Syllabus

Math 627, Summer Session II Number Theory 

Prerequisites: mastery of some upper-division undergraduate course:
(this could be anything from complex variables to linear algebra, just so
you can read and write proofs. Any graduate student who has survived the
whole year would surely qualify).

Course content:

Topics from chapters 1,2,4,6 and 7 from Niven, Zuckerman and Montgomery:
5th Edition: An Introduction to the theory of numbers, Wiley, ISBN
0471625469

Ch 1: background: divisibility, primes, unique factorization, binomial
theorem.

Ch 2: Congruences, Chinese Remainder theorem, RSA, primitive roots, 
[project topic: solving polynomial equations in modular arithmetic,
resource: Henri Cohen's book: A course in computational algebraic number theory, Springer (not a required text for the course).

Ch4: Greatest integer function, summatory function, M"obius function,
M"obius inversion formula [project topic: Lucas pseudoprime test]

Ch6. Farey fractions and irrational numbers. The array 

0/1,1/1

0/1,1/2,1/1

0/1,1/3,1/2,2/3,1/1

0/1,1/4,1/3,1/2,2/3,3/4,1/1

0/1,1/5,1/4,1/3,2/5,1/2,3/5,2/3,3/4,4/5,1/1

and its properties. Rational approximation. Blichfeldt's principle and Minkowski's theorem. 

Ch 7. Continued fractions. (The equation x^2-61*y^2=1 has solutions in positive integers. How would you know that, how would you find them, why do none show up on cursory search? ) Uniform distribution of sequences and the connection to continued fractions. [Project topics: random walks and continued fractions, counting questions, etc.]. 

The first three weeks will be devoted to pushing through the basics and developiog the methods of the discipline. After that, we will work on a variety of small-group course projects. The final few days will be given over to presentation, discussion and analysis of these projects. 

Lab: Almost all of the material has a rich computational component and can be illustrated and verified computationally. The algorithms can be implemented and checked as well. We will be using the computer algebra system Maple, which offers open-ended decimal precision and large integer arithmetic, field operations such as modular arithmetic, and so on. For instance, Maple will return 8 when given the command 1/5 mod 13. This means that 8 is the integer n between 1 and 12 so that 5*n-1 is divisible by 13. We'll see how it is that Maple can solve huge instances of this problem in a short time, yet cannot factor integers of comparable size. The class will meet at least once a week in a room equipped with terminals from which we can run Maple. 

Grades: Lab 1/4, Midterm 1/4, project 1/4, Final 1/4.