Math 4527 (Number Theory 2), Spring 2021
| Course Information | |||||
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| Instructor | Class Times | Office Hours | |||
| Evan Dummit edummit at northeastern dot edu |
MWR 10:30am-11:35am Online, via Zoom |
WR 3:00pm-4:15pm R 12:15pm-1:15pm Online, via Zoom |
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| We will use Piazza for any course-related discussion: here is the Piazza page. Links to the lecture recordings are hosted there. | |||||
| For detailed information about the course, please consult the 4527 Course Syllabus. Note: any information given in class or on this webpage supersedes the written syllabus. | |||||
| All homework assignments will be posted on this webpage (see below). Homework assignments will be collected via 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. The material we will cover in the course is drawn from a variety of sources, and will also depend somewhat on student interest; as such, it is difficult to give recommendations for a textbook. | |||||
| Homework Assignments + Exams | |||||||
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| Some Tips on Problem Solving are available as suggestions for the homework assignments. | |||||||
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Homework #1, due Thu Jan 28th. (solutions)
Homework #2, due Thu Feb 4th. (solutions) Homework #3, due Thu Feb 11th. (solutions) Homework #4, due Thu Feb 18th. (solutions) Homework #5, due Thu Feb 25th. (Some Mathematica code that may be helpful.) (solutions) Homework #6, due Thu Mar 4th. (Some Mathematica code that may be helpful.) (solutions) Homework #7, due Thu Mar 11th. (Some Mathematica code that may be helpful.) (solutions) Homework #8, due Thu Mar 18th. (solutions) Homework #9, due Fri Mar 26th. (solutions) Homework #10, due Thu Apr 1st. (solutions) Homework #11, due Thu Apr 8th. (solutions) Homework #12, due Thu Apr 15th. (solutions) Homework #13, due Fri Apr 23rd. For those not excused from the final exam, it will be emailed to you and due on Fri Apr 30th. | |||||||
| Lecture Slides | |||
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| These are the slides used during the lectures. | |||
| Date | Material | ||
| Wed, Jan 20th Thu, Jan 21st |
Lecture 1: Overview + Pythagorean Triples (Notes 6.1.3)
Lecture 2: Linear Diophantine Equations (Notes 6.1.1-6.1.3) |
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| Mon, Jan 25th Wed, Jan 27th Thu, Jan 28th |
Lecture 3: Farey Sequences (Notes 6.2.1) [Updated]
Lecture 4: Continued Fractions, Part 1 (Notes 6.2.2-6.2.3) Lecture 5: Continued Fractions, Part 2 (Notes 6.2.3-6.2.4) |
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| Mon, Feb 1st Wed, Feb 3rd Thu, Feb 4th |
Lecture 6: Rational Approximation and Irrationality (Notes 6.2.5-6.2.6)
Lecture 7: Transcendence + Pell's Equation, Part 1 (Notes 6.2.6-6.3.1) Lecture 8: Pell's Equation, Part 2 (Notes 6.3.1-6.3.2) |
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| Mon, Feb 8th Wed, Feb 10th Thu, Feb 11th |
Lecture 9: The Super Magic Box (Notes 6.3.3)
Lecture 10: Miscellaneous Diophantine Equations, Part 1 (Notes 6.4) Lecture 11: Miscellaneous Diophantine Equations, Part 2 (Notes 6.4) |
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| Mon, Feb 15th Wed, Feb 17th Thu, Feb 18th |
No classes; university holiday.
Lecture 12: Cubic Curves and the Addition Law (Notes 7.1.1-7.1.2) Lecture 13: Elliptic Curves Modulo p (Notes 7.1.2-7.1.3) |
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| Mon, Feb 22nd Wed, Feb 24th Thu, Feb 25th |
Lecture 14: Orders + Elliptic Curve Factorization (Notes 7.1.3-7.2.1)
Lecture 15: Elliptic Curve Factorization + Cryptography (Notes 7.2.1-7.2.2) Lecture 16: Elliptic Curve Cryptography, Part 2 (Notes 7.2.3-7.2.4) |
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| Mon, Mar 1st Wed, Mar 3rd Thu, Mar 5th |
Lecture 17: Cryptography Part 3 + Torsion Points (Notes 7.2.3-7.3.1)
Lecture 18: Elliptic Curves over C + Rational Points (Notes 7.3.1-7.3.2) Lecture 19: Integral Points + The Congruent Number Problem (Notes 7.3.3-7.3.4) |
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| Mon, Mar 8th Wed, Mar 10th Thu, Mar 11th |
Lecture 20: Rings and Ideals (Notes 8.1.1-8.1.2)
Lecture 21: Ideals and Quotient Rings + Maximal Ideals (Notes 8.1.2-8.1.3) Lecture 22: Prime Ideals + Arithmetic in Domains (Notes 8.1.3-8.1.4) |
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| Mon, Mar 15th Wed, Mar 17th Thu, Mar 18th |
Lecture 23: Quadratic Integer Rings + Euclidean Domains (Notes 8.1.5-8.1.6)
Lecture 24: PIDs and UFDs (Notes 8.1.7-8.1.8) Lecture 25: The Chinese Remainder Theorem + Factoring in OD (Notes 8.1.9-8.2.1) |
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| Mon, Mar 22nd Wed, Mar 24th Thu, Mar 25th |
Lecture 26: Factorization of Ideals in OD (Notes 8.2.1-8.2.2)
Classes cancelled ("CARE Day") Lecture 27: Computing Ideal Factorizations in OD (Notes 8.2.2-8.2.3) |
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| Mon, Mar 29th Wed, Mar 31st Thu, Apr 1st |
Lecture 28: Factoring in Z[i] and O√-2 (Notes 8.3.1-8.3.2)
Lecture 29: Factoring in O√-3, Diophantine Equations (Notes 8.3.2-8.3.3) Lecture 30: Fermat's Equation x3+y3=z3, Cubic Reciprocity (Notes 8.3.4) |
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| Mon, Apr 5th Wed, Apr 7th Thu, Apr 8th |
Lecture 31: Cubic and Quartic Reciprocity (Notes 8.3.4-8.3.5)
Lecture 32: Minkowski's Theorem + Sums of Two and Four Squares (Notes 9.1.1-9.1.2) Lecture 33: Sums of Three Squares + Ideal Class Groups (Notes 9.1.3-9.2.1) |
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| Mon, Apr 12th Wed, Apr 14th Thu, Apr 15th |
Classes cancelled ("CARE Day")
Lecture 34: Computing Ideal Class Groups (Notes 9.2.1-9.2.2) Lecture 35: Quadratic Forms (Notes 9.2.3) |
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| Mon, Apr 19th Wed, Apr 21st |
Lecture 36: Composition of Quadratic Forms (Notes 9.2.3-9.2.4)
Lecture 37: Quadratic Forms and the Class Group (Notes 9.2.4) |
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| Handouts / Lecture Notes | |||
|---|---|---|---|
| Handout | Topics | ||
| Chapter 6: Rational Approximation and Diophantine Equations (35pp, v2.10, updated 1/28) (updates in 2.10: typo fixes, added magic box description to 6.2.2, added examples to end of 6.2.4, added super magic box factorization to 6.3.3, added example to end of 6.4.1) |
6.1 ~ Simple Examples of Diophantine Equations (Linear Diophantine Equations, the Frobenius Coin Problem, Pythagorean Triples) 6.2 ~ Rational Approximation and Transcendence (Farey Sequences, Finite and Infinite Continued Fractions, Rational Approximation, Irrationality and Transcendence) 6.3 ~ Pell's Equation (Motivation and Examples, General Structure, The Super Magic Box) 6.4 ~ An Assortment of Other Diophantine Equations |
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| Chapter 7: Elliptic Curves (30pp, v1.00, posted 2/7) | 7.1 ~ Elliptic Curves and the Addition Law (Weierstrass Form, The Addition Law, Elliptic Curves Mod p, Orders of Points) 7.2 ~ Factorization and Cryptography With Elliptic Curves (Factorization, Elliptic Curve Encryption, Elliptic-Curve Diffie-Hellman, Elliptic Curve Signatures) 7.3 ~ Rational and Integral Points on Elliptic Curves (Torsion Points, Mordell's Theorem, the Nagell-Lutz Theorem, Mazur's Theorem, Siegel's Theorem, Congruent Numbers) |
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| Chapter 8: Quadratic Integer Rings (42pp, v2.02, updated 4/15) (updates in 2.02: fixed typos and chapter numbering) (updates in 2.00: fixed typos and reference errors, added 8.3.2-8.3.5) |
8.1 ~ Arithmetic in Rings and Domains (Ideals, Quotient Rings, Maximal and Prime Ideals, Integral Domains, Quadratic Integer Rings, Euclidean Domains, Principal Ideal Domains, Unique Factorization Domains, The Chinese Remainder Theorem) 8.2 ~ Factorization in Quadratic Integer Rings (Unique Factorization of Elements, Ideals in OD, Divisibility and Factorization of Ideals in OD, Calculating Factorizations) 8.3 ~ Applications of Factorization in Quadratic Integer Rings (Factoring in Z[i], O-2, O-3, Diophantine Equations, Cubic Reciprocity, Quartic Reciprocity) | ||
| Chapter 9: The Geometry of Numbers (27pp, v1.90, updated 4/18) (updates in v1.90: added 9.2.4, added examples + uniqueness result to end of 9.2.3) (updates in v1.80: added 9.2.3, fixed typos) |
9.1 ~ Minkowski's Convex-Body Theorem and Applications (Minkowski's Theorem, Sums of Two and Four Squares, Sums of Three Squares) 9.2 ~ Ideal Class Groups of Quadratic Integer Rings (The Ideal Class Group, Minkowski's Bound, Binary Quadratic Forms, Composition of Binary Quadratic Forms) |
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| 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 the notes and watch the lecture recording. You are responsible for all material covered in lecture. | ||
| Read the Lecture Notes | The lecture notes are a comprehensive source 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 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 | |||
|---|---|---|---|
| The schedule is subject to change! All sections refer to the course lecture notes. | |||
| Week | Schedule | ||
| Week of Jan 18 (class starts 1/20) |
§6.1: Pythagorean Triples No homework this week. |
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| Week of Jan 25 | §6.2: Continued Fractions §6.3: Farey Sequences Homework 1 due Thursday 1/28. |
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| Week of Feb 1 |
§6.4: Rational Approximation §6.5: Pell's Equation Homework 2 due Thursday 2/4. |
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| Week of Feb 8 |
§6.5: Pell's Equation §6.6: Miscellaneous Diophantine Equations Homework 3 due Thursday 2/11. |
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| Week of Feb 15 (no class 2/15) |
§7.1: Elliptic Curves and The Group Law Homework 4 due Thursday 2/18. |
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| Week of Feb 22 | §7.2.1: Elliptic Curve Factorization §7.2.2: Elliptic Curve Cryptography Homework 5 due Thursday 2/25. |
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| Week of Mar 1 |
§7.3: Rational Points on Elliptic Curves Homework 6 due Thursday 3/4. |
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| Week of Mar 8 |
§8.1: Arithmetic in Domains Homework 7 due Thursday 3/11. |
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| Week of Mar 15 |
§8.1: Arithmetic in Domains §8.2: Factorization in Quadratic Rings Homework 8 due Thursday 3/18. |
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| Week of Mar 22 (no class 3/24) |
§8.2: Factorization in Quadratic Rings Homework 9 due |
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| Week of Mar 29 | §8.3: Applications of Factorization in Quadratic Rings Homework 10 due Thursday 4/1. |
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| Week of Apr 5 | §8.3: Applications of Factorization in Quadratic Rings §9.1: Minkowski's Convex-Body Theorem and Applications §9.2: Ideal Class Groups of Quadratic Integer Rings Homework 11 due Thursday 4/8. |
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| Week of Apr 12 (no class 4/12) |
§9.2: Ideal Class Groups of Quadratic Integer Rings Homework 12 due Thursday 4/15. |
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| Week of Apr 19 (class ends 4/21) |
§9.2: Ideal Class Groups of Quadratic Integer Rings §9.3: Elliptic Curves With Complex Multiplication Homework 13 due Friday 4/23. |
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| Final exam due Friday 4/30. | |||