# Introduction to Number Theory

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## Book Description

**Introduction to Number Theory** is a classroom-tested, student-friendly text that covers a diverse array of number theory topics, from the ancient Euclidean algorithm for finding the greatest common divisor of two integers to recent developments such as 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.

Ideal for a one- or two-semester undergraduate-level course, this **Second Edition**:

- Features a more flexible structure that offers a greater range of options for course design
- Adds new sections on the representations of integers and the Chinese remainder theorem
- Expands exercise sets to encompass a wider variety of problems, many of which relate number theory to fields outside of mathematics (e.g., music)
- Provides calculations for computational experimentation using SageMath, a free open-source mathematics software system, as well as Mathematica® and Maple™, online via a robust, author-maintained website
- Includes a solutions manual with qualifying course adoption

By tackling both fundamental and advanced subjects—and using worked examples, numerous exercises, and popular software packages to ensure a practical understanding—**Introduction to Number Theory, Second Edition** instills a solid foundation of number theory knowledge.

## Table of Contents

**Introduction**What is number theory?

**The natural numbers**

**Mathematical induction**

**Notes**

**The Peano axioms**

**Divisibility**Basic definitions and properties

**The division algorithm**

**Representations of integers**

**Greatest Common Divisor**Greatest common divisor

**The Euclidean algorithm**

**Linear Diophantine equations**

**Notes**

Euclid

The number of steps in the Euclidean algorithm

Geometric interpretation of the equation

*ax + by = c*

**Primes**The sieve of Eratosthenes

**The fundamental theorem of arithmetic**

**Distribution of prime numbers**

**Notes**

Eratosthenes

Nonunique factorization and Fermat's last theorem

**Congruences**Residue classes

Linear congruences

Application: Check digits and the ISBN-10 system

The Chinese remainder theorem

**Special Congruences**Fermat's theorem

**Euler's theorem**

**Wilson's theorem**

**Notes**

Leonhard Euler

**Primitive Roots**Order of an element mod

*n*

**Existence of primitive roots**

**Primitive roots modulo composites**

**Application: Construction of the regular 17-gon**

**Notes**

Groups

Straightedge and compass constructions

**Cryptography**Monoalphabetic substitution ciphers

**The Pohlig-Hellman cipher**

The Massey-Omura exchange

**The RSA algorithm**

Notes

Computing powers mod

*p*

RSA cryptography

**Quadratic Residues**Quadratic congruences

Quadratic residues and nonresidues

**Quadratic reciprocity**

**The Jacobi symbol**

Notes

Carl Friedrich Gauss

**Applications of Quadratic Residues**Application: Construction of tournaments

**Consecutive quadratic residues and nonresidues**

**Application: Hadamard matrices**

**Sums of Squares**Pythagorean triples

Gaussian integers

Factorization of Gaussian integers

**Lagrange's four squares theorem**

**Notes**

Diophantus

**Further Topics in Diophantine Equations**The case

*n = 4*in Fermat's last theorem

**Pen's equation**

The

*abc*conjecture

Notes

Pierre de Fermat

The

*p*-adic numbers

**Continued Fractions**Finite continued fractions

**Infinite continued fractions**

**Rational approximation of real numbers**

**Notes**

Continued fraction expansion of

*e*

Continued fraction expansion of tan

*x*

Srinivasa Ramanujan

**Continued Fraction Expansions of Quadratic Irrationals**Periodic continued fractions

**Continued fraction factorization**

**Continued fraction solution of Pen's equation**

Notes

Three squares and triangular numbers

History of Pen's equation

**Arithmetic Functions**Perfect numbers

**The group of arithmetic functions**

**Mobius inversion**

**Application: Cyclotomic polynomials**

**Partitions of an integer**

**Notes**

**The lore of perfect numbers**

**Pioneers of integer partitions**

**Large Primes**Fermat numbers

**Mersenne numbers**

**Prime certificates**

**Finding large primes**

**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**

**Notes**

**Paul Erdős**

**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**

**Notes**

**Projective space**

**Associativity of the group law**

## Author(s)

### Biography

**Martin Erickson** (1963-2013) received his Ph.D in mathematics in 1987 from the University of Michigan, Ann Arbor, USA, studying with Thomas Frederick Storer. He joined the faculty in the Mathematics Department of Truman State University, Kirksville, Missouri, USA, when he was twenty-four years old, and remained there for the rest of his life. Professor Erickson authored and coauthored several mathematics books, including the first edition of *Introduction to Number Theory* (CRC Press, 2007), *Pearls of Discrete Mathematics* (CRC Press, 2010), and *A Student's Guide to the Study, Practice, and Tools of Modern Mathematics* (CRC Press, 2010).

**Anthony Vazzana** received his Ph.D in mathematics in 1998 from the University of Michigan, Ann Arbor, USA. He joined the faculty in the Mathematics Department of Truman State University, Kirksville, Missouri, USA, in 1998. In 2000, he was awarded the Governor's Award for Excellence in Teaching and was selected as the Educator of the Year. In 2002, he was named the Missouri Professor of the Year by the Carnegie Foundation for the Advancement of Teaching and the Council for Advancement and Support of Education.

**David Garth** received his Ph.D in mathematics in 2000 from Kansas State University, Manhattan, USA. He joined the faculty in the Mathematics Department of Truman State University, Kirksville, Missouri, USA, in 2000. In 2005, he was awarded the Golden Apple Award from Truman State University's Theta Kappa chapter of the Order of Omega. His areas of research include analytic and algebraic number theory, especially Pisot numbers and their generalizations, and Diophantine approximation.

## Reviews

Praise for the Previous Edition"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. … a valid and flexible textbook for any undergraduate number theory course."

—International Association for Cryptologic Research Book Reviews, May 2011"… a welcome addition to the stable of elementary number theory works for all good undergraduate libraries."

—J. McCleary, Vassar College, Poughkeepsie, New York, USA, fromCHOICE, Vol. 46, No. 1, August 2009"… a reader-friendly text. … provides all of the tools to achieve a solid foundation in number theory."

—L’Enseignement Mathématique, Vol. 54, No. 2, 2008The theory of numbers is a core subject of mathematics. The authors have written a solid update to the first edition (CH, Aug'09, 46-6857) of this classic topic. There is no shortage of introductions to number theory, and this book does not offer significantly different information. Nonetheless, the authors manage to give the subject a fresh, new feel. The writing style is simple, clear, and easy to follow for standard readers. The book contains all the essential topics of a first-semester course and enough advanced topics to fill a second. In particular, it includes several modern aspects of number theory, which are often ignored in other texts, such as the use of factoring in computer security, searching for large prime numbers, and connections to other branches of mathematics. Each section contains supplementary homework exercises of various difficulties, a crucial ingredient of any good textbook. Finally, much emphasis is placed on calculating with computers, a staple of modern number theory. Overall, this title should be considered by any student or professor seeking an excellent text on the subject.

--A. Misseldine, Southern Utah University, Choice magazine 2016