Reviewable: 
Yes
Include In BLL Rating: 
No
Jörn Steuding
Series: 
Discrete Mathematics and Its Applications
Publication Date: 
2005
Format: 
Hardcover
Category: 
Monograph
Edward B. Burger
03/29/2006

 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
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 Marko® 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 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

BIBLIOGRAPHY

INDEX

Publish Book: 
Modify Date: 
Wednesday, May 24, 2006

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