Math 3527 (Number Theory 1), Spring 2023
| Course Information | |||||
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| Instructor | Class Times | Office Hours | |||
| Evan Dummit edummit at northeastern dot edu 571 Lake Hall |
(Sec 1) MWR 1:35pm-2:40pm, 215 Shillman (Sec 2) MWR 10:30am-11:35am, 215 Shillman |
R noon-1:00pm WR 3:00pm-4:00pm 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.) | |||||
| Teaching Assistant | Problem Session | ||||
| Shengnan Huang huang.shengn at northeastern | Tuesday, 12:30pm-3:00pm, Forsyth 236 | ||||
| Nathan Zelesko zelesko.n at northeastern | Friday, 12:30pm-3:00pm, Forsyth 236 | ||||
| All homework assignments will be posted on this webpage (see below). | |||||
| 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. | |||||
| 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 17th. (solutions)
Homework #2, due Tue Jan 24th. (solutions) Homework #3, due Tue Jan 31st. (solutions) Homework #4, due Tue Feb 7th. (solutions) Homework #5, due Tue Feb 14th. (solutions) Homework #6, due Tue Feb 21st. (solutions), Mathematica code Homework #7, due Fri Mar 17th. (solutions), Mathematica code Homework #8, due Fri Mar 24th. (solutions) Homework #9, due Fri Mar 31st. (solutions) Homework #10, due Fri Apr 7th. (solutions) Homework #11, due Fri Apr 21st. (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 (17pp, v3.10, posted 1/1) |
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, v3.50, 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, v2.00, posted 2/14) | 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, v3.00, posted 2/27) | 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 (26pp, v3.50, posted 3/27) | 5.1 ~ Polynomial Congruences and Hensel's Lemma 5.2 ~ Quadratic Residues and the Legendre Symbol 5.3 ~ The Law of Quadratic Reciprocity 5.4 ~ The Jacobi Symbol 5.5 ~ Applications of Quadratic Reciprocity 5.6 ~ Generalizations of Quadratic Reciprocity |
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| Exam Information | |||
|---|---|---|---|
| Exam | Date, Time, Location | Topics | Review Material |
| Midterm 1 (Form A), (sols) (Form B), (sols) |
Mon, Feb 27th In Class |
Homeworks 1-6 Notes 1.1-3.4 |
Review Problems (answers) In-class review Thu Feb 23rd |
| Midterm 2 (Form A), (sols) (Form B), (sols) |
Mon, Apr 10th In Class |
Homeworks 7-10 Notes 4.1-5.1 |
Review Problems (answers) In-class review Thu Apr 6th |
| Final | Thu, Apr 27th 10:30am-12:30pm, Churchill 101 |
The final is COMPREHENSIVE! Homeworks 1-11 Notes Chapters 1-5 |
Review Problems (answers) Review sessions Tue Apr 25th, Wed Apr 26th, 1pm-2pm, Shillman 210 |
| 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 is a 3-hour weekly problem session run by the course TAs. This session runs from 9am-noon on Fridays in 544 Nightingale Hall, ahead of the time the homework is due. The goal of the problem session is 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. | ||
| Use Tutoring Services | Please consult the Northeastern Tutoring page for additional information on peer tutoring. Tutoring appointments can be made via MyNEU (which will provide lists of available appointments for the tutors for your specific classes); walk-in tutoring is very limited and tends to be unavailable near exam dates. | ||
| Course Schedule | |||
|---|---|---|---|
| The schedule is subject to change! All sections refer to the course lecture notes. | |||
| Week | Schedule | ||
| Week of Jan 9 (class starts 1/9) |
§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 §1.2.3: The Euclidean Algorithm No homework this week. |
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| Week of Jan 16 (no class 1/16) |
§1.2.3: The Euclidean Algorithm §1.3: Primes and Unique Factorization §1.4.1: Rings and Other Number Systems §1.4.2: Arithmetic in Rings, Units Homework #1 due Tuesday, Jan 17th. |
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| Week of Jan 23 | §2.1.1: Modular Congruences §2.1.2: Residue Classes §2.1.3: Modular Arithmetic §2.1.4: Units in Z/mZ §2.1.5: Zero Divisors in Z/mZ Homework #2 due Tuesday, Jan 24th. |
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| Week of Jan 30 |
§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 §2.3.3: The Euler φ-function and Euler's Theorem Homework #3 due Tuesday, Feb 1st. |
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| Week of Feb 6 |
§2.3.4: Primitive Roots §2.4: Repeating Decimals §3.1: Overview of Cryptography §3.2: Rabin Encryption Homework #4 due Tuesday, Feb 8th. |
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| Week of Feb 13 | §3.3: RSA Encryption §3.4: Zero-Knowledge Proofs §3.5: Primality and Compositeness Testing Homework #5 due Tuesday, Feb 15th. |
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| Week of Feb 20 (no class 2/20) |
§3.6: Factorization Algorithms Review for Midterm 1. Homework #6 due Tuesday, Feb 22nd. |
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| Week of Feb 27 | MIDTERM 1 in class on Mon, Feb 27th §4.1.1: Norms and Z[√D] §4.1.2: Integral Domains and Common Divisors §4.1.3: Irreducible and Prime Elements §4.1.4: Euclidean Domains and Division Algorithms No homework this week. |
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| Spring Break (no classes) from Mar 6 to Mar 10 | |||
| Week of Mar 13 | §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 §4.2.2: Arithmetic in R/rR Homework #7 due Friday, Mar 17th. |
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| Week of Mar 20 | §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 §4.3.2: Finite Fields Homework #8 due Friday, Mar 24th. |
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| Week of Mar 27 | §4.3.3: Primitive Roots §4.4.1: Residue Classes in Z[i] §4.4.2: Factorization in Z[i] §5.1: Polynomial Congruences and Hensel's Lemma Homework #9 due Friday, Mar 31st. |
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| Week of Apr 3 | §5.2: Quadratic Residues and Legendre Symbols §5.3: The Law of Quadratic Reciprocity Review for Midterm 2. Homework #10 due Friday, Apr 7th. |
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| Week of Apr 10 | MIDTERM 2 in class on Mon, Apr 10th §5.4: Jacobi Symbols §5.5: Applications of Quadratic Reciprocity §5.6: Generalizations of Quadratic Reciprocity No homework this week. |
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| Week of Apr 17 (no class 4/17, classes end 4/19) |
Review for Final Exam Homework #11 due Friday, Apr 21st. |
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| The FINAL EXAM will be held Thu Apr 27th from 10:30am-12:30pm in 101 Churchill | |||