Math 4400-001 (Howe), Spring 2022: Introduction to Number Theory
Tu/Th 9:10am - 10:30am in M LI 1725
Homework and quizzes to be submitted using Gradescope. Your cumulative class grade can be found on Canvas (this may lag slightly behind what is on Gradescope).
Office hours
Summer (LA) - Monday 3:00-4:30pm, Friday 11:00am-12:30pm. These are online - click here for access instructions for online office hours.
Sean (Instructor) - Monday 2:00pm-3:00pm in JWB 323 or by appointment (online or in person for appointments, please email me to schedule at sean dot howe at utah dot edu).
Syllabus and book
This is a flipped class. Please read the syllabus carefully for expectations.
Note:
The syllabus was updated on 1/13 - modified in-person office hour policy, added zoom access for class sessions, added that exams are open book and open notes, and added a section on course objectives.
Savin - Numbers, Groups, and Cryptography. Online PDF (there are different versions if you google; please use this one).
Exams
Video lectures and handouts
Timeline for updates:
- The weekly handout and the video for Tuesday's class will be posted by 9:10pm on Sunday; printed copies of the handout will be available in class.
- The video for Thursday's class will be posted by 9:10pm on Tuesday.
- Quizzes will be made available on Gradescope shortly after the corresponding video is posted and are due 30 minutes before the corresponding class (e.g., the quiz for the video attached to a Thursday session is due by 8:40am on Thursday)
- I will try to post a PDF of the whiteboard from a class session later that same day (starting 1/20).
- Week 15 (4/26) - Review
Note -- the final exam is Thursday 4/28, 8:00am-10:00am; see info in exams section above.
Tuesday 4/26 - Review session · Whiteboard
Please fill out the course feedback survey -- we will end class early on Tuesday to provide time for this
- Week 14 (4/19, 4/21) - Elliptic curves
The homeworks and quizzes for this week are not required and will count as bonus points.
Week 14 problem sheet
Complementary reading: Savin - Chapters 12 and 13
Thursday 4/21 - Elliptic curve cryptography · Video (14m53s) · Lecture notes · Whiteboard
Tuesday 4/19 - Elliptic curves · Video (24m11s) · Lecture notes · Whiteboard
- Week 13 (4/12, 4/14) - Shape numbers, Pell's equation, and rational approximation
Week 13 problem sheet
Complementary reading: Savin - Chapter 9
Thursday 4/14 - Rational approximation and Pell's equations · Video (29m10s) · Lecture notes · Whiteboard
Tuesday 4/12 - Shape numbers and Pell's equations · Video (22m36s) · Lecture notes · Whiteboard
- Week 12 (4/5, 4/7) - Gaussian integers and sums of two squares
Week 12 problem sheet
Complementary reading: Savin - Chapter 8
Thursday 4/7 - Gaussian primes and unique factorization · Video (39m03s) · Lecture notes · Whiteboard (note: there was an issue with the whiteboard export; the last two pages are from Thursday, but the first 8 are from Tuesday. Much of what we did on Thursday, e.g. examples of Euclidean algorithm in the Gaussian integers, are missing from the board, so you should consult your notes as well or attend office hours to ask any questions.)
Tuesday 4/5 - Sums of two squares and descent · Video (36m32s) · Lecture notes · Whiteboard
- Week 11 (3/29, 3/31) - Fermat primes and Mersenne primes
Week 11 problem sheet
Complementary reading: Savin - Chapter 7
Thursday 3/31 - Mersenne primes · Video (37m52s) · Lecture notes · Whiteboard
Tuesday 3/29 - Fermat primes · Video (32m47s) · Lecture notes · Whiteboard
- Week 10 (3/22, 3/24) - Explaining quadratic reciprocity
Week 10 problem sheet
Complementary reading: Savin - Chapter 6
Thursday 3/24 - Quadratic reciprocity · Video (61m16s) · Lecture notes · Video only; no class meeting and no quiz.
Tuesday 3/22 - A formula for the Legendre symbol · Video (24m50s) · Lecture notes · Whiteboard
- Week 9 (3/15, 3/17) - Squares mod p.
Week 9 problem sheet
Complementary reading: Savin - Chapter 6 (but we will do most of it next week actually)
Thursday 3/17 - The Legendre symbol · Video (28m16s) · Lecture notes · Whiteboard
Tuesday 3/15 - What else can I do? · Video (10m07s) · Lecture notes · Whiteboard
- Week 8 (3/01, 3/03) - Review and midterm.
Thursday 3/03 - Midterm (see info posted in exams section above).
Tuesday 3/01 - In-class review · Whiteboard 1 · Whiteboard 2
- Week 7 (2/22, 2/24) - Primitive roots and some big primes.
Week 7 problem sheet
Complementary reading: Savin - Chapters 4 and 5
Thursday 2/24 - Big primes and perfect numbers · Video (29m58s) · Lecture notes · Whiteboard
Tuesday 2/22 - Primitive roots + discrete logarithm · Video (25m57s) · Lecture notes · Whiteboard
- Week 6 (2/15, 2/17) - Diffie-Hellman and RSA.
Week 6 problem sheet
Complementary reading: Savin - Chapter 10
Thursday 2/17 - RSA · Video (26m08s) · Lecture notes · Whiteboard
Tuesday 2/15 - Shift cipher and Diffie-Hellman · Video (29m52s) · Lecture notes · Whiteboard
- Week 5 (2/8, 2/10) - Roots of polynomials and quadratic numbers
Week 5 problem sheet
Complementary reading: Savin - Chapter 3
Thursday 2/10 - Quadratic numbers · Video (31m56s) · Lecture notes · Whiteboard
Tuesday 2/8 - Roots of polynomials + modular arithmetic gone off the rails · Video (35m29s) · Lecture notes · Whiteboard
- Week 4 (2/1, 2/3) - Rings, groups, and modular arithmetic
Week 4 problem sheet
Complementary reading: Savin - Chapter 2, sections 1 and 4.
Thursday 2/3 - More Lagrange, refined CRT, and fields · Video (33m06s) · Lecture notes · Whiteboard
Tuesday 2/1 - Algebraic structures + Lagrange's theorem · Video (39m44s) · Lecture notes · Whiteboard
- Week 3 (1/25, 1/27) - Modular arithmetic
Week 3 problem sheet
Complementary reading: Savin - Chapter 2, sections 2, 3 and 5.
Thursday 1/27 - Inverses and the Chinese Remainder Theorem (CRT) · Video (32m55s) · Lecture notes · Whiteboard
Tuesday 1/25 - Modular arithmetic · Video (23m35s) (robovoice) · Lecture notes · Whiteboard
- Week 2 (1/18, 1/20) - The Euclidean algorithm
Week 2 problem sheet
Complementary reading: Savin - Chapter 1.
Thursday 1/20 - Continued fractions, the Euclidean algorithm, and uniqueness of prime factorization · Video (32m33s) (robovoice) · Lecture notes · Whiteboard
Tuesday 1/18 - The Euclidean algorithm · Video (39m51s). · Lecture notes
In the Tuesday video there is a mistake in the last line of the gcd(60,22) example; it's corrected in the notes.
- Week 1 (1/11, 1/13) - Wait, what's number theory?
Week 1 problem sheet
Thursday 1/13 - Wait, what's number theory? · Video (31m12s) · Lecture notes
Tuesday 1/11 - First day of class (discussed syllabus + class structure)