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Number Theory with Applications, 1/e

James A. Anderson, University of South Carolina-Spartanburg
James M. Bell, Milliken & Company

Published October, 1996 by Prentice Hall Engineering/Science/Mathematics

Copyright 1997, 566 pp.
Cloth
ISBN 0-13-190190-7


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Summary

Ideal for students of varying mathematical sophistication, this text provides a self-contained logical development of basic number theory, supplemented with numerous applications and advanced topics.

Features


Offers more flexibility in coverage than other texts. It allows instructors to first cover core ideas for writing proofs and basic number theory sections. It then branches into several directions as desired, including chapters that utilize abstract algebra.
Focuses on the axiomatic development of number theory — showing students how to prove theorems and understand the nature of number theory.
This is the most applications oriented text on the market:

  • Presents applications in each chapter where the supporting theory is developed.
  • Treats applications in depth — with substantive discussion of the context of each application.
  • Draws applications from many areas — e.g., physics, statistics, computer science, mathematics, astronomy, cryptography, and mechanics.
Features extensive, detailed worked examples that illustrate many number theoretic patterns — without the students' having to generate all of them themselves.
Provides over 1000 practice problems — of various types, covering all theory and applications presented.
Surveys the historical context of number theory and the people who developed the theorems.
Offers a self-contained treatment of tools necessary to understand and construct proofs — e.g., set theory and proof logic.
Covers Peano Postulates — a concise and historically important set of axioms from which number theory can be developed.


Table of Contents
0. SETS.
    1. Sets and Relations.
    2. Functions.
    3. Generalized Set Operations.

1. Elementary Properties of Integers.
    1. Introduction.
    2. Axioms for the Integers.
    3. Principle of Induction.
    4. Division.
    5. Representation.
    6. Congruence.
    7. Application: Random Keys.
    8. Application: Random Number Generation I.
    9. Application: Two's Complement.
2. PRIMES.
    1. Introduction.
    2. Prime Factorization.
    3. Distribution of the Primes.
    4. Elementary Algebraic Structures in Number Theory.
    5. Application: Pattern Matching.
    6. Application: Factoring by Pollard's r.
3. CONGRUENCES AND THE ˆ FUNCTION.
    1. Introduction.
    2. Chinese Remainder Theorem.
    3. Matrices and Simultaneous Equations.
    4. Polynomials and Solutions of Polynomial Congruences.
    5. Properties of the Function f
    6. The Order of an Integer.
    7. Primitive Roots.
    8. Indices.
    9. Quadratic Residues and the Law of Reciprocity.
    10. Jacobi Symbol.
    11. Application: Unit Orthogonal Matrices.
    12. Application: Random Number Generation II.
    13. Application: Hashing Functions.
    14. Application: Indices.
    15. Application: Cryptography.
    16. Application: Primality Testing.
4. ARITHMETIC FUNCTIONS.
    1. Introduction.
    2. Multiplicative Functions.
    3. The Möbius Function.
    4. Generalized Möbius Function.
    5. Application: Inversions in Physics.
5. CONTINUED FRACTIONS.
    1. Introduction.
    2. Convergents.
    3. Simple Continued Fractions.
    4. Infinite Simple Continued Fractions.
    5. Pell's Equation.
    6. Application: Relative Rates.
    7. Application: Factoring.
6. BERTRAND'S POSTULATE.
    1. Introduction.
    2. Preliminaries.
    3. Bertrand's Postulate.
7. DIOPHANTINE EQUATIONS.
    1. Linear Diophantine Equations.
    2. Pythagorean triples.
    3. Integers as Sums of Two Squares.
    4. Quadratic Forms.
    5. Integers as Sums of Three Squares.
    6. Integers as Sums of Four Squares.
    7. The Equation ax^2 + by^2 + cz = 0.
    8. The Equation x^4 + y^ 4 = z^2.
APPENDIX A. LOGIC AND PROOFS.
    1. Axiomatic Systems.
    2. Propositional Calculus.
    3. Arguments.
    4. Predicate Calculus.
    5. Mathematical Proofs.
APPENDIX B. PEANO'S POSTULATES AND CONSTRUCTION OF THE INTEGERS. APPENDIX C. POLYNOMIALS.


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