Math 3527 (Number Theory 1), Spring 2024
| Course Information | |||||
|---|---|---|---|---|---|
| Instructor | Class Times | Office Hours | |||
| Evan Dummit edummit at northeastern dot edu 571 Lake Hall |
(Sec 1) MWR 1:35pm-2:40pm, 033 Snell Library (Sec 2) MWR 10:30am-11:35am, 409 Robinson |
M 12:10pm-1:10pm MR 3:05pm-4:05pm Online via Zoom |
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| For detailed information about the course, please consult the 3527 Course Syllabus. (Note: any information given in class or on this webpage supersedes the written syllabus.) | |||||
| We will use Piazza for any course-related discussion: here is the 3527 Piazza page. | |||||
| Problem Session | Session Leader(s) | ||||
| Mondays 11:30am-1:30pm, Dodge 111 | Zoey Yelsky | ||||
| Tuesdays 3pm-5pm, Snell Library 115 | Toby Busick-Warner and Molly Sager | ||||
| Fridays 1pm-3pm, Behrakis 204 | Connor Anderson | ||||
| All homework assignments will be posted on this webpage (see below). Homework assignments will be collected via Gradescope on Canvas. Please submit scans of your homework pages by 11:59pm Eastern on the due date. Late assignments may be penalized at the grader's discretion. |
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| The instructor will write lecture notes for the course (see below) in lieu of an official textbook as the semester progresses. To a moderate degree, the course will follow the presentation in J. Silverman's "A Friendly Introduction to Number Theory", but we will also add substantial additional material, and it will not be necessary to purchase the textbook for this course. | |||||
| There will be final exam review sessions from 2pm-4pm on Tuesday April 23rd and Wednesday April 24th in Snell Library 123. The final exam will be held on Thursday April 25th from 3:30pm-5:30pm in West Village F 020. | |||||
| Homework Assignments | |||||||
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| Some Tips on Problem Solving are available as suggestions for the written assignments. | |||||||
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Homework #1, due Tue Jan 16th. (solutions)
Homework #2, due Tue Jan 23nd. (solutions) Homework #3, due Tue Jan 30th. (solutions) Homework #4, due Tue Feb 6th. (solutions) Homework #5, due Tue Feb 13th. (solutions), (Mathematica code) Homework #6, due Fri Mar 1st. (solutions), (Mathematica code) Homework #7, due Tue Mar 19th. (solutions), (Mathematica code) Homework #8, due Tue Mar 26th. (solutions) Homework #9, due Tue Apr 2nd. (solutions) Homework #10, due Fri Apr 12th. (solutions) Homework #11, due Fri Apr 19th. (solutions) | |||||||
| Homework assignments are to be submitted to Gradescope via the course's Canvas page. To submit an assignment, navigate to "Assignments" and select the appropriate homework assignment. Then attach scans of each page of your assignment (or a pdf) and click Submit. Please submit the pages in order and verify that all pages are included and uploaded correctly. You may resubmit as many times as you like. Assignments are due at 11:59pm eastern time. Late submissions, messy submissions, or otherwise unreadable submissions will be penalized at the grader's discretion. Ensure you mark all problem pages when submitting to Gradescope; failure to do so may result in point penalties. | |||||||
| Handouts / Lecture Notes | |||
|---|---|---|---|
| Handout | Topics | ||
| Chapter 1: The Integers (18pp, v3.60, updated 1/17) (updates in v3.60: added divisor formulas to 1.3, improved exposition of divisibility and gcd/lcm formulas in terms of prime factorizations) |
1.1 ~ The Integers, Axiomatically 1.2 ~ Divisibility and the Euclidean Algorithm 1.3 ~ Primes and Unique Factorization 1.4 ~ Rings and Other Number Systems |
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| Chapter 2: Modular Arithmetic (20pp, v4.00, posted 1/23) | 2.1 ~ Modular Congruences and The Integers Modulo m 2.2 ~ Linear Equations Modulo m and The Chinese Remainder Theorem 2.3 ~ Powers Modulo m: Orders, Fermat's Little Theorem, Wilson's Theorem, Euler's Theorem 2.4 ~ Repeating Decimals |
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| Chapter 3: Cryptography and Related Topics (26pp, v4.00, posted 2/7) | 3.1 ~ Overview of Cryptography 3.2 ~ Rabin Encryption 3.3 ~ RSA Encryption 3.4 ~ Zero-Knowledge Proofs 3.5 ~ Primality and Compositeness Testing 3.6 ~ Factorization Algorithms |
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| Chapter 4: Unique Factorization and Applications (34pp, v4.00, posted 2/23) | 4.1 ~ Integral Domains, Euclidean Domains, and Unique Factorization 4.2 ~ Modular Arithmetic in Euclidean Domains 4.3 ~ Arithmetic in F[x] 4.4 ~ Arithmetic in Z[i] |
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| Chapter 5: Squares and Quadratic Reciprocity (23pp, v4.00, posted 4/3) | 5.1 ~ Quadratic Residues and the Legendre Symbol 5.2 ~ The Law of Quadratic Reciprocity 5.3 ~ The Jacobi Symbol 5.4 ~ Applications of Quadratic Reciprocity 5.5 ~ Generalizations of Quadratic Reciprocity |
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| Exam Information | |||
|---|---|---|---|
| Exam | Date, Time, Location | Topics | Review Material |
| Midterm 1 (Form B), (sols) (Form C), (sols) |
Thu, Feb 22nd In Class |
Homeworks 1-5 Notes 1.1-3.4 |
Review Problems (answers) In-class review Wed Feb 21st |
| Midterm 2 (Form D), (sols) (Form E), (sols) |
Wed, Apr 3rd In Class |
Homeworks 6-9 Notes 3.5-4.4 |
Review Problems (answers) In-class review Mon, Apr 1st |
| Final | Thu, Apr 25th 3:30pm-5:30pm West Village F 020 |
The final is COMPREHENSIVE! Homeworks 1-11 Notes Chapters 1-5 |
Review Problems (answers) Review sessions TBA |
| On all exams, calculators are permitted though generally they are not needed, and you are allowed a 1-page note sheet (8.5in by 11in, both sides) on which you may write or type anything. | |||
| Tips For Success In This Course | |||
|---|---|---|---|
| Attend Lecture | Missing lecture is a bad idea! If for any reason you cannot make it to a class, you should review notes from someone who did attend. You are responsible for all material covered in lecture. | ||
| Read the Lecture Notes (or Textbook) | The lecture notes and the textbook are comprehensive sources of material for the course. The notes are intended as review material, although many students like to read them as preparation before attending the lecture on the corresponding topics. Please note that the electronic notes are not identical to the material covered in class: this is by design, so as to provide you a slightly different perspective on the material. | ||
| Solve Homework Problems | Much of the learning in this course will take place as you solve the homework problems. Like many other activities, problem-solving and proof-writing are things that are learned by doing them, not by hearing someone else tell you about them or reading about them in a book. As such, the homework assignments are an integral part of the course, and are fundamental to learning the material. It is highly recommended that you look over the homework assignments as soon as they are available, and work on them well in advance of the deadline: many problems will take substantial time and effort to solve, and you should expect to spend as much time as you need to finish the assignments. | ||
| Attend Problem Sessions | There are weekly problem sessions run by the course TAs. The goal of the problem sessions are to provide you a location where you can work collaboratively with other students on assignments, and also get TA assistance. | ||
| Attend Office Hours | Office hours are specifically reserved for you to receive individual, one-on-one help from the instructor. Office hours will be the most effective when you have already put in effort to learn the material on your own (including trying to solve the homework problems), and when you come in with a list of specific questions or topics you are struggling with. | ||
| Course Schedule | |||
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| The schedule is subject to change! All sections refer to the course lecture notes. | |||
| Week | Schedule | ||
| Week of Jan 8 (class starts 1/8) |
§1.1.1: The Integers, Axiomatically §1.1.2: Basic Arithmetic §1.1.3: Induction §1.2.1: Divisibility and Division with Remainder §1.2.2: Greatest Common Divisors No homework this week. |
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| Week of Jan 15 (no class 1/15) |
§1.2.2: Greatest Common Divisors §1.2.3: The Euclidean Algorithm §1.3: Primes and Unique Factorization Homework #1 due Tuesday, Jan 16th. |
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| Week of Jan 22 | §1.4.1: Rings and Other Number Systems §1.4.2: Arithmetic in Rings, Units §2.1.1: Modular Congruences §2.1.2: Residue Classes §2.1.3: Modular Arithmetic §2.1.4: Units in Z/mZ Homework #2 due Tuesday, Jan 23rd. |
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| Week of Jan 29 |
§2.1.5: Zero Divisors in Z/mZ §2.2: Linear Equations Modulo m and The Chinese Remainder Theorem §2.3.1: Orders of Elements Modulo m §2.3.2: Fermat's Little Theorem, Wilson's Theorem Homework #3 due Tuesday, Jan 30th. |
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| Week of Feb 5 |
§2.3.3: The Euler φ-function and Euler's Theorem §2.3.4: Primitive Roots §2.4: Repeating Decimals §3.1: Overview of Cryptography Homework #4 due Tuesday, Feb 6th. |
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| Week of Feb 12 | §3.2: Rabin Encryption §3.3: RSA Encryption §3.4: Zero-Knowledge Proofs Homework #5 due Tuesday, Feb 13th. |
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| Week of Feb 19 (no class 2/19) |
Review for Midterm 1. MIDTERM 1 in class on Thu, Feb 22nd No homework this week. |
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| Week of Feb 26 | §3.5: Primality and Compositeness Testing §3.6: Factorization Algorithms §4.1.1: Norms and Z[√D] §4.1.2: Integral Domains and Common Divisors Homework #6 due Friday, Mar 1st. |
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| Spring Break (no classes) from Mar 4 to Mar 8 | |||
| Week of Mar 11 | §4.1.3: Irreducible and Prime Elements §4.1.4: Euclidean Domains and Division Algorithms §4.1.5: Z[i] and F[x] as Euclidean Domains §4.1.6: Unique Factorization in Euclidean Domains §4.2.1: Modular Congruences and Residue Classes No homework this week. |
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| Week of Mar 18 | §4.2.2: Arithmetic in R/rR §4.2.3: Units and Zero Divisors in R/rR §4.2.4: The Chinese Remainder Theorem §4.2.5: Orders, Euler's Theorem, Fermat's Little Theorem §4.3.1: Polynomial Functions, Roots of Polynomials Homework #7 due Tuesday, Mar 19th. |
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| Week of Mar 25 | §4.3.2: Finite Fields §4.3.3: Primitive Roots §4.4.1: Residue Classes in Z[i] §4.4.2: Factorization in Z[i] Homework #8 due Tuesday, Mar 26th. |
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| Week of Apr 1 | Homework #9 due Tuesday, Apr 2nd. Review for Midterm 2. MIDTERM 2 in class on Wed, Apr 3rd §5.1: Quadratic Residues and Legendre Symbols |
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| Week of Apr 8 | §5.2: The Law of Quadratic Reciprocity §5.3: Jacobi Symbols §5.4: Applications of Quadratic Reciprocity Homework #10 due Friday, Apr 12th. |
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| Week of Apr 15 (no class 4/15, classes end 4/17) |
§5.5: Generalizations of Quadratic Reciprocity Review for Final Exam Homework #11 due Friday, Apr 19th. |
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| The FINAL EXAM will be held on Thu Apr 25th from 3:30pm-5:30pm in West Village F 020. | |||