MATH-UA 248, Fall 2019: Theory of Numbers
Syllabus
- Instructor: Robert Young (ryoung@cims.nyu.edu)
- Office: WWH 601
- Office hours: Mondays, 10-12, WWH 601
- Lectures: (check Albert for location) MW 3:30-4:45
- Recitations: (check Albert for location) F 3:30-4:45 (starting
September 13)
- TA: Donghyun Seo
- TA Office hours: Tuesdays, 11:30-12:30, WWH 811
- Textbook: Burton, Elementary Number Theory (6th or 7th edition
recommended; numbering may change in earlier editions)
Grading scheme
Assignments | 20% |
Quizzes | 15% |
Midterm | 25% |
Final exam | 40% |
Exam dates
- Quiz 1: Friday, October 4th, in recitation
- Midterm: Friday, October 25th, in recitation
- Quiz 2: Friday, November 15th, in recitation
- Quiz 3: Friday, December 6th, in recitation
- Final: Wednesday, December 18th, 4-5:50 PM
Policies
Assignments
Assignments will usually be given on Wednesdays and due at the
beginning of class the next Wednesday. You may also submit your
assignment by putting it under the door of my office by 6PM Tuesday
evening. If you do so, please let me know by email.
Collaboration is encouraged, but each student must write up their
own solutions in their own words. If you work closely with someone
else, please identify them on your assignment (e.g., "I worked with
__________").
Late assignments will not be accepted except in the case of an
emergency. At the end of the semester, your two lowest assignment
grades will be dropped from your average. This is meant to
accommodate non-emergency absences, so try not to use this unless
you have to.
Solving problems is important! Doing exercises and understanding
the assignments is the best way to master the material.
Computers
Laptop computers can be helpful for taking notes
and doing research, but please be aware of the distraction they may
cause to people behind you. If you choose to bring a computer to class, please follow these guidelines:
- Use your computer only for class-related material.
- Avoid bright colors. Lower the brightness of the screen as much as you can and consider setting your screen to black-and-white (On Windows, press CTRL+Win+C. On MacOS, go to the Accessibility control panel, select the Display tab, and toggle the "Use grayscale" box).
- Avoid animation and video.
How to do well in this class
- Come to class and recitation!
- Ask questions: Everyone gets confused or stuck somewhere. The
important thing is that you try to get unstuck. If there's
something you don't understand, ask me questions in class, after
class, at office hours, or by email. Ask the TA questions in
recitation or their office hours. Ask your classmates questions:
explaining something to someone else is one of the best ways to
learn.
- Solve problems, read actively and explore! One of the goals
of this course is to learn how to solve problems and prove theorems.
The only way to do that is by solving problems and proving theorems.
So, as you read the textbook or review your notes, read actively by:
- Testing out theorems on your own examples
- Filling in any gaps (like "We leave the proof as an exercise for the reader")
- Trying to prove a theorem yourself before looking at the proof
- Identifying the techniques used in proofs
- Reading the exercises and coming up with ideas on how to solve them
- Asking yourself questions like:
- What would happen if I changed one of the hypotheses of this theorem?
- What's a simple example of this definition?
- What's a complex example of this definition?
Problem sets
- Problem Set 1 (due Wednesday, Sept. 18)
- Problem Set 2 (due Wednesday, Sept. 25)
- Problem Set 3 (due Wednesday, Oct. 2)
- Problem Set 4 (due Wednesday, Oct. 9)
- Problem Set 5 (due Wednesday, Oct. 16)
- Problem Set 6 (due Wednesday, Oct. 23)
- Problem Set 7 (due Wednesday, Oct. 30)
- Problem Set 8 (due Wednesday, Nov. 6)
- Problem Set 9 (due Wednesday, Nov. 13)
- Problem Set 10 (due Wednesday, Nov. 20)
- Problem Set 11 (due Wednesday, Dec. 4)
- Problem Set 12 (due Wednesday, Dec. 11)
- Study guide
Course outline
09/04 |
Introduction, Well-Ordering Principle |
1.1 |
09/09, 09/11 |
Divisibility, GCD |
2.2, 2.3 |
09/16, 09/18 |
Euclidean Algorithm, \(ax+by=c\) |
2.4, 2.5 |
09/23, 09/25 |
Primes and prime factorization |
3.1-3.3 |
09/30, 10/02 |
Congruences and modular arithmetic |
4.1-4.3 |
10/07, 10/09 |
Linear congruences and systems of congruences |
4.4 |
10/15, 10/16 |
Fermat's Thm., Wilson's Thm. |
5.2-5.3 |
10/21, 10/23 |
Number-theoretic functions, perfect numbers |
6.1, 11.2 |
10/28, 10/30 |
Euler's theorem and the \(\phi\) function |
7.2-7.3 |
11/04, 11/06 |
The \(\phi\) function, public-key cryptography |
7.4, 10.1 |
11/11, 11/13 |
Order |
8.1-8.2 |
11/18, 11/20 |
Primitive roots |
8.1-8.2 Statistics of primitive roots |
11/25 |
Quadratic residues |
9.1 |
12/02, 12/04 |
Quadratic reciprocity |
9.2-9.3 Table of Legendre symbols |
12/09, 12/11 |
Continued fractions and rational approximation |
15.2-15.3 |