# 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 2024): by appointment or
T 9:30-10:00ORCH 3009N/AN/A
Th 9:30-11:00ORCH 3009 and on Zoom694 6667 3745914585
F 11:45-13:00at PIMS and on Zoom676 1308 4912139267

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

1. Jones and Jones, Elementary Number Theory, Springer.
2. Rosen, Elementary Number Theory and its applications, Addison Wesley (5th or 6th edition recommended).
3. Freitas and Gherga, Math 312 notes.
4. Rivest, Shamir and Adelman, A method for obtaining digital signatures and public-key cryptosystems, Comm. ACM 21 no. 2 (1978), 120–126.)

## Midterms

1. Test Information (updated 24/1/2021)
2. Solutions to Test 1.
3. 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.
4. 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.
5. 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.

## 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 MÃ¶bius 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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