Math 4527 (Number Theory 2), Spring 2025
| Course Information | |||||
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| Instructor | Class Times | Office Hours | |||
| Evan Dummit edummit at northeastern dot edu 571 Lake Hall |
MWR 10:30am-11:35am 165 Richards Hall |
M 12:05pm-1:05pm M 3:00pm-4:00pm Lake 571 |
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| 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.) | |||||
| We will use Piazza for any course-related discussion: here is the 4527 Piazza page, which can also be accessed through the course Canvas page. Access is restricted to students in Math 4527. | |||||
| Teaching Assistant | Problem Session | ||||
| Devin Brown | Tuesdays 11:45am-1:15pm, 141 Ryder Hall | ||||
| 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. | |||||
| 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 Tue Jan 14th.
Homework #2, due Tue Jan 21st. Problems labeled "Challenge" are optional. However, I highly recommend thinking about them, especially if you are considering applying to graduate school in mathematics, and especially especially if you might want me to write you a recommendation letter. | |||||||
| 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 6: Rational Approximation and Diophantine Equations (35pp, v3.50, posted 1/4) | 6.1 ~ Some 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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| 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 Problem Sessions | There are weekly problem sessions run by the course TAs. 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. Students are highly encouraged to attend the problem sessions and work on the homework assignments there. | ||
| 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 6 (class starts 1/6) |
§6.1.1: Linear Diophantine Equations §6.1.2: The Frobenius Coin Problem §6.1.3: Pythagorean Triples §6.2.1: Farey Sequences No homework this week. |
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| Week of Jan 13 | §6.2.2: Continued Fractions §6.2.3: Infinite Continued Fractions §6.2.4: Rational Approximation §6.2.5: Irrationality and Transcendence Homework #1 due Tuesday, Jan 14th. |
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| Week of Jan 20 (no class 1/20) |
§6.3.1: Pell's Equation Examples §6.3.2: Pell's Equation Properties §6.3.3: The Super Magic Box Homework #2 due Tuesday, Jan 21st. |
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| Week of Jan 27 |
§6.4.1: Assorted Diophantine Equations §6.4.2: The Fermat Equation §8.1.1: Ideals of Commutative Rings §8.1.2: Quotient Rings Homework #3 due Tuesday, Jan 28th. |
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| Week of Feb 3 |
§8.1.3: Maximal and Prime Ideals §8.1.4: Arithmetic in Integral Domains §8.1.5: Quadratic Fields and Quadratic Integer Rings §8.1.6: Euclidean Domains Homework #4 due Tuesday, Feb 4th. |
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| Week of Feb 10 | §8.1.7: Principal Ideal Domains §8.1.8: Unique Factorization Domains §8.1.9: The Chinese Remainder Theorem §8.2.1: Unique Factorization of Elements in OD Homework #5 due Tuesday, Feb 11th. |
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| Week of Feb 17 (no class 2/17) |
§8.2.2: Ideals in OD §8.2.3: Divisibility and Unique Factorization of Ideals in OD §8.3.2: Factorization in O√-2 and O√-3 Homework #6 due Tuesday, Feb 18th. |
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| Week of Feb 24 | §8.3.3: More Diophantine Equations §8.3.4: Cubic Reciprocity §8.3.5: Quartic Reciprocity Homework #7 due Tuesday, Feb 25th. |
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| Spring Break (no classes) from Mar 3 to Mar 7 | |||
| Week of Mar 10 | §9.1.1: Minkowski's Convex-Body Theorems §9.1.2: Sums of Two and Four Squares §9.1.3: Sums of Three Squares §9.2.1: The Ideal Class Group No homework this week. |
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| Week of Mar 17 | §9.2.2: Minkowski's Bound §9.3.1: Representations by Quadratic Forms §9.3.2: Equivalence of Quadratic Forms Homework #8 due Tuesday, Mar 18th. |
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| Week of Mar 24 | §9.3.3: Composition of Quadratic Forms §9.3.4: Dirichlet's Composition Law Homework #9 due Tuesday, Mar 25th. |
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| Week of Mar 31 | §9.3.5: Quadratic Forms and Ideal Class Groups §10.1.1: The Riemann Zeta Function §10.1.2: Motivation for Dirichlet's Theorem §10.1.3: Dirichlet Series Homework #10 due Tuesday, Apr 1st. |
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| Week of Apr 7 | §10.1.4: Group Characters and Dirichlet Characters §10.1.5: Dirichlet L-Series and Dirichlet's Theorem §10.2.1: Dedekind Zeta Functions Homework #11 due Tuesday, Apr 8th. |
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| Week of Apr 14 (classes end 4/15) |
§10.2.2: The Analytic Class Number Formula Homework #12 due Tuesday, Apr 15th. |
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| The takehome final exam is due on Thursday, April 24th. | |||