Syllabus for Numbers (Math 199 CHP). Fall, 2006.

Professor: Scott Ahlgren
Email: ahlgren@math.uiuc.edu
Phone: 244-1738
Office: 302 Altgeld Hall

Course Description: The natural numbers 1, 2, 3, 4, 5, ....... have fascinated humankind since the beginning of recorded history. Number Theory is the study of the profound and subtle relationships between these numbers. Number Theory is known as the ``Queen of Mathematics,'' as it is one of the most beautiful areas in all of mathematics. The subject is famous for vast numbers of elegant problems which are very simple to state (for example, how many prime numbers are there?) Some of these have simple solutions which have been known for thousands of years, while others have frustrated the attempts of the most brilliant thinkers for generations. Very recently, there have been very important practical applications of Number Theory (for example, in cryptography). This course will provide a hands-on introduction to this subject. There will be short lectures to introduce important concepts, and students will spend much of class time actively engaged in mathematics (this includes experimenting, formulating hypotheses, and proving these hypotheses). Emphasis will be placed on thinking in a way which is simultaneously creative, clear, elegant, and logical. This course is suitable for anyone with an interest in mathematics. There is no formal mathematical prerequisite. There are two prerequisites. The first is an intellectual interest in math and a willingness to engage new ideas. The second is NOT currently taking Math 347, and not having completed any math course at the 300 level.

Text. A friendly introduction to number theory, by Joseph Silverman.

Webpage. This syllabus and other course material and announcements, can be found on the course webpage: http://www.math.uiuc.edu/~ahlgren/math199/math199.html

Classes. Monday, Wednesday, 1:30-2:50, 186 Lincoln Hall (new location).

Grading. Your course grade will be computed as follows:

Attendance and participation: 20 points.
Problems: 50 points
Project: 30 points
Total: 100 points

Attendance and participation. You will be doing a lot of work during class. Therefore everyone should be at the class meetings. The first two meetings which you miss will not affect your grade. After the first two, each unexcused absence will reduce your score in this category by 5 points.

Problems. You will spend a lot of time thinking about problems, both in and out of class, both alone and with others. The 50 points in this category will be awarded based on your performance and effort on these problems, as reflected by your written solutions (and in some cases by your oral solutions).

Project. Each of you will work on a semester project, perhaps in a small group. I will suggest a number of topics related to the course material. You will prepare a written version and will also give a presentation to the class.

Office hours. For the first several weeks, my regular office hours will be:

Monday: 3-4.
Tuesday: 2-3.
Wednesday: 3-4

These will be adjusted during the semester as necessary; please check the course webpage for details. I am always available by appointment (grab me during class or email me). Or if my door is open stop in and I'll talk with you then or tell you when I can.

Books on reserve. There are many books in the mathematics library on elementary number theory. Feel free to browse (but remember that ``elementary'' can mean different things to different authors). There is also lots of material available on the web (let me know if you find something especially helpful!) If you are interested in a particular topic, please let me know and I will try to make a good suggestion. There are five books on reserve in the math library, which may or may not be helpful or interesting.......

Niven and Zuckerman, Introduction to the theory of numbers. A standard upper level undergraduate text, may be hard to understand depending on your background.
Hardy and Wright, Introduction to the theory of numbers. Another standard reference, probably more difficult to read than the last text.
Andrews, Number theory. An undergraduate text at a lower level than the last two.
Guy, Unsolved problems in number theory. As the name suggests, a long list of unsolved problems, many of which are easy to state!
Burn, Pathway into number theory. Lots of problems with solutions worked out.

Final note. I am always open to productive suggestions about the course. Please feel free to contact me with any suggestions or concerns.

Best wishes for a productive and enjoyable semester!