Math 507: Advanced Number Theory

Spring 2008

Course Description:
Number Theory is one of the oldest disciplines in mathematics. Every major result treated in this course has a simple statement involving terminology that can be understood by a high school mathematics student. Each of the proofs uses no more mathematical information than a science student is exposed to in a thorough calculus course and in a thorough discrete mathematics course. The main feature making this a graduate level course is the degree of ingenuity with which these elementary methods are used to derive serious mathematical results.  The primary intangible objectives of this course are to:
1.  increase the sophistication of the student's skill in using elementary materials in clever ways to create mathematics,
2. enrich the student's mathematical culture by attention to historical context,
3. prime the student's professional etiquette by giving attention to proper written presentation, exposition and attribution of mathematical work,
4. inspire the student's taste for searching for aesthetically pleasing mathematics, 

The theoretical high points of the course will include some of the following:
a) A proof of Chebyshev's prime number theorem.
b) Chebyshev's proof of Bertrand's postulate.
c) A proof of Brun's Twin Prime theorem.
d) A proof of Landau's theorem on infinite series involving the Mobius function.
e) A heuristic justification and analysis of a subexponential factoring algorithm.
f) A proof of Lorentz's theorem on summation in modular arithmetic.

Prerequisites:
 MATH 306.

Text:

The course will not strictly follow a specific text book. However, Dr. Victor Shoup's text

 A computational introduction to Number Theory and Algebra (Version 1)

will be a reference for the course and is freely available from Dr. Shoup's website. Be sure to also download the errata and supplement to the book.

Evaluated work in the course.

There will be homework (200), a test (100) and a final examination(200).

Calendar:
Sept. 1:    Labor Day. No classes
Sept. 8:    Last day to add, change to audit, drop without a "W"
Oct.  16:  Midterm Exam.
Nov. 24:  Thanksgiving break until Nov. 28.
Dec. 16:   3:30 pm - 5:30 pm. Final Examination.