Fundamental Number Theory with Applications, Second Edition

Fundamental Number Theory with Applications, Second Edition

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Features

  • Covers all aspects of basic number theory
  • Provides a rigorous mathematical presentation
  • Explores numerous applications to cryptography
  • Presents biographies of notable contributors to the field
  • Includes almost 400 exercises, a bibliography, an extensive index, and appendices of background material
  • Offers a solutions manual for qualifying instructors
  • Summary

    An update of the most accessible introductory number theory text available, Fundamental Number Theory with Applications, Second Edition presents a mathematically rigorous yet easy-to-follow treatment of the fundamentals and applications of the subject. The substantial amount of reorganizing makes this edition clearer and more elementary in its coverage.

    New to the Second Edition

    •          Removal of all advanced material to be even more accessible in scope

    •          New fundamental material, including partition theory, generating functions, and combinatorial number theory

    •          Expanded coverage of random number generation, Diophantine analysis, and additive number theory

    •          More applications to cryptography, primality testing, and factoring

    •          An appendix on the recently discovered unconditional deterministic polynomial-time algorithm for primality testing

    Taking a truly elementary approach to number theory, this text supplies the essential material for a first course on the subject. Placed in highlighted boxes to reduce distraction from the main text, nearly 70 biographies focus on major contributors to the field. The presentation of over 1,300 entries in the index maximizes cross-referencing so students can find data with ease.

    Table of Contents

    Preface
    Arithmetic of the Integers
    Induction
    Division
    Primes
    The Chinese Remainder Theorem
    Thue’s Theorem
    Combinatorial Number Theory
    Partitions and Generating Functions
    True Primality Tests
    Distribution of Primes
    Modular Arithmetic
    Basic Properties
    Modular Perspective
    Arithmetic Functions: Euler, Carmichael, and Möbius
    Number and Sums of Divisors
    The Floor and the Ceiling
    Polynomial Congruences
    Primality Testing
    Cryptology
    Primitive Roots
    Order
    Existence
    Indices
    Random Number Generation
    Public-Key Cryptography
    Quadratic Residues
    The Legendre Symbol
    The Quadratic Reciprocity Law
    Factoring
    Simple Continued Fractions and Diophantine Approximation
    Infinite Simple Continued Fractions
    Periodic Simple Continued Fractions
    Pell’s Equation and Surds
    Continued Fractions and Factoring
    Additivity—Sums of Powers
    Sums of Two Squares
    Sums of Three Squares
    Sums of Four Squares
    Sums of Cubes
    Diophantine Equations
    Norm-Form Equations
    The Equation ax2 + by2 + cz2 = 0
    Bachet’s Equation
    Fermat’s Last Theorem
    Appendix A: Fundamental Facts
    Appendix B: Complexity
    Appendix C: Primes 9547 and Least Primitive Roots
    Appendix D: Indices
    Appendix E: The ABC Conjecture
    Appendix F: Primes Is in P
    Solutions to Odd-Numbered Exercises
    Bibliography
    List of Symbols
    Index

    Editorial Reviews

    This is an introductory text in number theory from a well-known name … it covers most of the material traditionally expected in such a course. … One of the most interesting features of this book is the extensive (and crunchy) biographical sketches of relevant mathematicians (both living and dead). … I heartily recommend this book to undergraduates and the passing layman, as it is the work of a master and is lucidly explained. …
    —IACR book reviews, March 2010

    The second edition of this very interesting book includes a revision of its contents and a pledge for the publication of a second volume with advanced material for a second course in number theory.
    —Panayiotis Vlamos,  Zentrablatt Math, 1175

    Praise for the First Edition
    …a very useful addition to the many books on number theory with applications, and it is meant to be accessible to anyone from the novice to the research scientist … [it] provides an excellent supplementary source of information for the reader, not least in the many biographical footnotes on the mathematicians involved in the subject matter, and there are also more than a thousand exercises and examples in the text.  …
    —P. Shiu, Zentralblatt MATH, Vol. 943

     
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