# Math 412 - Introduction to Number Theory - Fall 2012

## General Information:

Meeting Time: MWF 10 - 10:50 Morton 203 Ryan Vinroot 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). An Introduction to the Theory of Numbers, Fifth Edition, by Ivan Niven, Herbert S. Zuckerman, and Hugh L. Montgomery 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. 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). 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.
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 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):

 Midterm Week of Oct 19 TBA TBA Final Exam Mon, Dec 10 2-5 TBA
• All relevant announcements will be listed here. Check back frequently (don't forget to refresh your browser) for updates.
• Important Dates and Class Holidays:
• 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.

## 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.

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.

 Assignment Problems Due Date 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 Extra: 55* 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.