Number theory is, simply put, the study of properties of the integers. It is
an extremely old subject, and in fact one of the oldest branches of
mathematics. Naturally, this means that number theory covers many topics, and
we will have to choose only a few of these topics to cover during this course.
|Meeting Time:||MWF 10 - 10:50
|Location: ||Morton 203|
Office: Jones 130
Office Hours: Mon 3:30 - 4:30, Wed 2:30 - 3:30, Thurs 9:30 - 11 and 3:30 - 5 (also by appt).
|Textbook:||An Introduction to the Theory of Numbers, Fifth Edition,
by Ivan Niven, Herbert S. Zuckerman, and Hugh L. Montgomery|
| Grade |
| Midterm - 30%, Homework - 35%, Final Exam - 35%. The
grading scale will be the standard 10 percentage point scale, so that a final
score of 93 or higher is an A, 90-92 is an A-, 87-89 is a B+, 83-86 is a
B, 80-82 is a B-, 77-79 is a C+, 73-76 is a C, 70-72 is a C-, etc. |
|Attendance & Lecture Policy:||You are expected to attend all lectures. Attendance
is crucial in order to succeed in the course. Any legitimate absence for an exam
must be discussed with me prior to the test date. Please do not email or text
during lecture, and keep all cell phones/hand held devices/laptops put away
during lecture (especially during exams).
|Prerequisites: || Math 214 - Foundations of Mathematics. Also, as this is a 400-level class, it will be
assumed that you are comfortable with mathematical proofs beyond the level
of Math 214. In particular, I will lecture with the assumption that
everyone has had either Math 307 or Math 311. ||
We will cover topics in Chapters 1-5 in the text (but not every section in
every chapter), and if time allows, some topics from Chapters 6 and 7. We will
begin by reviewing some topics which should be familiar with everyone, which
include divisibility, prime numbers, and congruences. We will apply some of
these concepts to learn to solve several types of Diophantine equations, which
are equations where we are only interested in integer solutions. This basic
question, of finding integer solutions to equations, is a huge motivating
factor for many branches of number theory. We will also cover the
famous quadratic reciprocity law, another old theorem which has
far-reaching generalizations. We will also discuss some
basic number-theoretic functions, which are functions defined on the
integers based on (mostly) divisibility properties of that integer. Hopefully,
we will have some time at the end of the semester to talk about continued
fractions and rational approximation to irrational numbers.
Dates & Course Announcements:
Midterm and Final Exams:
The midterm will consist of a take-home portion, as well as an in-class
portion. The final exam will have, at least, a take-home portion, although the
specific structure of the final will be determined and discussed as the
semester moves forward.
Exam Calendar (Tentative):
||Week of Oct 19
||Mon, Dec 10
- All relevant announcements will be listed here. Check back frequently (don't forget
to refresh your browser) for updates.
- Important Dates and Class Holidays:
- Fri, Sep 7: ADD/DROP DEADLINE
- Sat, Oct 13 - Tues, Oct 16: NO CLASS (Fall Break)
- Fri, Oct 26: WITHDRAW DEADLINE
- Wed, Nov 21 - Sun, Nov 25: NO CLASS (Thanksgiving Break)
- Mon, Dec 10, 2:00 - 5:00 - FINAL EXAM
- (8/29) I will determine my weekly scheduled office hours after the first
week or so of classes. During this first (short) week of classes, I will be
available in my office for any questions on Wed, Aug 29, 2:30-4:30, and
Thurs, Aug 30, 10-11:30 and 3-4:30.
- (9/5) My office hours for today and tomorrow will be: Wed Sep 5, 2-3:30,
Thurs Sep 6, 9-11 and 3:30-5.
- (9/10) I have posted, at the top of this page, my regular weekly office
hours. They are M 2:30-3:30, W 3:30-4:30, and Th 9:30-11, 3:30-5.
As is the case with all of mathematics, the only way to learn it well is to do
as many problems as possible. So, homework problems will be a very important
part of the course, and there will be homework assigned almost every week (other than
the week of the midterm). Completion of all homework problems is required, and your
grade on a homework assignment will be based on completeness, as well as on the
details of the solutions of the problems graded. In particular, I will not
necessarily grade every homework problem assigned, but part of your score for an
assignment will be for the completion of all problems. Individual homework
assignment should be completed by the student alone, although I am always open
for questions, either in office hours or by email.
For each homework problem assigned, a complete solution with each step
explained should be written up clearly and neatly. Be sure to completely explain your steps and reasoning
for calculations as well as for proofs. This is especially important in
enumerative problems, as there can be many ways to arrive at the same answer,
and what I am interested in is your thought process.
Homework is due at the beginning of
class on the due date of the assignment. Late homework will be marked off 20%
for every day late (10% off if it is turned in after the beginning of class,
but it is in my hands on the day it is due). Homework turned in after class on the due date is
considered one day late, and the next weekday after that 2 days late, and so on. Everything
is easier, of course, if you turn in the homework on time!
The textbook has many challenging problems. On each assignment, I will give
some extra problems (marked by *), which are not required. Work on them if you want. These
are for your own challenge.
|1|| Sec. 1.2 (pgs 17-20) #1(a,b), 6, 11, 16, 24, 34, 35, 43
Extra: 50*, 51*
|Fri, Sep 7
|2|| Sec. 1.3 (pgs 29-33) #9, 10, 12, 16, 26, 44
Extra: 39*, 40*
|Fri, Sep 14
|3|| Sec. 2.1 (pgs 56-59) #6, 12, 14, 23, 30, 47
|Fri, Sep 21
|4|| Sec. 2.1 #38, 51, Sec. 2.2 #2, 5(e), Sec. 2.3 #2, 4
Extra: Sec. 2.1 #56*
|Fri, Sep 28
|5|| Sec. 2.3 #14, 15, 26, 27, Sec. 2.5 #1, 2
Extra: Sec. 2.3 #46*
|Fri, Oct 5
|6|| Sec. 5.1 #4(b,f), 6, 10, Sec. 5.3 #4, 7, 8
Extra: 5.3 #15*
|Fri, Oct 12
|7|| Sec. 5.4 #1, 4, 6, 12, Sec. 5.2 #1
||Fri, Oct 19
|8|| Sec. 4.1 #8, Sec. 4.2 #12, 16, 20, Sec. 4.3 #5, 7, 8
||Mon, Nov 12
|9|| Sec. 2.6 #2, 7, 10, Sec. 3.1 #4, 9, 10, 13
||Mon, Nov 19
|10|| Sec. 3.2 #6, 7, 8, 14, Sec. 3.3 #2, 5, 6, 7
||Fri, Nov 30
Math Major Writing Requirement (Math 300):
If you are a math major, and you would like to complete your major writing
requirement through a writing assignment in this class, please let me know in
the first week of class. This writing assignment will not count towards your
grade in this class, but will rather just serve as your Major Writing
Requirement. If you decide to do this, you must write your paper on a topic in
Number Theory approved by me, and you must keep to a schedule of turning in
drafts that is set at the beginning of the semester in order to get credit.
You are also encouraged to sign up for Math 300 during this semester if you
fulfill the writing requirement through this class.