Math 312: Introduction to Number Theory

Winter Term, 2021
Lior Silberman

General Information

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.
  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.

Homework

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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