MA341: Elementary Number Theory Projects There Are Going to Be Six

MA341: Elementary Number Theory Projects

There are going to be six different projects. Five of the projects will have three participants and the sixth will have only two. I will choose the groups. Each project will consist of an original treatment of important number-theoretic material. Each will ask you to learn and digest new material and then report on it. By far, the most important part of the project will be a final paper. That will due be on the final day of class: Friday May 1st. For the rest of what will be due, please see the section “Other Deliverables”. Your project choices are due on Thursday March 20th. Please use the piece of paper at the end of this document to: • Rank your choices 1–6. • List up to three other people you would want to work with. • Provide me with information which will help me assess which project to assign you. I will also post a PDF of the two documents to the website.

Project timeline For your convenience, I have digested the below information into a timeline.

Table 1. What you need to do

Mar 18 Projects announced Mar 20 Project choices turned in Mar 21 Project groups announced Mar 26–Apr 2 Meet with John to discuss project Apr 4 Project outlines due (as a writing assignment) Apr 18 Rough drafts due for peer assessment Apr 18–Apr 25 Meeting with John to discuss rough drafts Apr 25 Peer assessments due back (in lieu of a writing assignment) May 1 Final projects due May 1 Participation report due

Paper guidelines Before I begin with guidelines, the number one rule you must follow is: If you have any question about the project, you must ask me. Great, now that we have that cleared up, I will explain what I want out of the papers. What I expect is a coherent report that will explain some theorems. In order to explain your theorems well, you will have to put them in context, explain their proof (or at least give a plausible argument) and illustrate some use of the theorems. Here are the main ingredients for your paper: • A strong1 introduction. What are you going to report on? Why is it interesting? What kind of problems will it help solve? Are there any deficiencies? In the introduction you can state the result you are going to prove, leaving the definitions and terms for the main text. • History. Put your results in context. Is there an interesting history to the problem2? How long have these ideas been around? Are people still interested in this type of mathematics today? • Background. What are the important definitions and concepts? Is there anything subtle about what you are about to do? What are some good examples to keep in mind?

1As a practicing mathematician, I can say that the introduction of a technical paper is by far the most important part and should receive the most critical attention. 2This will always be “yes”. 1 • The main results. What are the major results your are proving? Which ones can you prove? If you can prove them, how? If you cannot, why not? • Applications of your result(s). Are there some good applications of your results (e.g. exercises from your textbook)? Or maybe there is some novel theorem you can prove as a corollary? • Conclusion. What is a reasonable next step to the project? What’d you learn? What do you wish you had learned better?

Your audience. I expect you to write as if the person reading your paper is a fellow classmate taking MA341. In fact, one of your classmates will read (a version of) your paper! (See below in “Other Deliverables”.)

Absolute must. I expect each project to state and prove at least two theorems. The best way to know which results are “theorems” is to ask me.

Page length. I don’t have a hard rule for page length. I expect that your projects will be between 5 and 15 pages, leaning towards the longer end. Don’t be afraid of writing so much about mathematics! The introduction, history and background of your results can easily take up a couple of pages each!

Originality. Originality in this kind of project is a tricky issue. After all, the reports you are going to write will detail theorems that were understood (and in most cases, proven) well before the end of the 19th century. In that sense, nothing you discuss in your papers will be original material. That being said, what I expect from you is an original exposition. So how do you go about doing that? The same way3 you went about writing your papers on divisibility tests! You learn the material and then you explain it in your own words. You surround the relevant results with your thoughts on what it means, how beautiful (or not?) it is, etc. It may happen that you come across a result or some discussion (most likely, a proof of a theorem) which you cannot understand, and thus cannot rewrite in your own words. You should not be afraid of this situation! Do not copy the discussion blindly. Instead, understand what you can (e.g. the statement of the theorem) and explain that. Then, calmly cite the proof with a proper citation. I do have some expectation that you will prove something in your paper so here is a general guideline: • If the proof is at the level of this course, I expect you to digest the proof and give it yourself. Your explanation should be originally written. • If you need to cite a proof, please find a bona fide source for it. I prefer it be found in a textbook but if you have a question about a source you found, please ask me. I can probably find a better source. (More on this below.)

Sources. Our textbook contains a little material on each project. I provided those references below. You should read that material to get a gentle introduction into the circle of ideas, but I expect you’ll need to go beyond what Silverman discusses. The Science & Engineering library will contain many references. Quite a few of them will be written at a level beyond this course, but if you look hard enough you will ﬁnd a number of sources pitched perfectly at the material. I encourage you to jump Google and Wikipedia4 to read more about each project! The Internet has made learning mathematics a ton easier and you should take advantage of it. You will likely ﬁnd personal notes written by mathematicians on each project. Use these notes to learn the material! However, if there is a source that you use then I ask that you provide a proper citation.

Final thought. Let me ﬁnish by reminding you of the most important rule If you have any question about the project, you must ask me.

Other deliverables Along with the ﬁnal paper you will produce a few other documents.

3One small difference is that I am explicitly giving you some references. 4The Wikipedia articles on mathematics are some of the best articles on that site. 2 Outline. An outline for your paper will be due (as a writing assignment) on Friday April 4th. This will give you two weeks from when your project is announced. It should be no more than one page in length, but should include: • A description of each section of your paper. • A clear statement of the main results you are planning to prove You do not need to detail each little lemma that you are going to prove. I will read your outlines and make suggestions. You should expect that you will have to expand the material you want to cover. Rough draft. Because this experience will be new for all of you, I am asking you to produce a rough draft by Friday April 18th. Thus you will have two weeks from when your outline is due to have a rough draft written. On the day it is due, please bring four copies of the draft. All four will go to me, I will read one and three of your classmates will read the others. The rough draft should read as a coherent piece of mathematics. It should be written perhaps with the level of attention you’ve paid your writing assignments during the course. If you cannot finish understanding a proof for the rough draft, it is not a big deal. Simply state the proof will be in the final draft of the paper and that the reader should take statements on faith. Peer assessment. Each of you will be expected to read and report on one of the other projects’ rough5 draft. This will be due Friday April 25th (one week after the rough drafts are due). It is not your job to correct spelling and grammar (though your classmates might appreciate it). Instead, I’d like a one page summary of what you read: • What were the main ideas of the paper? • What kind of techniques are employed? • Is there anything you didn’t understand? Could some of the ideas be better explained? You should be critical and constructive in your assessment. When you come to class on the Friday these are due, you should bring four copies: one for me and one for each of the members whose project you read. Participation report. Each of you will write a short summary on how your group worked. You should include a brief discussion of what you did on the project and what others did.

Meetings Along with written feedback, I’d like each group to meet with me at least twice during the process. For each meeting, you should propose four diﬀerent one hour slots in which to meet. In order to give you a wide range of choice, I’ve given a table below of times you can choose from. If your group cannot come during one of these times, we can try to arrange for something else (either later in the evening or on a weekend–you should alert me to this as soon as possible.).

Table 2. Times I can meet

Monday 9:30am–1:30pm Tuesday 12:30pm–6:00pm Wednesday 9:30am–6:00pm Thursday 11:00am–6:00pm Friday 9:30am–2:00pm

First meeting. The ﬁrst meeting will take place between Wednesday March 26th and Wednesday April 2nd, prior to the outlines being due. Each project group is expected to come and discuss with me what they want to write about, which results they are going to try to learn, etc. Second meeting. The second meeting will be between Monday April 21–Friday April 25. Prior to the meeting I will read your rough drafts and then you will come and discuss them with me.

5I wish I could do this for a final draft, but wouldn’t you rather be done with your work before finals week starts? 3 Grades As far as your grade in the class is concerned, 30% comes from the final project. I will calculate the grade for your final project as follows: (a) 60% will be the paper itself. (b) 40% will be participation. I fully expect everyone to earn all the possible participation points but I will have my discretion at assigning marks for: • Turning everything in on time. • Attending meetings with me and your group. • Your peer assessment. • The participation reports written at the end of the project. You can ensure a good participation score by being actively involved in each area of your project.

Project outlines On the following pages I outline six projects. Please read them as well as you can and turn in your choices to me on Thursday March 20th.

Project #1: Fermat’s Last Theorem Fermat’s Last Theorem was one of the longest, and most frustrating, outstanding problems in mathematics at the time it was solved in the early nineties. It says: Theorem. If n ≥ 3 then the equation xn + yn = zn has only trivial solutions in the integers. The methods used to ultimately prove this theorem are beyond this course. However, few problems have so consistently inspired top-rate mathematics. I would like you to investigate the history of the problem, going back to Fermat, and report on some special cases which are of historical interest. Here are some possible questions which might inspire you: • What does the theorem mean? For example, what is a “trivial solution”? Why do we need n ≥ 3? • Did Fermat prove any cases? How does his method of “descent” help deal with the case of n = 4? Is the “descent” method applicable to other problems? Can you explain the descent method? • Beyond Fermat, who else worked on the problem? How many special cases can be done “by hand”? • Are there any quick reductions? Are prime exponents particularly important? If so, why? If not, why not? Here are some possible sources which may inspire you: • Chapter 30 of your book discusses Fermat’s theorem for n = 4, and Chapter 46 discusses Fermat’s Last Theorem in general. The Internet contains a lot of material, and there are many “pop math” accounts. For example, Simon Singh’s “Fermat’s Enigma” is a fun read (I have a copy).

Project # 2: Rational, irrational and transcendental numbers When building the arithmetic world we live in, we roughly go through the following progression. First, we start with 0 and a way to add +1. This helps us build the natural numbers 0, 1, 2, 3,... . We then learn to subtract and we end up with Z. Taking fractions we get the rational numbers Q. At this point, there are various things one can do. • Use algebra. That is, we construct numbers by taking roots of polynomial equations. For example, solutions to the equations x2 −2 and x2 +1 both produce non-rational numbers. In general this gives us huge set of complex numbers: the algebraic numbers Q ⊂ C. √ • Use analysis. The rational numbers naturally come with “holes”6 in them like 2. The real numbers R ﬁll in those holes. √ 6For example, in decimal form we have 2 = 1.41421356237... Thus the sequence of rational numbers 14 141 14142135 1, , ,..., ,... 10 100 10000000 √ converges to a “hole” in Q, which we call the real number 2 ∈ R. 4 Now, it turns out that these lend themselves to two ways of getting to the complex numbers C. Doing these two processes in diﬀerent orders we arrive at the same number system! Here is a picture which I hope explains what I mean. algebra Q / Q

analysis analysis R / C algebra But there is a question which is begging to be asked in this discussion: Do we need both analysis and algebra? √ Well, I can say for sure that we need algebra: you cannot build i = −1 by taking a limit of rational numbers. Don’t be silly! But what about analysis? Do we need to take limits at all, or can we do everything only using roots of polynomials. Unlike the previous situation, it is not so easy for me to produce a counter-example. Let me thus re-phrase our previous question as the following one: Are there interesting numbers which don’t satisfy polynomial equations? This project is supposed to show that, yes, there are such numbers. And in fact, many of your favorite numbers behave this way! Here are some questions which you might try to understand: • First, what are algebraic numbers? Is every number algebraic? If not, how can you decide? Are there more algebraic numbers, or non-algebraic ones? • Are there interesting examples of non-algebraic numbers? Can you prove they are not algebraic? • Maybe you have heard that some numbers are not algebraic. But maybe you cannot prove that. Maybe you can prove that these numbers are irrational at least. A ﬁrst possible source for this material is • Chapters 37 and 33 (especially Theorem 33.2) of your book. I know where to ﬁnd proofs that e is not algebraic, and that π is not irrational which you should be able to understand with just calculus. √ Project #3: The ring Z[ 2] We are back to our old tricks with the number system √ n √ o Z[ 2] = a + b 2 ∈ R: a, b ∈ Z .

The integers Z have a number of features which we developed in sufficient generality to gain an interesting world in which to work. In no particular order, here are some features of the integers: (a) Prime numbers: those that cannot be divided. (b) Euclidean division: given m and n we can find q, r such that m = nq + r where r satisfies 0 ≤ r < n. (c) Greatest common divisors: these have a “functional definition” in Z but can also be defined as the terminating remainder in the Euclidean algorithm. (d) Factorization: every integer factors uniquely into prime factors. (e) Units: this is not very interesting, only ±1 are units. In class we will go through the process of developing the parallel theory for the number system Z[i]. That is: we will define analogs of primes, Euclidean division, greatest common divisors and factorization. The units in Z[i] will be easy to see as well: they will be ±1 and ±i. √ This project asks you to develop the theory a third time, but for the number system Z[ 2]. You’ll want to know: • How do we divide? What are prime numbers? • Can we perform a version of Euclidean division? What kind of ambiguity are there in “greatest common divisors”? • Is there a unique factorization theorem? • How many units are there? You might also want to consider the following question: 5 2 What is the relationship between√ solutions to the congruence x ≡ 2 mod p in the integers, and the behavior of p in Z[ 2]? [This will become more clear soon] Here are some possible sources which may inspire you: • Chapters 31 and 36 of your book.

Project #4: Arithmetic progressions For this project, your goal will be to understand the statement of Dirichlet’s theorem: Theorem (Dirichlet). Suppose that gcd(a, m) = 1. Then there exists infinitely many primes p such that p ≡ a mod m. Thus, for example, 3+4k is prime for infinitely many integers k (this is a = 3, m = 4). Dirichlet’s theorem itself requires techniques that outpace this course. For example, it requires complex analysis in the same way that the prime number theorem requires complex analysis. While the proof goes beyond the scope of this class, there are many examples which can be done by hand. For example, the case of m = 1 is the statement “there are infinitely many primes”. Here are some possible questions you might ask: • Does the proof that there are infinitely many primes give way to proofs of Dirichlet’s theorem for special modulus ? • What kind of techniques are helpful in proving special cases? • Fix m. For each a, there are infinitely many p such that p ≡ a mod m. Are there more or less if you change the value of a? • Can you prove consequences of Dirichlet’s theorem without proving the theorem itself? For example, it follows from Dirichlet’s theorem that if m ≥ 2 then there are infinitely many primes such that p 6≡ 1 mod m. Can you prove this without Dirichlet’s theorem? • What other kind of sets of primes might it be interesting to know are infinite, or finite? Perhaps you can look into “twin primes” or the classes of primes discussed in your book: Fermat, Mersenne, Germain, etc. A useful source to begin with will be • Chapters 12 and 13 of your textbook.

Project #5: Continued fractions In the second project I discussed how analysis plays a role in passing from Q or R. This project is about starting a a real number x ∈ R and approximating it by rationals. There is a lot to write down to do this, so let me just explain in an example. Start with the number x = π = 3.1415926 ... . At each step, we are going to think about rational numbers approximating π to different digits. But the key is that we want to use integers always. 1) 3 is the closest rational to π without any decimal place. 2) π − 3 = 0.1415926. And now approximating 0.1415926 is boring by integers, but 1/0.1415926 = 7.0625159 ... is better. The number 7 closely approximates it. So now we look at 1 1 3 + 0.1415926 = 3 + ∼ 3 + . 7 + 0.0625159 ... 7 Here ∼ is meant to be read “roughly”. 3) If we continue on, we’d find ourselves looking at 0.0625159. Again, we can invert it and get 1/0.0625159 = 15.99593. And now we have 1 π ∼ 3 + 7 + 0.0625159 ··· 1 ∼ 3 + 1 7 + 15+0.99593··· 1 ∼ 3 + 1 . 7 + 16 6 I’m going to stop here, but this expansion goes on forever. Ok, what I want to point out now is what these first two approximations are. The first approximation of π is 3 + 1/7 = 22/7 = 3.14. This is a well-known approximation of π, and let me point out that we achieve two decimal places of exactness with only inverting a single digit integer. The second approximation is 1 1 355 3 + 1 = 3 + 113 = = 3.141592920 ··· 7 + 16 16 113 And so this gives another approximation of π to the sixth decimal place! And we only had to invert a three-digit number. Ok, so this whole project is about continued fraction expansions (like the one above) and their properties. Here are some questions you might think about: • What is the continued fraction expansion? • If continued fractions are supposed to be rational approximations of real numbers, what are the continued fraction expansions of rational numbers? • What are the continued fraction expansions of some famous numbers, like π or e or whatever. • Can the continued fraction expansion become “periodic”? If so, what kind of numbers have periodic continued fraction expansions? Good beginning sources for this material are • Chapters 47 and 48 of your book (which is online at the website http://www.math.brown.edu/~jhs/frint.html).

Project #6: Fibonacci numbers mod m

The Fibonacci numbers are a famous sequence which begins with F0 = 1 and F1 = 1 and then for n ≥ 2,

Fn = Fn−1 + Fn−2.

Fix a modulus m. Then consider the sequence of numbers Fn mod m. Since the Fibonacci numbers are deﬁned recursively it happens that Fn mod m must be periodic (this is essentially the fact that there are only ﬁnitely many congruence classes mod m). At this point you should grab some paper and work this out for a low value of m, like m = 3 or m = 4. In any case, there is some least j such that Fn mod m ≡ Fn+j mod m for all n. Note that j = j(m) depends on m. It is easy to see that, like the φ-function, the computation of j(m) reduces to the computation of j(pk) where p is a prime (this is essentially the Chinese remainder theorem). For this project, I’d like you to explore the Fibonacci periods, starting with what I’ve already mentioned, and building upward. Here are some questions you might start thinking about. • Why does the Fibonacci sequence repeat modulo m? • Why does the period of the sequence reduce to only prime power cases? • One famously knows that the Fibonacci numbers are closely linked with the golden ratio by the following two formulas: √ F 1 + 5 lim n+1 = ϕ := n→∞ Fn 2

and √ n √ n 1+ 5 1− 5 2 − 2 Fn = √ . 5 √ Does the appearance of 5 have anything to do with the period? For example, does the period have depend in some way on whether or not the number 5 is a square mod m? A good beginning source for this material is • Chapter 39 of your book.