## Number Theory: Math 522, Fall 2004 Schedule

### Schedule

A best approximation.

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. .