# MATH 4400/6400: Elementary number theory

## Spring 2013

### MWF 11:15 AM-12:05 PM, Boyd Graduate Studies Building, Room 303Instructor: Paul Pollack

 < URL: http://www.math.uga.edu/~pollack/4400s13/ >This page is an online copy of the syllabus. The web page will occasionally be updated with additional course materials.

Current assignments/other course material

Course summary

Number theory concerns itself with the properties of the positive integers 1, 2, 3, ..., and related objects that arise in their study. It is tempting to assume that something as simple as the counting numbers must be rather well understood. But this is far from the case! Indeed, number theory owes much of its charm to the wealth of easily-stated problems -- possible to explain to middle-school age children -- which have so far defied solution, despite hundreds of years of effort by great mathematicians.

This course is designed to introduce you to the foundational results, with particular attention paid to some the pathbreaking accomplishments of the 18th and 19th centuries. We will start with the basic theory (factorization into primes) before turning to Gauss's theory of congruences. The theory of congruences culminates with a proof of Gauss's Aureum Theorema (golden theorem), the law of quadratic reciprocity.

More advanced topics will be discussed later in the course. Possibilities include theorems about sums of squares, results on arithmetic functions, and results on the distribution of prime numbers.

Textbook (required)

Elementary number theory
by Jones and Jones (Amazon link)

Number theory, more than some other areas of mathematics, is largely an experimental science. It is recommended (but not required) that you download software that allows you to play with very large numbers. This is a great way to get an intuitive handle on the results we discuss. Commercial packages of this sort include Maple and Mathematica, but there are also free (even open source!) options, such as gp/PARI.

Office hours

 Instructor: Paul Pollack E-mail: pollack at math dot uga dot edu Office: 318 Boyd Graduate Studies Building Office hours: Monday, 2:30--3:25 PM Wednesday, 2:30--3:30 PM Thursday, 2--3:00 PM

Exam dates

There are three in-class midterm exams as well as a final exam.

• Midterm #1: Friday, Feb. 8
• Midterm #2: Friday, March. 8
• Midterm #3: Friday, April 12
• Final exam: Monday, May 6, 12:00 -- 3:00 pm (usual classroom)
No make-up exams will be given. The final exam is cumulative.

Attendance/Homework/Exam Policies

• Each midterm: 16% (total of 48%)
• Homework: 22%
• Final exam: 30%

HW assignments will contain problems required of all students, as well as more challenging problems required only for MATH 6400 students. Undergraduate participants may do these additional problems for extra credit.

You are expected to participate in class. In particular, attendance in this course is required. More than four unexcused absences may result in you receiving a WF. Keep me posted whenever you have a conflict that requires you to miss class and this should not be an issue.

All exams are in-class, closed book and closed notes.

Homework will be collected in class, about once each week. Late homework will not be accepted. (If you have a need to turn in HW early, that can be arranged.) Your lowest HW score will be dropped at the end of the term.

On homework, collaboration is allowed and in fact is very much encouraged. Mathematics wouldn't be nearly as much fun if we couldn't talk about it with other people! However, copying (from a textbook or another student) and web searches are not allowed, and you must write your own final solutions independently. Keep in mind that by entering UGA, you have already agreed to abide by the UGA Honor code described in detail at <URL: http://honesty.uga.edu/ahpd/culture_honesty.htm>.

In practice, what this means that you may discuss homework problems and their solutions with your classmates, but you may not turn in a solution unless you understand it yourself. A reasonable rule of thumb is that you should be able to explain your solutions verbally to me (in all their gory detail) if requested to do so.

The withdrawal policy for this course is that if you withdraw within a week of the date the first midterm is handed back, you automatically qualify for a WP (assuming you are eligible by UGA's standards). After that point, it is at the discretion of the instructor (i.e., me) who will take your performance in the class to-date into account.

Special accommodations

Students with disabilities who may require special accommodations should talk to me as soon as possible. Appropriate documentation concerning disabilities may be required. For further information, please visit the Disabilities Resource Center page at <URL: http://drc.uga.edu/>.

Disclaimer

This course syllabus provides a general plan for the course; deviations may be necessary.