Introduction to Number Theory  book cover
2nd Edition

Introduction to Number Theory

ISBN 9781498717496
Published December 1, 2015 by Chapman and Hall/CRC
426 Pages 23 B/W Illustrations

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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

What is number theory?
The natural numbers
Mathematical induction
The Peano axioms

Basic definitions and properties
The division algorithm
Representations of integers

Greatest Common Divisor
Greatest common divisor
The Euclidean algorithm
Linear Diophantine equations
The number of steps in the Euclidean algorithm
Geometric interpretation of the equation ax + by = c

The sieve of Eratosthenes
The fundamental theorem of arithmetic
Distribution of prime numbers
Nonunique factorization and Fermat's last theorem

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
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
Straightedge and compass constructions

Monoalphabetic substitution ciphers
The Pohlig-Hellman cipher
The Massey-Omura exchange
The RSA algorithm
Computing powers mod p
RSA cryptography

Quadratic Residues
Quadratic congruences
Quadratic residues and nonresidues
Quadratic reciprocity
The Jacobi symbol
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

Further Topics in Diophantine Equations
The case n = 4 in Fermat's last theorem
Pen's equation
The abc conjecture
Pierre de Fermat
The p-adic numbers

Continued Fractions
Finite continued fractions
Infinite continued fractions
Rational approximation of real numbers
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
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
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
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
Projective space
Associativity of the group law

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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.


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, from CHOICE, 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, 2008

The 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