| Day | Topics | Preparation | |
|---|---|---|---|
| Wed. 8/31 | Introduction. Numbers: integer, rational, algebraic, real and complex. |
Sec. 1.1 | |
| Fri. 9/2 | Induction. Sequences, summation notation. |
Sec. 1.2-3 | |
| Wed. 9/7 | Induction proofs. | Sec. 1.3 | |
| Fri. 9/9 | Recursion, Fibonacci numbers. | Sec. 1.4 | |
| Mon. 9/12 | Divisiblity, quotient remainder theorem. | Sec. 1.5 | |
| Wed. 9/14 | Floor and ceiling functions. Base r representation of an integer. |
Sec. 1.1,5 Sec. 2.1 | |
| Fri. 9/16 | MAPLE: Meet in GMCS 405 | Getting started | |
| Mon. 9/19 | Base r conversions and arithmetic, computational complexity. | Secs. 2.1-3 | |
| Wed. 9/21 | Prime numbers. | Sec. 3.1 | |
| Fri. 9/23 | The prime number theorem. Greatest common divisior. | Sec. 3.2 Sec. 3.3 | |
| Mon. 9/26 | Greatest common divisor. Euclidean algorithm. |
Sec. 3.3,4 | |
| Wed. 9/28 | Extended Euclidean algorithm.
Gcd of several ints. Least common multiple. |
Sec. 3.3,4 | |
| Fri. 9/30 | EXAM. | Secs. 1.1-5; 2.1-2 3.1-3 | |
| Mon. 10/3 | Unique factorization. | Sec. 3.5 | |
| Wed. 10/5 | Applications of unique factorization. | Sec. 3.5 | |
| Fri. 10/7 | MAPLE: Meet in GMCS 405. | Prime numbers | |
| Mon. 10/10 | Linear Diophantine equations. | 3.6. | |
| Wed. 10/12 | Linear Diophantine equations in 3 variables. Unique factorization: Application to roots of polynomials. |
Sec. 3.4, 6 | |
| Fri. 10/14 | Modular arithmetic. | Sec. 4.1 | |
| Mon. 10/17 | Solution of congruences, linear and quadratic.
Units and zero-divisors in Z/m . |
Sec. 4.1-2 | |
| Wed. 10/19 | Chinese remainder theorem. | Sec. 4.3 | |
| Fri. 10/21 | MAPLE: meet in GMCS 405. | Modular arithmetic | |
| Mon. 10/24 | Extensions of the Chinese remainder theorem. | Sec. 4.3 Sec 5.5 | |
| Wed. 10/26 | Applications: divisiblity tests, tournaments. | Secs. 5.1, 5.3 | |
| Fri. 10/28 | Check digits. | Sec. 5.5 | |
| Mon. 10/31 | Systems of linear congruences. | Sec. 4.5 | |
| Wed. 11/2 | Linear algebra modulo m . | Sec. 4.5 | |
| Fri. 11/4 | Linear cryptosystems. | Sec. 8.1, 8.2 | |
| Mon. 11/7 | Questions? Nonlinear Diophantine equations: Pythagorean triples, Fermat's last theorem |
Sec. 13.1-2 | |
| Wed. 11/9 | EXAM. | Sec. 3.5,7 Sec. 4.1-3,5; Sec. 5.1,5; Sec. 8.1-2 | |
| Fri. 11/11 | MAPLE: meet in GMCS 405 Show me your work! |
. | |
| Mon. 11/14 | Fermat's little theorem.
Exponentiation cryptosystems |
Sec. 6.1 Sec. 8.3 | |
| Wed. 11/16 | Euler's theorem | Sec. 6.3 | |
| Fri. 11/18 | RSA Cipher. | Sec. 8.4 | |
| Mon. 11/21 | Multiplicative functions:
Euler phi function, number of divisors, sum of divisors |
Sec 7.1,2 | |
| Wed. 11/23 | Multiplicative functions: The summatory function, number of divisors, sum of divisors | Sec 7.1,2 | |
| Mon. 11/28 | Polynomials over Z/m | Sec. 9.2 | |
| Wed. 11/30 | Unique factorization in Z/p [x] . | Secs. 9.2, 9.4 | |
| Fri. 12/2 | Primitive elements in Z/p , index arithmetic. | Sec. 9.1, 9.4 | |
| Mon. 12/5 | Discrete log cryptosystems. | Sec. 10.2 | |
| Wed. 12/7 | The Gaussian integers. | . | |
| Fri. 12/9 | MAPLE: meet in GMCS 405 Show me your work! |
. | |
| Mon. 12/12 | A domain without unique factorization. | . |