# Math 3110 (Spring 2016) Number Theory

## Goals in this course

Over the course of the semester, we will...
1. rediscover some beautiful mathematics independently.
2. learn to communicate mathematics effectively.
3. become comfortable in alternate number systems.

## References

There is no required textbook, but readings, exercises, and inspiration will be drawn from the following sources:

## Office hours

Office: Math 204
Phone: (303) 492-3018
Office hours: calendar

## Outline of the course

Below is a list of a few topics in number theory that we may investigate. I don't promise we will get to all of these—we can't cover them all in a single semester—so I plan to let your interests guide us once we cover the foundations.
• Numbers, divisibility, and primes (Silverman, Ch. 5)
• irreducible implies prime in Z, prime factorization, Chinese remainder theorem, division algorithm, Euclidean algorithm

• Topographical arithmetic (Conway; Hatcher)
• Number systems
• Z, Z/nZ, Z[i], C, R, other number rings, Zp, Qp, surreal numbers, the game of set

• Sums of squares, Pell's equations, quadratic reciprocity, Hasse-Minkowski theorem

• Elliptic curves (Silverman, Ch. 41-46)
• Generating functions (Silverman, Ch. 50)
• Fibonacci sequence, Lucas sequences, Riemann's zeta function

• Cryptography
• public key crytosystems (RSA, elliptic curves)

## Assignments and assessments

Assignments will vary depending on how we cover material in class. Your grade will be determined from homework assignments, writing assignments, as well as some in-class quizzes. Many of these will be assigned to be completed in groups. I will also assign some exercises regularly, but unlike the other kinds of assignments, these exercises will not be graded. However, you may ask me questions about unassigned problems—and I encourage you to do so!

 Final paper 40% Paper 1 25% Paper 2 25% Quizzes 10%

Exercises, reading, and homework assignments will be below:

In class on April 8, we discussed 7 possible final paper topics. For April 11, please pick 3 of these topics. For each of the topics you've picked, make up 2 or 3 questions you can study that will help you towards understanding it. One very good thing to try is to come up with computations that will improve your intuition for the topic. You don't have to answer these questions (yet), but you should give serious thought to finding good questions, because you will have to answer some of them soon.

Here are the suggested paper topics:

1. Classification of integer binary quadratic forms up to change of variables
2. Suggested references: Hatcher, Topology of Numbers; Conway, The sensual (quadratic) form

4. Suggested references: Lemmermeyer, Chapter 10

5. Construct all finite fields
6. Suggested references: K. Conrad, "Finite fields"

7. Find all integer solutions to x2 + xy + y2 = n, or a similar equation
8. Suggested references: Hatcher, Topology of numbers, Chapter 6

9. Study recurrence sequences, such as the Fibonacci sequence, in modular arithmetic
10. Suggested references: Silverman, A friendly introduction to number theory, Chapter 39.

11. Find a formula for Euler's totient function φ(n) = | (ℤ / n ℤ)* |
12. Suggested references: Silverman, A friendly introduction to number theory, Chapters 11 and 15

13. Primitive roots: Given an integer n, can you find an integer m such that every integer relatively prime to n is congruent modulo n to a power of m?
14. Suggested references: Silverman, A friendly introduction to number theory, Chapter 28

#### Exploration 4: Quadratic forms [pdf] [tex] [overleaf]

Suggested reading: Hatcher, Topology of numbers, Chapter 5. Conway, The sensual (quadratic) form, Chapter 1.

#### Exploration 3: Quadratic reciprocity [pdf] [tex] [overleaf]

Every elementary number theory text has a section on quadratic residues and quadratic reciprocity. You may want to look at Chapters 20—25 of Silverman (note: Chapter 21 is available for free). You might also want to look at Chapter 4 of William Stein's Elementary number theory: primes, congruences, and secrets, or Chapter 4 of Pete Clark's Number theory: a contemporary introduction, or Chapter 6 of Hatcher's Topology of numbers. Note that you will find different proofs of quadratic reciprocity in all of these texts.

#### Exploration 2: Number systems [pdf] [tex] [overleaf]

Find partners and propose a paper topic by Friday, 3/4. Your partner or partners should be different from the partners you had last time. The ideal size for a group is 2, but I will accept groups of size 3 as well. Your proposal should indicate whom you plan to work with and a brief indication of what you plan to write about.

This paper should discuss the ideas we have encountered in the second exploration. The paper should provide some synthesis of those ideas and how they are related to one another, as well as those we've enocountered in the first exploration. Including examples is useful, but make sure also to explain the general principle those examples are examples of. Finally, there should be some discussion of an question or problem left open in class. Some suggestions for this question: Can you generalize the Chinese remainder theorem to more equations, or to more general situations? Which numbers can be the size of a finite field?

Suggested reading: K. Conrad, Modular arithmetic; The Art of Problem Solving; Silverman, §8, Stein; §2.1; Clark, §C.2

A question to consider for Wednesday, February 17: Can you build a field with 4 elements? You need to find a set with 4 elements, addition, subtraction, multiplication, and division laws (except division by 0), where addition and multiplication are both commutative and associative, and multiplication distributes over addition.

Another question for Wednesday, February 17: Can you find a simple formula for (p-1)! in 𝔽p?

#### Exploration 1: The Euclidean algorithm [pdf] [tex] [overleaf]

The exploration is still being updated frequently, so please download it regularly and press refresh if necessary.

Suggested reading: Silverman, §§5-6; Clark, §2.4

Suggested problems: Silverman, §5, #1, 3; §6, #1, 2; there are many more in the exploration

Suggested reading: Hatcher, pp. 27-37, 60-61. Hatcher's perspective is a little bit different from ours, in that he constructs Conway's topograph from the Farey triangulation. However, if you can understand this translation (see pp. 60-61), the discussion on pp. 27-37 is very close to what we are doing in class.

Assignment (due Wednesday, January 27): Write up a proof of one theorem from the exploration. You may choose a theorem that we have already discussed in class or one that we have not. You may work in groups of up to 3 people. Submit one document per group.

## Latex resources

Latex is a programming language for typesetting that is particularly useful for typesetting mathematics. Until recently, getting started in Latex required a little bit of work: you need to download a compiler, a text editor, and a document viewer, not to mention learn the programming language. However, new tools are making this easier. There are now several ways to edit and typeset Latex documents within a web browser, so you don't need to download or install anything. I have chosen to recommend Overleaf to the class, but there are many other options that appear equally useful: ShareLatex, Papeeria, Authorea.

Once you are able to create a basic Latex document, the best way to learn Latex is to get a template and start modifying it to do what you want. You can find a number of templates at Overleaf. You can also find tutorials online with a Google search.

We will do a tutorial in class on Wednesday, February 3. Here is the document we will be using: [tex] [overleaf].

To find the Latex code for a symbol, try Detexify.

If you have a question about Latex, try asking the hive mind. Chances are your question has been asked before, so make sure to search before posting a question: the hive is not always receptive to repetitions.

## How to succeed in this course (and others)

Here are a few tips for making the most of this course:
• Learning math takes time, so at the beginning of the semester, you should make sure you have enough time available for this course. The Boulder Faculty Assembly has stated:

An undergraduate student should expect to spend approximately 3 hours per week outside of class for each credit hour earned.

This is a 3-credit course, so you should make sure you have 9 additional hours available each week to spend on this class. (Of course, this number is approximate, and doesn't by itself guarantee success in the course.)
• Homework isn't the only way to work at home. Spend time reading textbooks, studying your notes, visiting office hours, etc.
• Study actively. To learn math you need to develop intuition about abstract concepts that are inherently unintuitive. The only way to do this is to wrestle actively with those concepts until they organize themselves in your mind. This doesn't always come naturally—at least not at first—so here are a few things you can try:
• Try to relate new concepts to things you already know: is the statement saying something you already know, at least in some special cases?
• Make up examples: if some statement is supposed to be true for all integers, make up a few integers and try the statement out on them.
• Make up counterexamples: if some statement is only true in certain situations, think about the statement in other situations to figure out what is special about the situations where it is true.