Math 312: Introduction to Number Theory
Winter Term, 2021
Lior Silberman
General Information
- Office: MATX 1112, 604-827-3031
- Email: "lior" (at) Math.UBC.CA (please include the course number in
the subject line, if applicable)
- Office hours (Winter 2025):
Time | Location | Zoom Meeting ID | Zoom Password |
---|
TTh 12:30-13:00 | ESB 4127 and on Zoom | 684 2674 1683 | 101671 |
F 10:30-11:30 | My office and on Zoom | | |
This is an introductory course in number theory, intended for
math majors. The book by Jones and Jones is available for free
download through the UBC library (you need to be on campus or loggen on to the
VPN for that). That said, any book titled "elementary number theory" or the
like would be good. You can also look at the notes by Freitas and Gherga.
References
- Jones and Jones, Elementary Number Theory, Springer.
- Rosen, Elementary Number Theory and its applications, Addison Wesley (5th or 6th edition recommended).
- Freitas and Gherga, Math 312 notes.
- Rivest, Shamir and Adelman, A method for obtaining digital signatures and public-key cryptosystems, Comm. ACM 21 no. 2 (1978), 120–126.)
Midterms
- Test Information
(updated 24/1/2021)
- Solutions to Test 1.
- Test 2 will held on Wednesday, February 10, and be broadly similar to
Test 1.
- It will cover material up to the discussion of linear congruences on
Friday, February 5, including practice Problem Set 3.
- Here's a sketch of a few solutions to problem 2c from my office hours after the exam.
The key point is that the usual laws of arithmetic apply to modular
arithmetic, so everything we learned previously about arithmetic still
applies -- including row reduction, substitution, and other methods of
solving equations. Only tricky point is that before dividing we need
to find an inverse -- if it exists.
- Solutions to Test 2.
- Test 3 will held on Wednesday, March 10, and be broadly similar to Test 1.
It will cover material up to Euler's Theorem
and primality testing on Friday, February 26,
including practice Problem Set 4.
- Test 4 will held on Wednesday, March 24, and be broadly similar to Test 1.
(subject to revision) It will cover material up to the lectures on RSA
on Wednesday, March 17,
including practice Problem Set 5.
Homework
- ComPAIR
- Read the General instructions.
- ComPAIR FAQ.
- We expect typeset solutions; two options include installing
LyX to your computer or using
Overleaf online. Here are
some notes on LaTeX
from a past MATH 220 instructor, which include links to further
information.
source files below.
- To download the solutions you need to be logged on to Canvas.
- LaTeX macro file.
- ComPAIR Set 1:
PDF,
LyX,
TeX
(submission due 20/1, comparisons due 24/1).
Solutions.
- ComPAIR Set 2:
PDF,
LyX,
TeX
(submission due 3/2, comparisons due 7/2).
Solutions.
- ComPAIR Set 3:
PDF,
LyX,
TeX
(typo in problem 1(b) fixed) (submission due 3/3, comparisons due 7/3).
Solutions.
- ComPAIR Set 4:
PDF,
LyX,
TeX
(questions (b),(d),(f) clarified) (submission due 17/3, comparisons due 21/3).
Solutions.
- ComPAIR Set 5:
PDF,
LyX,
TeX
(submission due 7/4, comparisons due 11/4).
Solutions.
- Practice problem sets (not for submission!)
- Problem Set 1: PDF,
Solutions.
- Problem Set 2: PDF,
Solutions.
- Problem Set 3: PDF,
Solutions.
- Problem Set 4: PDF,
Solutions.
- Problem Set 5: PDF,
Solutions.
- Problem Set 6: PDF,
Solutions.
- More Practice Problem Sets and Solutions by Freitas and Gherga.
Lecture-by-Lecture information
Week |
Date |
Material |
Reading |
Scan |
Notes |
Jones^2 |
Rosen |
1 |
M 11/1 |
The Integers: Induction |
§1.1 |
§1.3, §1.5 |
Scan |
Slides |
W 13/1 |
Divisibility |
§1.1 |
§1.3, §1.5 |
Scan |
|
F 15/1 |
The GCD; Euclid's Algorithm |
§1.2 |
§3.3, §3.4 |
Scan |
|
2 |
M 18/1 |
(continued) |
|
|
Scan |
|
W 20/1 |
primes |
§2.1 |
§3.1 |
Scan |
CP1 due |
F 22/1 |
Unique factorization |
§2.2 |
§3.2, §3.5 |
Scan |
|
3 |
M 25/1 |
Diophantine equations |
§1.5 |
§3.7 |
Scan |
|
W 27/1 |
Test 1 |
|
|
|
Info |
F 29/1 |
(continued) |
|
|
Scan |
|
4 |
M 1/2 |
Congruence |
§3.1 |
§4.1 |
Scan |
|
W 3/2 |
Divisibility tests |
|
§5.1, §5.5 |
Scan |
CP2 due |
F 5/2 |
Linear Congruences |
§3.2 |
§4.2 |
Scan |
|
5 |
M 8/2 |
The CRT |
§3.3 |
§4.3 |
Scan |
|
W 10/2 |
Test 2 |
|
|
|
|
F 12/2 |
(continued) |
|
|
Scan |
|
6 |
M 22/2 |
Wilson's Theorem |
§4.1 |
§6.1 |
Scan |
|
W 24/2 |
Fermat's Little Theorem |
§4.2 |
§6.2 |
Scan |
|
F 26/2 |
Euler's Theorem and Pseudoprimes |
§§5.1-2 |
§6.3 |
Scan |
|
7 |
M 1/3 |
Arithmetic Functions |
§8.1 |
§7.1, §7.2 |
Scan |
|
W 3/3 |
(continued) |
|
|
Scan |
CP3 due |
F 5/3 |
Möbius Inversion; Mersenne Primes |
§8.3 |
§7.4, § 7.3 |
Scan |
|
8 |
M 8/3 |
Character & block ciphers |
Wiki:
1,
2,
|
§8.1 |
Scan |
|
W 10/3 |
Test 3 |
|
|
|
|
F 12/3 |
(continued) |
|
|
Scan |
|
9 |
M 15/3 |
RSA |
Wiki |
§8.4, §8.6 |
Scan |
|
W 17/3 |
(continued) |
|
|
Scan |
CP4 due |
F 19/3 |
Primitive Roots |
§6.2, §6.3 |
§9.1, §9.2 |
Scan |
|
10 |
M 22/3 |
(continued) |
|
|
Scan |
|
W 24/3 |
Test 4 |
|
|
|
|
F 26/3 |
Existence mod p |
|
|
Scan |
|
11 |
M 29/3 |
Quadratic residues |
§§7.1-3 |
§9.4, §10.2, §11.1 |
Scan |
|
W 31/3 |
(continued) |
|
|
Scan |
|
W 7/4 |
Quadratic reciprocity |
§7.4 |
§11.1, §11.2 |
Scan |
CP5 due |
F 9/4 |
(continued) |
|
|
Scan |
|
12 |
M 12/4 |
The Gaussian Integers |
|
|
Scan |
|
W 14/4 |
Elliptic Curves |
|
|
Scan |
|
|
W 21/4 |
Final Exam: 8:30am-11am |
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