General Information:
| Meeting Time: | TTh 3:30 - 4:50 |
| Location: | Small Physics Lab 233
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| Instructor: | Ryan Vinroot
Office: Jones 130
Office Hours: M 10-11 and 4-5, W 10-11 and 3-4:30, Th 10-12 (also by appt).
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| Textbook: | An Introduction to the Theory of Numbers, Fifth Edition,
by Ivan Niven, Herbert S. Zuckerman, and Hugh L. Montgomery |
Grade
Breakdown: | Class Participation - 5%, Midterm - 30%, Homework - 35%, Final Exam - 30%. 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: | It is expected that you attend all
lectures, with exceptions minimized. It is greatly appreciated when you
are on time. Please do your best to stay awake and attentive during
lecture, please do not email or text during lecture, and keep all cell
phones/hand held devices/laptops put away during lecture. While it is
understandable that you may miss a lecture here and there, or be sleepy in
class once in awhile, repeated absences, late arrivals, naps, or general
non-attentiveness will negatively affect your class participation score.
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| 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. |
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Syllabus:
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.
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 be a take-home exam which will be due on the day
of the scheduled final exam.
Exam Calendar (Tentative):
| Midterm |
In class: Tues, Oct 22 |
Take home: Due Tues, Oct 29 |
| Final Exam |
Due Mon, Dec 16, 5 PM |
Turn in exam in Jones 130 |
- 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 6: ADD/DROP DEADLINE
- Sat, Oct 12 - Tues, Oct 15: NO CLASS (Fall Break)
- Fri, Oct 25: WITHDRAW DEADLINE
- Wed, Nov 27 - Sun, Dec 1: NO CLASS (Thanksgiving Break)
- Mon, Dec 16, 2:00 - 5:00 - FINAL EXAM
- (8/29) I will determine my weekly scheduled office hours after the first
week or two of classes. In the meantime, I will be available in my office
this Fri, Aug 30, 10-11, and next Mon, Sept 2, 10-11 and 2:30-4.
- (9/2) This week, I will be available at the following times (apart from
the hours for Mon, Sept 2, listed above): Tues, Sept 3, 9:30-11; Wed, Sept
4, 10-11 and 3-4:30; Thurs, Sept 5, 10:30-12.
- (9/5) My office hours next week will be: Mon, Sept 9, 10-11; Tues, Sept 10, 10-11:30, Wed, Sept 11, 10-11 and
3-4:30; Thurs, Sept 12, 10:30-12.
- (9/9) I had to slighlty shift my Tues, Sept 10 office hours to 10-11:30 AM
(rather than 9:30-11 AM). I apologize for any inconvenience this may
cause.
- (9/13) I have set my weekly office hours for the semester as: Mon 10-11
and 4-5, Wed 10-11 and 3-4:30, Thurs 10-12.
- (10/7) My office hours this Wed afternoon (10/9) have to be shifted
earlier, and will be 2:15-3:45 (instead of 3-4:30).
- (10/16) The midterm will have a timed in-class portion, and a take-home
portion. The in-class part will be on Tues, Oct 22, and you will pick up
your take-home when finishing the in-class part. The take-home will be due
the following Tues, Oct 29.
- (10/17) I will have extra office hours on Mon, Oct 21, and will be in my office 10-11 and 2:30-5.
- (11/4) My afternoon office hours today, Nov 4, will be 3-4 instead of 4-5.
- (12/6) I will have your last homework graded and ready for you to pick up
in my office on Monday, Dec 9. You can come pick it up any time during the
hours 12-2 or 3-5 in my office (Jones 130).
- (12/6) During the first week of finals (prior to the due date of our
take-home final) I will be available in my office on the following days and
times: Mon, Dec 9, 12-2 and 3-5; Thurs, Dec 12, 9-11, 12-2, and 3-5; Fri,
Dec 13, 9-11, 12-2, and 3-4:45.
- (12/12) My office hours on Fri, Dec 13, are ending at 4:45 instead of 5.
Homework:
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. You should not, under any
circumstances, attempt to look up solutions or hints to problems online. I
will consider this plagiarism, an honor offense.
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. When you are in doubt
whether you should explain something, then explain it. If you are tempted to
use words like "clearly" or "obviously", then instead explain the statement in
a short sentence.
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 several assignments, I will give
some extra problems (marked by *), which are not required. Work on them if you want. These
are for your own challenge, but only attempt them if you have completed the
rest of the assignment.
| Assignment |
Problems |
Due Date |
| 1 | Sec. 1.2 (pgs. 17-20), #1(b,c), 6, 11, 16, 24, 34, 35, 43
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Thurs, Sep 5 |
| 2 | Sec. 1.3 (pgs. 29-33), #9, 10, 12, 26, 33, 44
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Thurs, Sep 12 |
| 3 | Sec. 2.1 (pgs. 56-58), #6, 14, 18, 23, 29, 34, 47
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Thurs, Sep 19 |
| 4 | Sec. 2.1 (pgs. 57-59), #12, 38, 51, Sec. 2.2 (pgs. 62-63), #2,
3, 5(b,d,e)
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Thurs, Sep 26 |
| 5 | Sec. 2.3 (pgs. 71-73), #2, 4, 14, 18, 26, 35, Sec. 2.5 (pg. 86), #2
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Thurs, Oct 3 |
| 6 | Sec. 5.1 #4(a,e), 10, Sec. 5.3 #7, 8, Sec. 5.4 #1, 4, 12
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Thurs, Oct 10 |
| 7 | Sec. 4.1 #8, Sec. 4.2 #12, 16, 19, Sec. 4.3 #5, 7, 8
*Extra Credit: Give a complete proof of the
"day of the week"
formula at the end of Section 4.1.
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Thurs, Nov 7 |
| 8 | Sec. 3.1 #4, 10, 13, Sec. 3.2 #4, 6, 7, 9
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Thurs, Nov 14 |
| 9 | Sec. 3.2 #10, 13, 14, Sec. 3.3 #2, 5, 6, Sec. 3.4 #7
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Thurs, Nov 21 |
| 10 | Sec. 3.4 #1, 3, 8, Sec. 3.5 #1, 3, 5, 9
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Thurs, Dec 5 |
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.
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