| Day | Topics | Preparation | |
|---|---|---|---|
| Wed. 9/3 | Introduction, countability, the rationals, the reals. | Sec. 1.1 | |
| Fri. 9/5 | Induction, sums and products of sequences, recursion. | Sec. 1.2-3 | |
| Mon. 9/8 | Fibonacci numbers. | Sec. 1.3 | |
| Wed. 9/10 | Divisiblity, greatest integer function. | Sec. 1.4 | |
| Fri. 9/12 | base r representation of an integer. | Sec. 2.1 | |
| Mon. 9/15 | Base r conversions and arithmetic, computational complexity. | Secs. 2.1-3 | |
| Wed. 9/17 | Prime numbers. | Sec. 3.1 | |
| Fri. 9/19 | Prime number conjectures.
Some exercises. |
Sec. 3.1 | |
| Mon. 9/22 | Greatest common divisor. | Sec. 3.2 | |
| Wed. 9/24 | Using Maple: See First worksheet | Sec. 3.3 | |
| Fri. 9/26 | Euclidean algorithm
Gcd of several ints. Least common multiple. |
Sec. 3.3 | |
| Mon. 9/29 | Unique factorization | Sec. 3.4 | |
| Wed. 10/1 | Unique factorization Linear Diophantine equations. |
3.6. | |
| Fri. 10/3 | Linear Diophantine equations. | 3.6. | |
| Mon. 10/6 | Exam. | Secs. 1.1-4; 2.1-2; 3.1-4, 6; | |
| Wed. 10/8 | Unique Factorization:
Application to primes in arithmetic progression Application to roots of polynomials. |
Sec. 3.4 | |
| Fri. 10/10 | MAPLE: meet in GMCS 405 Primes in arithmetic progression; worksheet Adjoining roots of an irreducible polynomial; worksheet |
Sec. 3.4 | |
| Mon. 10/13 | Nonlinear Diophantine equations:
Pythagorean triples, Fermat's last theorem |
Sec. 13.1-2 | |
| Wed. 10/15 | Modular arithmetic. | Sec. 4.1 | |
| Fri. 10/17 | MAPLE: meet in GMCS 405: Modular arithmetic in MAPLE. |
Sec. 4.1 | |
| Mon. 10/20 | Units and zero-divisors in Z/m .
Complete system of residues. |
Sec. 4.1-2 | |
| Wed. 10/22 | Solution of congruences, linear and quadratic. | Sec. 4.1-2 | |
| Fri. 10/24 | Applications: divisiblity tests, tournaments. | Secs. 5.1, 5.3 | |
| Mon. 10/27 | Class cancelled due to fire. | . | |
| Wed. 10/29 | Class cancelled due to fire. | . | |
| Fri. 10/31 | Chinese remainder theorem and extensions. | Sec. 4.3 | |
| Mon. 11/3 | Extensions of the Chinese remainder theorem. | Sec. 4.3 Sec 5.5 | |
| Wed. 11/5 | Check digits. | Sec. 5.5 Sec. 4.5 | |
| Fri. 11/7 | Systems of linear congruences. | Sec. 5.5 Sec. 8.1-2 | |
| Mon. 11/10 | Linear Algebra in Maple, meet in computer lab | Sec. 4.5 | |
| Wed. 11/12 | Questions? Linear Cryptosystems. |
Sec. 8.1, 8.2 | |
| Fri. 11/14 | EXAM | Secs 4.1-3; 4.5; 5.1; 5.5; 13.1-2. | |
| Mon. 11/17 | Fermat's little theorem. | Sec. 6.1 | |
| Wed. 11/19 | Euler's theorem
Exponentiation cryptosystems |
Sec. 6.3 Sec. 8.3 | |
| Fri. 11/21 | RSA Cipher. | 8.4 | |
| Mon. 11/24 | Multiplicative functions:
Euler phi function, number of divisors, sum of divisors |
Sec 7.1,2 | |
| Wed. 11/26 | Order in Z/m , primitive elements | Sec. 9.1 | |
| Mon. 12/1 | Polynomials over Z/m | Sec. 9.2 | |
| Wed. 12/3 | Unique factorization in Z/p [x] . | Secs. 9.2, 9.4 | |
| Fri. 12/5 | Primitive elements in Z/p , index arithmetic. | Sec. 9.4 | |
| Mon. 12/8 | Discrete log cryptosystems. | Sec. 10.2 | |
| Wed. 12/10 | MAPLE. Show me your work. | . | |
| Fri. 12/12 | . | . |