Math 580, Fall 2023

Elementary Number Theory -- Mathematics 580

Frank Thorne - Fall 2023

University of South Carolina

Instructor: Frank Thorne, thorne [at] math [dot] sc [dot] edu

Office Hours: Tue 10:30-12:00 and Wed 3:30-5:00, in LeConte 447.

Course objectives/learning outcomes:

Successful students will:

Master concepts which are foundational in number theory: congruences, Diophantine equations, and the like.
Understand various properties of the integers, what they mean, where they come from, and why they are important. To that end you will learn about different systems of numbers: The integers, the rationals, the "Dudley numbers", the p-adic integers, finite fields, Gaussian integers, and the quaternions. These will be developed for their own interest, and you will also see stunning applications to classical problems involving the ordinary integers.
Put some elements of "recreational math" on a firm footing. Do you know that if you want to test an integer for divisibility by 3, you can add the digits and test that for divisibility by 3? We will prove it.
Learn more about the history of mathematics. Number theory is rivalled only by geometry in the depth of its history, and it has attracted the attention of the most brilliant mathematicians who have ever lived. Very significant results were proved as early as 300 BCE and as recently as this May. We will see both.
Learn how to think in terms of algebraic structures. This is a great class to take for anyone interested in Math 546, or who is taking it at the same time.
Study some of the jewels of mathematics. Some of the most beautiful proofs in all of mathematics concern number theory, and they illustrate ways of thinking that apply everywhere in mathematics, and indeed everywhere in rigorous thinking of any sort. We will see Fermat's infinite descent, Euclid's parameterization of pythagorean triples, Gauss's quadratic reciprocity law, and more.
Each of these ideas can be taken very far: see here, here, and here. (worry not: these topics will not be on the exam.)

In addition, students will also:

Thoroughly understand what definitions and theorems are. The student will be able to precisely state definitions and theorems and understand how they are applied.
Practice writing proofs. It is expected that the student will have some, but not a lot, of experience writing proofs. The student will gain more practice.
Practice good mathematical writing. In mathematics, as indeed in everything else, it is important not only to be correct but to explain yourself clearly and as simply as possible.

Warning. You should expect 5-8 hours of homework a week in this class, which is more than most other instructors assign; in my experience there is no other way to learn the material. Your consistent effort will certainly lead to improved understanding, and it will almost certainly lead to you earning high grades.

Text : Dudley, Elementary Number Theory, buy it here or at the bookstore.

This introductory textbook is a little bit old-fashioned, but it is delightful, it is fun to read, it is geared towards beginners, and unlike some other books you don't need to remortgage your house to buy it.

Lectures : MWF 9:40-10:30, LeConte 348.

The class will mix lecture and classroom discussion. If you have Covid, the flu, etc. please contact me and when possible I will livestream lecture. Note that any in-class discussions

Microsoft Teams :

We will use Microsoft Teams for announcements, discussion, and (if anyone gets sick) livestreaming.

As part of your participation grade, you are required to participate with at least one mathematical comment, question, or answer to someone else's question. Logistical questions (e.g. "when will we get the homework back?") are allowed but don't count towards this requirement.

In class participation is also required. Participation is an integral part of the course, and it is part of your grade. Please plan on active participation!

Midterm Exams : There will be two two-hour take-home, closed-book, pledged midterms. After they are handed out you will have at least a 72-hour period in which to finish them. Dates will be announced later in the term.

Final Project : You will be asked to work on a final project -- consisting of some sort of mathematical investigation, a writeup of your results, and a class presentation. Presentations will be made during the scheduled final exam period. Group work is encouraged, but you are free to work on your own if you prefer.

More details will be announced.

LaTeX : You are encouraged, but not required, to write up your final projects using software called LaTeX. Compared to something like Word, there is more of a learning curve, but it's easier to write math in once you get the hang of it. It is what every professional mathematician uses to write their papers.

Various software packages package LaTeX together with a text editor -- see for example TeXShop for Macs or MiKTeX for Windows, Mac, or Linux.

Here is a sample LaTeX source file, which compiles to this. To get started, download your choice of LaTeX software, and start off by compiling the .tex file. Make sure it matches the PDF linked above. Then, start editing the source, and see what happens.

To create your own file, it is recommended that you copy this .tex file, remove the content, and add your own content. Unless you want to change the look and feel, you don't need to change any of the stuff at the top.

To create presentations with LaTeX you can use Beamer. Here is a sample file, which produces this. The image files are here, here, and here. To recreate the presentation, download the TeX file and all the image files to the same directory, and then use whatever LaTeX compiler you were using for the paper.

There will not be a final exam.

Homework :

Warning. I assign a lot of homework.

The homework is intended to take 5-8 hours a week. That is a lot. Please count on making a consistent effort to do well in this class! Starting the night before is a bad idea.

If homework takes you more than 10 hours on any given week, then that is more than I intended; please let me know.

There might be a bonus problem on some of the homeworks, good for extra credit.

Grading :

Please note. You will be graded both on correctness and on quality of exposition. Indeed, a major focus of Math 580 is the ability to communicate mathematical ideas clearly. The standard is that someone who doesn't know the answer should be able to easily follow your work. In particular, please write in complete English sentences and draw clear diagrams where appropriate. Any work that is confusing, ambiguous, or poorly explained will not receive full credit.

Grading scale: You are guaranteed at least the following: A = 88+, B+ = 83+, B = 75+, C+ = 70+, C = 60+, D = 50+.

Please note that my grading scale is more generous than the usual 10-point scale. However, I am a (slightly) stricter grader than most. This is intended to balance out.

Grade component % of grade

Homework: 40%

Two take-home midterm exams 15% x 2

Participation 15%

Final project: 15%

Contacting me : Please contact me if you have any questions about the course, about my expectations, about my lectures, about the homeworks, about the reading, or about anything else. The syllabus is demanding and it is my job to help you succeed. The best ways to get help are to come to office hours (no appointment necessary) or to e-mail me (during the week, I will almost always reply within 24 hours). If neither of these work for you then please e-mail me to set up an appointment.

Make-up policy :

In case of illness or emergency, please let me know immediately; alternative arrangements will be made. Beyond this, late homework will be accepted once per student, up to one week late, with no excuse needed.

Academic honesty and attendance are expected of all students.

Calculators will not be needed or allowed.

Accommodations : If you require special accommodations, please talk to the Student Disability Resource Center as soon as possible. If is your responsibility to advise me of any needed accomodations a week in advance.

Important. If you have difficulty seeing the board or hearing the lectures, or a related problem, let me know ASAP, and I will do something about it. Similarly if anyone is engaged in harassing or bullying behavior.

It is very important to me that everyone feel comfortable in class and have the opportunity to succeed.

Schedule and homeworks :

To be announced, and posted either here or on Microsoft Teams.

Grade component	% of grade
Homework:	40%
Two take-home midterm exams	15% x 2
Participation	15%
Final project:	15%