Math 329: Number Theory

Spring 2018

Instructor: Enrique Treviño
Lectures: MWF 11:00 - 11:50 pm in Young Hall 207
Office Hours: MWF between 10 to 10:50 am. You can also arrange a meeting by appointment.
Office: Young Hall 105
Email:
Phone Ext.: #6187

Announcements

Lecture Notes. I include some notes on some things we've covered and some things we'll soon cover.

Self-Explaining Booklet, is a short file with tips that can help you read proofs better.

The homework assignments can be found below:
Homework Assignments

Textbook

There will be no textbook.

However, the book Elementary Theory of Numbers by LeVeque will be in reserve in the library as an extra source (besides the lecture notes).

Class Content

The goal of the class is to give an introduction to the theory of numbers. A tentative list of topics is below:

Divisibility and the Euclidean Algorithm

Euclidean Algorithm
Prime numbers
Fundamental Theorem of Arithmetic
Greatest common divisor and least common multiple

Congrences

Fermat's little theorem
Wilson's theorem
Euler's φ function
Euler-Fermat theorem
Systems of congruences
Chinese Remainder Theorem

Diophantine equations

Linear Diophantine equations
Pythagorean triples
Pell equations
Continued fractions

Quadratic Reciprocity

Quadratic residues
Legendre symbol
Jacobi symbol
Euclid-Mullin sequences

Arithmetic functions

Multiplicative functions
Möbius function
The floor (or greatest integer) function

List of other possible topics that can be covered depending on the pace of the class:

Primitive roots of unity
Cyclotomic polynomials
Bertrand's postulate
Hardy-Ramanujan theorem on the number of distinct prime divisors
Algebraic integers
The arithmetic of complex numbers

Student Learning Outcomes

Practice how to solve problems.
Learn how to write solutions.
Learn to write number theory proofs.
Learn the topics covered in class (described above).
Learn to solve some computational problems.

Homework

There will be ten homework assignments. Each homework assignment will include 5 "easy" problems, 4 "medium" problem and one "hard" problem. The easy problems will usually test computational aspects of the theory, the medium ones will test whether the student can do proofs, the hard ones will test whether the student can go beyond what was covered. The easy problems are worth 1 point, the medium ones 2 points and the hard one 3 points.

The student will submit up to 5 problems for each assignment. The student will get quick feedback on the assignment from the instructor and then the student can revise up to two problems to submit together with the next homework assignment. The revisions for the final homework assignment can be turned in at the scheduled time for the final exam, which is Saturday May 5 between 1:30pm and 4:30pm.

When working on a homework assignment, students cannot work with each other, cannot search up answers on the internet (outside this website) and cannot ask for help from sources other than the instructor. The only available resources are the instructor, lecture notes, anything posted on this website, and the book "Elementary Theory of Numbers" by LeVeque.

Grading

The course grade will be based entirely on the points the student receives from the homework assignments. Here's a tentative breakdown of the points needed for a particular grade:

For an A, the student will need 90 points or more.
For a B, the student will need 70 points or more.
For a C, the student will need 50 points or more.
For a D, the student will need 30 points or more.

Special Office Hours

Besides the office hours listed in the course description, I will reserve two hours a week (Tuesdays from 9:45am to 11:45am) for the purpose of helping students with homework assignments. I expect you to use these office hours often, since success in the class is highly dependent on working hard on the homework assignments.

Videos and Questionnaire

The week of April 9 to 13, I will be attending the European Girls Mathematical Olympiad. There will be no class sessions. As a substitution, I will assign you to watch several videos related to Number Theory. As part of the week, you will have to answer a questionnaire that will have questions related to the videos you will watch.

Videos to watch:

Questionnaire.

Class Participation

I expect you to come to every class, pay attention, and to ask questions when you have questions. Some classes will have in-class activities, when these occur, you should be working on the task at hand. I expect you to refrain from using smartphones in class.

Academic Honesty Policy

The student is expected to be honest with their work. I will penalize any student that uses unapproved resources for the homework assignments. I encourage students to be familiar with the college's Academic Honesty Policy to see the possible sanctions for academic dishonesty.

Accommodations Statement

If you believe that you need accommodations for a disability, please consult with The Learning and Teaching Center. Since accommodations may require early planning and are not retroactive, please contact the center as soon as possible. For details about the services for students with disabilities and the accomodations process, visit http://www.lakeforest.edu/academics/resources/disability/.

You are also welcome to contact me privately to discuss your academic needs. However, all disability-related accommodations must be arranged through Teryn Robinson at the Learning and Teaching Center.

Last modified on April 7, 2018.