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.
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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.
This is the most applications oriented text on the market:
Features extensive, detailed worked examples that
illustrate many number theoretic patterns without the
students' having to generate all of them themselves.
- 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,
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
Covers Peano Postulates a concise and historically
important set of axioms from which number theory can be developed.
1. Sets and Relations.
3. Generalized Set Operations.
1. Elementary Properties of Integers.
2. Axioms for the Integers.
3. Principle of Induction.
7. Application: Random Keys.
8. Application: Random Number Generation I.
9. Application: Two's Complement.
3. CONGRUENCES AND THE FUNCTION.
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.
4. ARITHMETIC FUNCTIONS.
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.
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.
5. CONTINUED FRACTIONS.
2. Multiplicative Functions.
3. The Möbius Function.
4. Generalized Möbius Function.
5. Application: Inversions in Physics.
6. BERTRAND'S POSTULATE.
3. Simple Continued Fractions.
4. Infinite Simple Continued Fractions.
5. Pell's Equation.
6. Application: Relative Rates.
7. Application: Factoring.
7. DIOPHANTINE EQUATIONS.
3. Bertrand's Postulate.
1. Linear Diophantine Equations.
APPENDIX A. LOGIC AND PROOFS.
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.
1. Axiomatic Systems.
APPENDIX B. PEANO'S POSTULATES AND CONSTRUCTION OF THE
APPENDIX C. POLYNOMIALS.
2. Propositional Calculus.
4. Predicate Calculus.
5. Mathematical Proofs.