# 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.

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:
1. Divisibility and the Euclidean Algorithm
• Euclidean Algorithm
• Prime numbers
• Fundamental Theorem of Arithmetic
• Greatest common divisor and least common multiple
2. Congrences
• Fermat's little theorem
• Wilson's theorem
• Euler's φ function
• Euler-Fermat theorem
• Systems of congruences
• Chinese Remainder Theorem
3. Diophantine equations
• Linear Diophantine equations
• Pythagorean triples
• Pell equations
• Continued fractions
• Legendre symbol
• Jacobi symbol
• Euclid-Mullin sequences
5. 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.

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.