Math 115B, Number Theory
Winter 2019
MWF 12:10-1PM, 147 Olson
ANNOUNCEMENT: Our final is on Friday, March 22, 1-3PM. It will be in our usual classroom and will cover everything from the beginning of the course till the end, with no intended bias towards the first or second half of the course. Please refer to the detailed plan of our lectures below for specifics of what was covered.
You are not allowed to use your book, calculator, phone, or anything other than writing utensils and two sides of a regular sheet of paper of handwritten notes. This cheat sheet can contain anything you'd like.
To help you prepare, both Austin (our TA) and I are having extra office hours the week of the final. Austin's office hours will be Tuesday, 1-3PM. My office hours will be Wednesday, 12-2PM and Thursday, 2:30-3:30PM. There will also be a review session on Tuesday, 3-4 PM in 147 Olson.
Here are some practice problems to help you study. Here are the solutions. In case you want them, here are the solutions to the practice MIDTERM.. Note that solutions to homeworks and the midterm are also available on Canvas.
Finally, don't forget that if you want to complete the extra credit project, you should turn it in by the last day of class.
Welcome to the course webpage for Math 115B. Here you will find some general info about the course. This is also the place to look for homework assignments, occasional course notes, and other things that might interest you.
Textbook Exams Homework Assignments Material by Day Various Notes Back to Main Page
Course Syllabus
Professor: Elena Fuchs
Office: MSB 3109
Email: efuchs at math dot ucdavis dot edu
Office hours: M 11AM-12PM, W 1:10PM-2:10PM.
TA: Austin Tran
TA office: 2204 MSB
TA office hour: Tuesdays 1-2PM
Textbook and Prerequisites:
The textbook we will be using is "Elementary Number Theory" by K. Rosen, 6th Edition.
The most important prerequisite for this course is Math 115A. I will assume that you are comfortable with the material covered in the 115A syllabus, aside from the section on cryptography which will not be necessary for this course.
Homeworks, Exams, and Grading:
Your grade for the course is determined as follows:
- 15% for homework,
- 35% for the midterm,
- 50% for the final.
There will be no make up exams.
Homework (along with occasional supplementary notes) will be posted here every Wednesday. The homework will be collected on Wednesdays in class. No late homework will be accepted, but one homework score will be dropped.
Our midterm will be in class on Friday, February 15. More details about it will be announced in class and posted here closer to the date.
Our final exam code is W. It will take place on Friday, March 22, 1:00-3:00PM.
Basic Plan:
As far as the book goes, we will strive to cover parts of Chapter 7, 9, 11, 12, and 13. For a detailed syllabus as suggested by the department of mathematics, please click here.
The goal of this course is to expose students to various beautiful number theoretic questions and their answers (when known). We will scratch the surface of some subjects, and go deeper into others, but the hope is to impress upon everyone how multifaceted and fascinating number theory is.
Detailed Plan:
The following is a rough outline of what we will be doing in lecture every day, along with the relevant sections in the book (note that the reading for a given lecture should be taken to mean the relevant section of the mentioned chapter). In reality, we may move faster or slower. It will be updated on a regular basis.
- 1/7: Introduction to the course; arithmetic functions and multiplicativity. Reading: Section 7.2.
- 1/9: Continuing discussion of tau and sigma. Starting perfect numbers and Mersenne primes. Reading: Sections 7.2 and 7.3.
- 1/11: Continuing perfect numbers and Mersenne primes, beginning talking about Moebius function. Reading: Sections 7.3 and 7.4.
- 1/14: Moebius inversion. Reading: Section 7.4.
- 1/16: Moebius inversion continued, convolution of arithmetic functions. Reading: Section 7.4.
- 1/18: Quadratic residues and noonresidues modulo primes, Legendre Symbol. Reading: Section 11.1.
- 1/21: Martin Luther King Jr. Day: no class.
- 1/23: Euler criterion, beginning Gauss's Lemma. Reading: Section 11.1.
- 1/25: Gauss's Lemma, starting qudratic reciprocity. Reading: Section 11.2.
- 1/28: Quadratic reciprocity: some examples, beginning proof (Eisenstein's Lemma). Reading: Section 11.2.
- 1/30: Proof of quadratic reciprocity, beginning quadratic residues modulo non-primes, Jacobi symbol. Reading: Section 11.2, 11.3.
- 2/1: Some more on Jacobi symbol, beginning primitive roots. Reading: Section 11.3, Section 9.1.
- 2/4: Finishing up Jacobi symbols. Primitive roots and order mod m. Reading: Section 9.1.
- 2/6: Primitive roots continued: number of primitive roots mod m, primitive roots modulo primes. Reading: Section 9.2.
- 2/8: Classification of m for which there exist primitive roots modulo m. Reading: Section 9.3.
- 2/11: Fermat's Last Theorem movie
- 2/13: Review for the midterm.
- 2/15: Midterm exam.
- 2/18: President's day, no class.
- 2/20: Diophantine equations: primitive Pythagorean triples. Reading: Section 13.1.
- 2/22: Primitive Pythagorean triples, a bit about Apollonian circle packings. Reading: Section 13.1 and, if you'd like, the MAA article on Apollonian packings below in the supplementary notes section.
- 2/25: Finishing Apollonian packings. Sums of squares: sums of two squares. Reading: Section 13.3.
- 2/27: Sums of three and four squares, and beyond. Beginning Pell's equation. Reading: Section 13.3, 13.4.
- 3/1: Pell's equation continued. Reading: Section 13.4.
- 3/4: Pell's equation, continued fractions, rational approximation. Reading: Sections 13.4, 12.2.
- 3/6: Finite continued fractions continued: convergents. Reading: Section 12.2.
- 3/8: Infinite continued fractions: existence of limit and how to produce an infinite continued fraction for an irrational number. Reading: Section 12.3.
- 3/11: Infinite continued fractions continued; quality of approximation using continued fractions. Reading: Section 12.3.
- 3/13: Periodic continued fractions. Reading: Section 12.4.
- 3/15: Wrapping up periodic continued fractions, touching on Pell's equation. Reading: Section 12.4 and 13.4.
Some supplementary notes:
Homework assignments: