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Publisher:

Chapman&Hall/CRC

Publication Date:

2008

Number of Pages:

521

Format:

Hardcover

Series:

Discrete Mathematics and Its Applications

Price:

89.95

ISBN:

978-1584889373

Category:

Textbook

[Reviewed by , on ]

Underwood Dudley

06/9/2008

This is a text for advanced undergraduates. It is notable for including *everything* that anyone would expect to find in an elementary number theory text, and more. For example, besides chapters on analytic number theory and elliptic curves, there is an outline of the resolution of Hilbert’s tenth problem. Continued fractions get a chapter, the gaussian integers are developed to answer the question of which integers are the sums of two squares, the Dirichlet convolution, of which I have always been fond, is given its due, and Hadamard matrices make an appearance.

Of course, this makes for a long book, more than 520 pages, that could not possibly be covered in a semester. As the authors point out, chapters 1–9, omitting chapter 4 on cryptography, give the basics, and then some, in 320 pages.

A feature of the book of which, for good or ill, prospective users should be aware is the inclusion of code, for both Mathematica and Maple, for various algorithms. Reading it did very little for me and , since the code can be found on the authors’ website, students would not have to retype it. Devotees of one or other of the programs may well feel differently.

The exposition is straightforward, written in good mathematical prose without distractions or infelicities. There is a good selection of exercises at the end of each of the five to ten sections in each chapter but no answers or hints at the back of the book, not even to problems whose numbers are congruent to 7 (mod 8). I would have liked to see more examples, but the authors are not aiming at weak students (who will find the book slow going, written as it is in lemma-proposition-theorem-proof style) and making the book longer would be a mistake.

The book is remarkably free of errors and misprints, for which the authors and their editor deserve great credit. Reviewers, however, are entitled to two free shots: there is a grammatical error on page 346, line 3, and on page 80 Lagrange is given a Spanish flavor (“Joseph-Luis”).

The book is pleasing to the eye except for the tables, which have too many heavy lines. The authors are to be congratulated for avoiding that uncouth usage, which unfortunately may be spreading, of “d*t*” in integrals.

It contains no color pictures, no sidebars, its margins are of the standard size, and though the authors include “real-life” applications, they do not claim that number theory will lead to lucrative careers.

Erickson and Vazzana have written an admirable text.

Woody Dudley’s number theory text was published forty years ago. The field has changed since then, slightly.

*Core Topics*

**Introduction**

What is number theory?

The natural numbers

Mathematical induction

**Divisibility and Primes**

Basic definitions and properties

The division algorithm

Greatest common divisor

The Euclidean algorithm

Linear Diophantine equations

Primes and the fundamental theorem of arithmetic

**Congruences**

Residue classes

Linear congruences

Application: Check digits and the ISBN system

Fermat’s theorem and Euler’s theorem

The Chinese remainder theorem

Wilson’s theorem

Order of an element mod *n*

Existence of primitive roots

Application: Construction of the regular 17-gon

**Cryptography**

Monoalphabetic substitution ciphers

The Pohlig–Hellman cipher

The Massey–Omura exchange

The RSA algorithm

**Quadratic Residues**

Quadratic congruences

Quadratic residues and nonresidues

Quadratic reciprocity

The Jacobi symbol

Application: Construction of tournaments

Consecutive quadratic residues and nonresidues

Application: Hadamard matrices

*Further Topics*

**Arithmetic Functions**

Perfect numbers

The group of arithmetic functions

Möbius inversion

Application: Cyclotomic polynomials

Partitions of an integer

**Large Primes**

Prime listing, primality testing, and prime factorization

Fermat numbers

Mersenne numbers

Prime certificates

Finding large primes

**Continued Fractions**

Finite continued fractions

Infinite continued fractions

Rational approximation of real numbers

Periodic continued fractions

Continued fraction factorization

**Diophantine Equations**

Linear equations

Pythagorean triples

Gaussian integers

Sums of squares

The case *n* = 4 in Fermat’s last theorem

Pell’s equation

Continued fraction solution of Pell’s equation

The *abc* conjecture

*Advanced Topics*

**Analytic Number Theory**

Sum of reciprocals of primes

Orders of growth of functions

Chebyshev’s theorem

Bertrand’s postulate

The prime number theorem

The zeta function and the Riemann hypothesis

Dirichlet’s theorem

**Elliptic Curves**

Cubic curves

Intersections of lines and curves

The group law and addition formulas

Sums of two cubes

Elliptic curves mod *p*

Encryption via elliptic curves

Elliptic curve method of factorization

Fermat’s last theorem

**Logic and Number Theory**

Solvable and unsolvable equations

Diophantine equations and Diophantine sets

Positive values of polynomials

Logic background

The negative solution of Hilbert’s tenth problem

Diophantine representation of the set of primes

**APPENDIX A: Mathematica Basics**

**APPENDIX B: Maple Basics**

**APPENDIX C: Web Resources**

**APPENDIX D: Notation**

**References**

**Index**

*Notes appear at the end of each chapter.*

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