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Number Theory with Applications
ISBN-10: 0-13-190190-7
ISBN-13: 978-0-13-190190-2
Published by Pearson
© 1997
Pub. Date: Oct 11, 1996
Format: Paper
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 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 ax2 + by2 + cz = 0.
8. The Equation x4 + y 4 = z2.
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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