## Key Idea

The goal of the writing assignment is to dig deeper into a topic of interest to you, and write up a survey of the topic. It is exceedingly important that you write from your understanding, not from your sources. **So you may read and take notes to understand your topic, but all your materials, including your notes, must be closed during your writing periods. Not following this rule is a violation of the Honor Code.**

## Scaffolded process

This method of writing is very challenging, and so I will scaffold the process by providing you with a series of regular tasks (e.g. picking a topic, choosing references, identifying core understandings, writing an outline, etc.) that take you through my suggested process, and these tasks will be graded.

## Option to skip the line

However, your process may differ from mine, so you have an option to “skip the line.” If at any time you believe you have written a top quality assignment, you can hand that in and I will grade it. If it receives an A, you are exempt from all the rest of the scaffolded tasks and you are done with the assignment. If not, you must continue handing in the tasks as required and continue to work on your assignment.

**Important note: ** If you want to exercise this option, you must hand in your final project **4 business days** ahead of the next Task Due Date, because I need time to decide if you are exempt and let you know. You are only exempt from the next Task if you have received confirmation from me that you are exempt. Otherwise you must hand in the task.

## What you will produce

The goal of the writing assignment, in the end, is to write approximately 4-5 pages of LaTeX’d mathematics covering a topic you find interesting (if there are figures, these will lengthen the writeup). The writeup should describe the key definitions, give illustrating examples, state and prove key results, and provide philosophical and heuristic surrounding discussion to motivate the material and place it in context. Every reference you use will be cited.

## Grading

There are 6 scaffolded tasks, which will account for 30% of the grade (5% each). The other 70% will be the final grade on the project (unless you “skip the line” — see above — in which case the final grade will be proportionally more). The final project grade rubric is:

- Grade of A (100%):
- Entirely mathematically correct, except for at most typos or insignificant errors.
- Fluidly written, easy to read, precise in its content.
- Organized into a linear logical progression that is helpful to the reader, with helpful remarks and discussion as appropriate. Appropriately chosen examples that illustrate important points.
- Includes sophisticated mathematics, in the form of significant proofs, which are correct.
- The sources for the mathematical arguments in the proofs are cited. Examples are generated by the student, or, if that isn’t possible, their source is cited. (Citations do not need to be in a specific style, as long as they are tidy.)

- Grade of B (80%):
- May have some significant mathematical errors, but only a few, and they do not derail the content.
- Writing can be followed, and is not vague or wandering.
- Organization is sufficient, and concepts are introduced before they are used. Relevant examples and discussion are provided.
- Includes some sophisticated mathematics, in the form of significant proofs, which are mostly correct.
- The sources for the mathematical arguments in the proofs are cited. Examples are generated by the student, or, if that isn’t possible, their source is cited.

- Grade of C (60%):
- A larger number of mathematical and/or logical errors and misunderstandings that obstruct the ability of the reader to understand the paper.
- Writing hard to follow at times, and precision is lacking.
- Organization suffers from deficiencies, for example, concepts used before they are introduced. Examples and discussion may not be useful and relevant, or be misplaced.
- Includes correct mathematics, but may not include more sophisticated or interesting arguments beyond basic use of definitions. Significant mathematical errors may be present.
- The sources for the mathematical arguments in the proofs are cited. Examples are generated by the student, or, if that isn’t possible, their source is cited.

- Grade of F (40% or lower):
- Any of the following may be grounds for this grade:
- More errors than correct mathematics.
- Writing not possible to follow.
- Organization lacking.
- No interesting content.
- Citations are missing.
- Multiple sources are used without synthesizing.
- Evidence of not following the honor code.
- Only 1 or 2 pages in length.

- Any of the following may be grounds for this grade:

## Tasks

#### TASK A: Choosing a topic (MAR 18)

**What you will hand in. **With the above in mind, you task is to choose a topic and hand in a PDF on canvas with this info:

- Paper title
- 200 words describing what you will cover.
- A list of 2-3 references you will use, with specific page numbers. More is not better, but 1 is too few. I prefer books (textbooks are good) or academic articles, not webpages. You will have to carefully read all the pages you choose, so keep it to perhaps 10-20 pages total across all resources.

I will grade this on whether what you write is coherent and correct, and the resources are appropriate.

**Details: **You will need to do enough reading to be able to write that 200 word digest of your topic from your understanding. Here is where the internet is useful; wikipedia, blog posts, etc. sometimes have good overviews. **You may not have any resources open when writing your 200 words.** You must have done enough investigation to understand what you will be writing about, so that you do not need resources for this 200 words.

For any topic you choose, you will have to pick and choose the portion you want to cover in depth to aim for 4-5 pages final writeup. Don’t bite off more than you can chew. You can always state some lemmas without proof to get to the core idea of something.

You have free choice of topic, but you should not have the same topic as anyone else you interact with in class. Here I suggest appropriate topics, as examples:

- Proofs that pi and/or e are irrational and/or transcendental.
- Prove Liouville’s Theorem, which is a weaker form of Thue-Siegel-Roth that relies on methods from calculus.
- Generalizations of the Ford circles.
- p-adic integers (a different number system with a metric, like an alternative reality instead of real numbers, but based on a prime p)
- Hensel’s Lemma (this tells you how to “lift” solutions of a congruence modulo p to solutions modulo p^2; it is a kind of “modular” version of approximating roots with Newton’s method.)
- The Mobius inversion formula. This is a combinatorial tool (a variation on the inclusion-exclusion principle) with applications to number theory such as a new proof of a formula for Euler’s phi function, counting squarefree numbers (similar to our proof about counting coprime integers, but with a new combinatorial tool).
- Variations on Euclid’s algorithm: e.g. there is something called the binary Euclidean algorithm. There are also variations for the Gaussian integers (complex numbers a+bi where a,b are integers; Chapter 4 in text), and other rings.
- Continued Fractions. This is an interesting way to express real numbers that is related to their approximation by rationals.
- Elliptic curves. These are curves like y^2 = x^3 + k which have a geometrically constructed addition law that make them into number systems with interesting properties. A lot of cryptography is based on these.
- Diffie-Hellman Key Exchange (a basic cryptographic protocol in very wide usage based on modular arithmetic) and the associated hard cryptographic problems (Diffie-Hellman problem, Discrete Log Problem) and known algorithms.
- Factoring algorithms: quadratic sieve, for example.
- How a quantum computer can factor numbers: Shor’s algorithm.
- Prime counting. There are some results you can prove that approximate the prime counting function, but aren’t as strong as the Prime Number Theorem.
- The Fast Fourier Transform – this enables high speed computer multiplication.
- Error-correcting codes (requires thinking about finite fields).
- Minkowski’s convex body theorem and its use to prove Lagrange’s theorem. (This is basic geometry of lattices, used to prove every integer can be expressed as a sum of four squares.)
- The Quaternions. Another number system. These relate to the question of how to represent numbers as sums of four squares.
- Hilbert’s Tenth Problem. (There is no algorithm to solve Diophantine equations in general.) Doing the whole proof is too much, but you could explain some related concepts and key tools such as the idea of Diophantine sets. This is a good topic if you like logic/foundations.
- Sturmian Words. These are integer sequences that arise from the geometry of a line of irrational slope, and have interesting properties.
- Pseudo-random number generators. Some of these are based on modular arithmetic.
- Zsigmondy’s Theorem and its applications to Diophantine equations.
- Some interesting Diophantine equations and their solutions (there are many methods and choices here; feel free to ask).
- There are many more options! If something we did in class interested you, feel free to ask for ideas about something related.

#### Task B: Taking notes (APR 1)

**What you will hand in:** You will hand in a scanned PDF (or photos transformed to PDF, there are apps for that) of your reading notes to canvas. I won’t read them for details/correctness, but I want to see that they represent substantial reading/understanding/progress. I will grade them on organization & useful content. Please hand in at least 5 pages of notes, including one page outline/overview.

**Details: ** The next stage is to read your chosen resources and take notes from them. This is practice in active reading. Everything you want to know from these resources should go into your notes, in your own words, the way you understand the material. Some suggestions:

- Number the sheets of paper. Keep them in a binder or folder.
- Mark each section of your notes with the corresponding reference and page number of that reference that you are using.
- Rewrite in your words. Don’t copy.
- Keep a separate page for “Outline/Overview” where you can organize the key definitions, ideas, and facts/theorems. This page can reference page numbers of your other notes.
- Rewrite as needed. I rewrite multiple times while organizing ideas, since when I understand something, I usually understand it in “my way” and then rewrite notes according to that.
- If you take messy notes while first understanding something, tidy them up with a second round to figure out what to keep. Notes that you can’t understand later are useless.

#### Task C: LaTeX’d Outline (APR 8)

**What you will hand in:** A LaTeX’d outline of your project’s content, as a PDF to canvas. This outline should include:

- Title, your name, date
- Section titles (including Introduction)
- Key definitions (instead of full definitions, placeholders indicating the terms to be defined are ok)
- Key lemma, proposition, theorem statements, possibly stated informally
- placeholders for proofs
- placeholders for examples
- placeholders for remarks and discussions
- references in a bibliographic format

I will grade based on whether the outline is appropriate and well thought-out.

**Details: ** When I say “placeholders” I mean things like “Definition of the term *prime*“, “Proof of Theorem A,” or “Example of Definition A in small dimension” or “Example illustrating the necessity of hypotheses in Theorem A.” or “Discussion of possible generalizations to other rings such as complex numbers” etc. You don’t need to have it written, but you need to indicate what you plan to write. Writing just “Definition”, “Example” or “Discussion” isn’t enough detail. You must state precisely what the specific attribute or purpose of the example or discussion etc is.

**Do not** hand in a draft of your final project, or a partial draft. I want an outline only. If you want to “skip the line,” remember the 4-business-day rule.

#### Task D: 1-2 pages filled out (Apr 15)

**What you will hand in: ** Your outline LaTeX’d PDF, with approximately 1-2 pages of the content filled in, including at least one substantial proof (more is ok too).

**Details:** You must follow the instructions in the next paragraph about closed-book writing, which makes writing very time-consuming. The purpose of this task is to make you realize how much time it takes and not leave the whole thing to the last minute.

**Important:** For this class, it is a violation of the honor code if you write with your resources (including your notes) open. They must be closed. (You may use your Task C outline.) You can switch between the **Reading/Learning Phase** and the **Writing Phase** at most every 15 minutes. You must be able to understand a proof completely before you write it. This is **difficult**. Leave time for it. The Writing Phase can include scribbling on paper and then typing it into LaTeX. The point is that there’s a brick wall between the resources you used and the writing you produce that is bridgeable only by your own understanding (as if on a closed-book test).

#### Task E: Complete Final Draft (APR 22)

**What you will hand in: ** A complete final draft in PDF of your report, approximately 4-5 pages LaTeX’d. The same restrictions on closed-book writing apply.

If I give it an A, you are done. If I give it a B or lower, you will hand in another improved draft based on my comments.

#### TASK F: Possible Revision (APR 29)

**What you will hand in:** If you didn’t already receive an A, you will hand in a newer version that takes into account my suggestions for improvements. This is your last chance to improve the grade.

**Details: **When revising, you may have your resources open, unless you are adding substantial new content or re-writing wholesale, in which case you should write that content closed-book, as before.