Math 412 - Introduction to Number Theory - Fall 2013

General Information:

Meeting Time: TTh 3:30 - 4:50 Small Physics Lab 233 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). An Introduction to the Theory of Numbers, Fifth Edition, by Ivan Niven, Herbert S. Zuckerman, and Hugh L. Montgomery 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. 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. 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 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:
• 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 Thurs, Sep 5 2 Sec. 1.3 (pgs. 29-33), #9, 10, 12, 26, 33, 44 Thurs, Sep 12 3 Sec. 2.1 (pgs. 56-58), #6, 14, 18, 23, 29, 34, 47 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) Thurs, Sep 26 5 Sec. 2.3 (pgs. 71-73), #2, 4, 14, 18, 26, 35, Sec. 2.5 (pg. 86), #2 Thurs, Oct 3 6 Sec. 5.1 #4(a,e), 10, Sec. 5.3 #7, 8, Sec. 5.4 #1, 4, 12 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. Thurs, Nov 7 8 Sec. 3.1 #4, 10, 13, Sec. 3.2 #4, 6, 7, 9 Thurs, Nov 14 9 Sec. 3.2 #10, 13, 14, Sec. 3.3 #2, 5, 6, Sec. 3.4 #7 Thurs, Nov 21 10 Sec. 3.4 #1, 3, 8, Sec. 3.5 #1, 3, 5, 9 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.