Winter Term, 2021

Lior Silberman
- 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 2022): by appointment or
Time Location Zoom Meeting ID Zoom Password MF 11:30-12:30 My office and Zoom 691 7826 7667 761818 Wed 21:30-22:30 Zoom only 682 2985 1665 155350

- Classes: MWF 9:00-9:50 on Zoom
- Syllabus.
- (Rough) lecture notes.
- Canvas.

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.

- 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.)

- 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.

- 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!)
- More Practice Problem Sets and Solutions by Freitas and Gherga.

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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