426 Pages 23 B/W Illustrations
    by Chapman & Hall

    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.

    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


    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