### Features

Presents a modern treatment of the fundamentals of number theory, including primes, congruences, and Diophantine equations Contains an entire chapter on cryptography Covers special topics in number theory, including elliptic curves and Hilbert’s tenth problem Uses *Mathematica* and Maple calculations to elucidate and expand on the theory covered Applies number theory to real-world problems, such as the ISBN system, RSA codes, and the construction of tournaments Provides a supplemental web page with *Mathematica* notebooks, Maple worksheets, and links to Internet resources Includes over 100 worked examples and over 500 exercises, along with a solutions manual for qualifying instructors
### Summary

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.

### Table of 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.*
### 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