## Number Theory: Math 522, Fall 2003 Schedule

### Schedule

I will try to keep to this schedule, but will update it as needed.

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 . .