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Introduction to Number Theory

By Anthony Vazzana, Martin Erickson, Anthony Vazzana, David Garth

Series Editor: Kenneth H. Rosen

Chapman and Hall/CRC – 2007 – 536 pages

Series: Textbooks in Mathematics

Purchasing Options:

  • Add to CartHardback: $104.95
    978-1-58488-937-3
    October 30th 2007

Description

One of the oldest branches of mathematics, number theory is a vast field devoted to studying the properties of whole numbers. Offering a flexible format for a one- or two-semester course, Introduction to Number Theory uses worked examples, numerous exercises, and two popular software packages to describe a diverse array of number theory topics.

This classroom-tested, student-friendly text covers a wide range of subjects, from the ancient Euclidean algorithm for finding the greatest common divisor of two integers to recent developments that include cryptography, the theory of elliptic curves, and the negative solution of Hilbert’s tenth problem. The authors illustrate the connections between number theory and other areas of mathematics, including algebra, analysis, and combinatorics. They also describe applications of number theory to real-world problems, such as congruences in the ISBN system, modular arithmetic and Euler’s theorem in RSA encryption, and quadratic residues in the construction of tournaments. The book interweaves the theoretical development of the material with Mathematica® and Maple™ calculations while giving brief tutorials on the software in the appendices.

Highlighting both fundamental and advanced topics, this introduction provides all of the tools to achieve a solid foundation in number theory.

Reviews

Introduction to Number Theory is a well-written book on this important branch of mathematics. … The authors succeed in presenting the topics of number theory in a very easy and natural way, and the presence of interesting anecdotes, applications, and recent problems alongside the obvious mathematical rigor makes the book even more appealing. I would certainly recommend it to a vast audience, and it is to be considered a valid and flexible textbook for any undergraduate number theory course.

—IACR Book Reviews, May 2011

Erickson and Vazzana provide a solid book, comprising 12 chapters, for courses in this area … All in all, a welcome addition to the stable of elementary number theory works for all good undergraduate libraries.

—J. McCleary, Vassar College, CHOICE, Vol. 46, No. 1, August 2009

… reader-friendly text … 'Highlighting both fundamental and advanced topics, this introduction provides all of the tools to achieve a solid foundation in number theory.'

L’Enseignement Mathématique, Vol. 54, No. 2, 2008

Contents

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.

Name: Introduction to Number Theory (Hardback)Chapman and Hall/CRC 
Description: By Anthony Vazzana, Martin Erickson, Anthony Vazzana, David GarthSeries Editor: Kenneth H. Rosen. One of the oldest branches of mathematics, number theory is a vast field devoted to studying the properties of whole numbers. Offering a flexible format for a one- or two-semester course, Introduction to Number Theory uses worked examples, numerous...
Categories: Number Theory, Combinatorics, Cryptology