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Publisher:

Champman&Hall/CRC

Publication Date:

2005

Number of Pages:

261

Format:

Hardcover

Series:

Discrete Mathematics and Its Applications

Price:

69.95

ISBN:

1-58488-482-7

Category:

Monograph

[Reviewed by , on ]

Edward B. Burger

03/29/2006

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*.

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

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