Instructor: Mark Sing (they/them)

Office: Kassar 012

Office Hours: Monday 1-2 (Zoom) and Wednesday 1-2 (in-person). Also by appointment.

Email: mark_sing@brown.edu

Math 420 is an introduction to number theory. This branch of mathematics has a long and rich history, with centuries of contributions and developments by mathematicians around the world.

The main object of study in number theory is the natural numbers 0, 1, 2, 3, .... In this course, we will explore these numbers from many angles, both classical and modern. We will begin with a careful study of the natural numbers before expanding our world to larger “systems of numbers”. The course’s motivating theme is, roughly, to understand how expanding the notion of number allows us to better understand the natural numbers, by exploring how these new numbers interact with the natural numbers and each other. At the end of the course we will cover more specialized topics based on class interest.

There are no pre-requisites to this course (not even calculus). Non-concentrators are strongly encouraged to take this course. All you need is a willingness to engage actively with the course- work. My goal is for you to learn some mathematics and mathematical thinking, and especially developing your ability to formalize proofs in number theory.

A brief list of topics we aim to cover, in no particular order:

- Natural numbers and integers (especially factorization)
- Diophantine equations (pythagorean triples, etc)
- Modular arithmetic (including quadratic reciprocity)
- Gaussian integers
- Real, complex, an p-adic numbers
- Cryptography
- Additional topics according to student interest

The textbook we will use is *A Friendly Introduction to Number Theory, 4th edition* by Joseph H. Silverman. The book has a website where Chapters 1-6 are available for free in PDF format: https://www.math.brown.edu/johsilve/frint.html. This site also includes errata, bonus exercises, and some chapters unavailable in the print version of the book.

We will also use the free software Sage (https://www.sagemath.org/) in this course. Numerical experimentation is an important part of mathematics, and a valuable complement to the abstract theory!!

Each week there will be two lectures (TR 10:30 - 11:50).

To help organize the material and your learning, most lessons are structured around a worksheet based on what we are reading in the book at that time. During the lesson there will be some lecture and some time spent working on the problems from the worksheets in small groups. Groups will be asked to explain their work and questions to each other. Mathematics is a collaborative activity!!

Homework is crucial to success in this course. All assignments are mandatory

One of the best ways to learn math is by doing math. It is nearly impossible to learn and master any subject without spending some time working on problems. You are always welcome to come to office hours with questions!!

Each homework set is short and two are due each week to ensure you are continuously engaging with the material without overwhelming your schedule. Problem sets and their due dates will be posted on the course website. Homework will typically be released Fridays and Mondays, and due during class on Tuesday and Thursday (sometimes moved slightly to accommodate holidays and exams).

Homework grading will emphasize conceptual understanding and a willingness to put effort into the assignment.

Late homework

More details later: this will be a small individual research assignment, possibly with a presentation component if time permits.

Math 420 is meant to be a fun and interesting introduction to aspects of mathematics that are outside the typical curriculum. It should not and will not be stressful!!

The breakdown of your grade will be as follows:

- 80% Homework
- 20% Final Project

We take academic integrity very seriously. Our goal, above everything else, is that you work hard and learn the material as well as you are able.

At the same time, mathematics is a collaborative activity. You are strongly encouraged to work together on your homework assignments, but we ask that you write and submit your own solutions, rather than copying those of other students. One good way to follow this policy is to start problems on your own and work until you get stuck before consulting others to get unstuck. If you are badly stuck and cannot consult your peers or the instructor, you can look for hints online as long as you carefully cite your sources. Violation of this policy, or any other form of academic dishonesty is prohibited by Brown’s Academic Code and may have serious consequences.

Here is our course calendar and all the files (class notes, worksheets, homework assignments, and solutions). Each day has a brief description of the topics covered, the assigned reading and homework due that day, and the worksheet that day. Solutions will also be posted.

In this calendar, the homework file is located at

Homework due Tuesday will be posted the Friday before, and homeworks due on Thursday will be posted on the Monday before. Typically, homework due on Thursday will include concepts from Tuesday's class, so skimming it ahead of time might be useful!

Topics, especially for later weeks, are subject to change.

Here is a link to the longer cumulative assignment for this course: cumulative homework.

Week 1: 1/25, 1/27 | |||
---|---|---|---|

Tuesday | Thursday | ||

no class!! | First class. Syllabus and some general comments. | ||

Week 2: 2/1, 2/3 | |||

Tuesday | Thursday | ||

Discuss Chapter 1. Pythagorean triples from an algebraic viewpoint, divisibility and congruence. | More on Pythagorean triples from an algebraic viewpoint. Introduce the geometric approach. | ||

Read: Intro, Chapter 1, start of Chapter 2. | Read: rest of Chapter 2. | ||

worksheet | solutions | see previous | see previous |

Week 3: 2/8, 2/10 | |||

Tuesday | Thursday | ||

A geometric approach to Pythagorean triples. | More on the geometry of Pythagorean triples. | ||

Read: Chapter 2, start of Chapter 3 | Read: rest of Chapter 3. | ||

Ch1: 1-4, Ch2: 1-2 | solutions | homework | solutions |

Week 4: 2/15, 2/17 | |||

Tuesday | Thursday | ||

Fermat's Last Theorem, the Mordell Conjecture. | GCD and divisibility | ||

Read: Chapter 4 | Read: Chapter 5 | ||

Ch2: pick three from 3-8, Ch3: 2-4 | solutions | homework | solutions |

worksheet | solutions | worksheet | solutions |

Week 5: 2/12, 2/24 | |||

Tuesday | Thursday | ||

University Holiday | Algorithms for GCD, extended GCD | ||

Read: Chapter 5, 6 | |||

Week 6: 3/1, 3/3 | |||

Tuesday | Thursday | ||

Emergency cancellation | Modular arithmetic | ||

Read: Chapter 8 | |||

Week 7: 3/8, 3/10 | |||

Tuesday | Thursday | ||

Powers mod primes | More on powers mod primes | ||

Read: Chapter 8, 9 | Read: modular arithmetic summary | ||

Homework: 8.1, 8.2, 8.8 (optional), 8.10, email bonus | solutions | solutions | |

worksheet | solutions | worksheet | solutions |

Week 8: 3/15, 3/17 | |||

Tuesday | Thursday | ||

Complex integers, examples and norm | Divisibility in complex integers | ||

Read: none | Read: none | ||

solutions | Homework: finish previous worksheet | solutions | |

worksheet | solutions | ||

Week 9: 3/22, 3/24 | |||

Tuesday | Thursday | ||

More applications. Sums of squares. | Modular arithmetic in Z[i] | ||

Review these notes | Read: none | ||

homework | solutions | Homework: finish previous worksheet | solutions |

worksheet | solutions | ||

Most of my past students have enjoyed seeing my bunnies, Cujo and Lula, so a few photos are attached here. They make a welcome break from mathematics for me, and I hope for you as well!!

One or the other occasionally visits my office - I'll send out email in advance if you'd like to meet them (only if you are not allergic to rabbits or hay!!)

This is Cujo's friend Peter, whose friend David contributed to the design of this website. Thanks!