This text presents a logical development of number theory, focusing on the axiomatic development of number theory, showing how to prove theorems and understand the nature of number theory. Drawing applications from physics, statistics, computer science, mathematics, astronomy, cryptography and mechanics, this book features extensive worked examples which illustrate many number theory patterns. It treats applications in depth with substantive discussion of the context of each application.
Sprache
Verlagsort
Verlagsgruppe
Zielgruppe
Maße
Höhe: 242 mm
Breite: 185 mm
Dicke: 25 mm
Gewicht
ISBN-13
978-0-13-190190-2 (9780131901902)
Copyright in bibliographic data is held by Nielsen Book Services Limited or its licensors: all rights reserved.
Schweitzer Klassifikation
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 Moebius Function.
4. Generalized Moebius 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 + y4 = 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.
