This book effectively integrates computing algorithms into the number theory curriculum using a heuristic approach and strong emphasis on proofs. Its in-depth coverage of modern applications considers the latest trends and topics, such as elliptic curvesa subject that has seen a rise in popularity due to its use in the proof of Fermat's Last Theorem.
Offers a wealth of platform-independent computer activities (stressing the fundamental algorithms), allowing students to discover interesting properties of integers and work toward proving these properties.
Discusses applications to cryptography in detail, examining all of the commonly used public-key cryptographic methods.
Provides a clear presentation of primality testing and factoring, guiding students through numerous extended examples.
Includes coverage of advanced topics not treated in most other texts, such as binary quadratic forms and elliptic curves.
Encourages cooperative learning through over 50 group projects.
Features over 500 worked examples, over 1,000 exercisesmany of which are computer projectsand over 50 extended end-of-chapter exercises which are suitable for group projects.
Web site offering instruction in Maple and Mathematica for the text computer projects.
2. Divisibility and Primes.
3. Modular Arithmetic.
4. Fundamental Theorems of Modular Arithmetic.
6. Primality Testing and Factoring.
7. Primitive Roots.
9. Quadratic Congruences.
11. Continued Fractions.
12. Factoring Methods.
13. Diophantine Approximations.
14. Diophantine Equations.
15. Arithmetical Functions and Dirichlet Series.
16. Distribution of Primes.
17. Quadratic Reciprocity Law
18. Binary Quadratic Forms.
19. Elliptic Curves.
Appendix A: Mathematical Induction.
Appendix B: Binomial Theorem.
Appendix C: Algorithmic Complexity and O-notation.
Answers and Hints.
Index of Notation.