Elementary Number Theory -- Mathematics 580

Frank Thorne - Fall 2013

University of South Carolina

Instructor: Frank Thorne, LeConte 400G, thorne [at] math [dot] sc [dot] edu

Office Hours: Tuesdays, 10:30-11:30 and Wednesdays, 9:30-10:30 and 1:30-2:30.

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.)

• 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 : LeConte 121, TR, 2:50-4:05.

• Exam schedule : In-class midterms will be on Thursday, October 3 and Thursday, November 7.

The final exam will on Wednesday, December 11 at 12:30 p.m.

Here are the previous midterms, with solutions.

• 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 will be at least one bonus problem on each homework, each worth one or two points, up to a maximum score of 11/10 on each week's homework. This is the only way to earn extra credit; please note that bonus problems will not be accepted late.

Graduate credit: If you want graduate credit, you are required to do at least one bonus problem from every other homework (i.e., do one from HW1 or HW2, etc.) Bonus problems on top of that count for extra credit.

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: A = 90+, B+ = 85+, B = 78+, C+ = 73+, C = 66+, D = 52+.

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 Two in-class exams 20% x 2 Final exam: 35% Homework: 25%

Please note: If you come to office hours, please don't be shy about interrupting me! I'm usually in the middle of something, but it can wait.

• Make-up policy :

If you have a legitimate conflict with any of the exams it is your responsibility to inform me at least a week before the exam. Otherwise makeup exams will be given only in case of documented emergency.

Late homework will be accepted once per student, up to a week late; after that, no late homework please except in case of emergency.

Academic honesty and attendance are expected of all students.

Calculators will not be needed or allowed.

• Schedule and homeworks :

(Note: Daily topics are listed only for some days, and when in the future are subject to change.)

• (R) August 22: Hanc marginis exiguitas non caperet.

• (T) August 27: Disibility and the Euclidean algorithm (Chapter 1)

• (R) August 29: Induction and prime factorizations (Chapter 2)

• (T) September 3: Uniqueness of prime factorizations (Chapter 2)

• (R) September 5: Linear Diophantine equations (Chapter 3)

• (T) September 10: Congruences (Chapter 4)

• Homework 1, due Tuesday, September 2.

• Homework 2, due Thursday, September 11.

• Homework 3, due Friday, September 26.

• Homework 4, due Friday, October 25.

• Homework 5, due Monday, November 11.

• Homework 6, due Thursday, November 21.

• Homework 7.