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

ISBN-10: 0131901907
ISBN-13: 9780131901902

Publisher: Prentice Hall
Copyright: 1997
Format: Paper; 566 pp
Published: 10/11/1996
Status: Available on Demand What's this?

Suggested retail price: $73.33
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For undergraduate courses in Number Theory for mathematics, computer science, and engineering majors.

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.

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. Pg.___

This is the most applications oriented text on the market:
- Presents applications in each chapter where the supporting theory is developed. Pg.___
- Treats applications in depth — with substantive discussion of the context of each application. Pg.___
- Draws applications from many areas — e.g., physics, statistics, computer science, mathematics, astronomy, cryptography, and mechanics. Pg.___

Features extensive, detailed worked examples that illustrate many number theoretic patterns —without the students' having to generate all of them themselves. Pg.___

Provides over 1000 practice problems — of various types, covering all theory and applications presented. Pg.___

Surveys the historical context of number theory and the people who developed the theorems. Pg.___

Offers a self-contained treatment of tools necessary to understand and construct proofs — e.g., set theory and proof logic. Pg.___

Covers Peano Postulates — a concise and historically important set of axioms from which number theory can be developed. Pg.___

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 + 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.

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