| Day | Topics | Preparation | |
|---|---|---|---|
| Mon. 8/30 | Introduction. Numbers: integer, rational, algebraic, real and complex. |
Sec. 1.1 | |
| Wed. 9/1 | Induction. Sequences. | Sec. 1.2 | |
| Fri. 9/2 | Induction proofs. | Sec. 1.2 | |
| Wed. 9/8 | Recursion, Fibonacci numbers. | Sec. 1.3 | |
| Fri. 9/10 | Divisiblity, greatest integer function. | Sec. 1.4 | |
| Mon. 9/13 | base r representation of an integer. | Sec. 2.1 | |
| Wed. 9/15 | Base r conversions and arithmetic, computational complexity. | Secs. 2.1-3 | |
| Fri. 9/17 | Some exercises. | . | |
| Mon. 9/20 | Prime numbers. | Sec. 3.1 | |
| Wed. 9/22 | Prime number conjectures. | Sec. 3.1 | |
| Fri. 9/24 | Using Maple: See First worksheet | Sec. 3.1-3 | |
| Mon. 9/27 | Greatest common divisor. Euclidean algorithm. |
Sec. 3.2,3 | |
| Wed. 9/29 | Extended Euclidean algorithm.
Gcd of several ints. Least common multiple. |
Sec. 3.3 | |
| Fri. 10/1 | Unique factorization | Sec. 3.4 | |
| Mon. 10/4 | Linear Diophantine equations. | 3.6. | |
| Wed. 10/6 | Linear Diophantine equations in 3 variables. Unique Factorization: Application to primes in arithmetic progression |
Sec. 3.4, 6 | |
| Fri. 10/8 | Maple | . | |
| Mon. 10/11 | Unique Factorization: Application to roots of polynomials.
Pythagorean triples. |
3.4, 13.1 | |
| Wed. 10/13 | EXAM. | Secs. 1.1-4; 2.1-2; 3.1-4, 6; | |
| Fri. 10/15 | Nonlinear Diophantine equations:
Pythagorean triples, Fermat's last theorem |
Sec. 13.1-2 | |
| Mon. 10/18 | Modular arithmetic. | Sec. 4.1 | |
| Wed. 10/20 | Solution of congruences, linear and quadratic.
Units and zero-divisors in Z/m . |
Sec. 4.1-2 | |
| Fri. 10/22 | MAPLE: meet in GMCS 405: Modular arithmetic in MAPLE. |
Sec. 4.1-2 | |
| Mon. 10/25 | Chinese remainder theorem. | Sec. 4.3 | |
| Wed. 10/27 | Applications: divisiblity tests, tournaments. | Secs. 5.1, 5.3 | |
| Fri. 10/29 | Extensions of the Chinese remainder theorem. | Sec. 4.3 Sec 5.5 | |
| Mon. 11/1 | Divisiblility tests Check digits. |
Sec. 5.1 Sec. 5.5 | |
| Wed. 11/3 | Check digits and review. | Sec. 5.5 | |
| Fri. 11/5 | Systems of linear congruences. | Sec. 4.5 | |
| Mon. 11/8 | Linear algebra modulo m . | Sec. 4.5 | |
| Wed. 11/10 | Linear cryptosystems. | Sec. 8.1, 8.2 | |
| Fri. 11/12 | MAPLE: meet in GMCS 405 | Sec. 4.1-2 | |
| Mon. 11/15 | Fermat's little theorem.
Exponentiation cryptosystems |
Sec. 6.1 Sec. 8.3 | |
| Wed. 11/17 | EXAM. | Secs 4.1-3; 4.5; 5.1; 5.5; 11.1; 13.1-2. | |
| Fri. 11/19 | Euler's theorem | Sec. 6.3 | |
| Mon. 11/22 | RSA Cipher. | 8.4 | |
| Wed. 11/24 | Multiplicative functions:
Euler phi function, number of divisors, sum of divisors |
Sec 7.1,2 | |
| Mon. 11/29 | Polynomials over Z/m | Sec. 9.2 | |
| Wed. 12/1 | Unique factorization in Z/p [x] . | Secs. 9.2, 9.4 | |
| Fri. 12/3 | Primitive elements in Z/p , index arithmetic. | Sec. 9.4 | |
| Mon. 12/6 | Discrete log cryptosystems. | Sec. 10.2 | |
| Wed. 12/8 | MAPLE. Show me your work. | . | |
| Fri. 12/10 | The ABC conjecture. | . |