Introduction to Number Theory

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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.

    Editorial 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