## Colorado State University, Pueblo Math 319 — Number Theory — Spring 2015

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Lectures: MWF 2-2:50pm in PM 116      Office Hours: M1-1:50pm and T$\Theta$ 12-1:50pm, or by appointment

Instructor: Jonathan Poritz     Office: PM 248     E-mail: jonathan.poritz@gmail.com
Phone: 549-2044 (office — any time); 357-MATH (personal;please use sparingly)

Text: Yet Another Introductory Number Theory Textbook, v2.0, by Poritz, available here.

Prerequisites: A satisfactory grade (C or higher) in Math 307 (Introduction to Linear Algebra) or Math 320 (Introductory Discrete Mathematics). The point of these prerequisites is to ensure that you have started the process of becoming comfortable with a certain level of abstraction in mathematics, such reading and writing proofs.

Postrequisites: This course is required for mathematics majors with secondary certification.

Course Content/Objective: The Catalog simply gives a grab-bag of topics to be covered:

Divisibility, prime numbers, linear congruences, multiplicative functions, cryptology, primitive roots, and quadratic residues.
This is vast underselling of the subject of this course: Number Theory is one of the oldest of the "true" mathematical disciplines (= areas of mathematical investigation done in a way we would recognize today) ... perhaps the 1.9th oldest. It has astonishingly simple yet astonishingly beautiful results. It has rich structure and depth which work smoothly together like an intricate machine, and yet has consequences for statements simple enough that an elementary school student could understand.

One delightfully ironic aspect of Number Theory is that it was thought of for a couple thousand years as the purest of pure mathematics, and it should be therefore be innocent of good and bad application. The great English mathematician G.H. Hardy wrote in his book A Mathematician's Apology (published in 1940):

No one has yet discovered any warlike purpose to be served by the theory of numbers or relativity, and it seems very unlikely that anyone will do so for many years.
Hardy would probably be terribly disappointed to know that Number Theory underlies the great majority of techniques of ensuring security and privacy on the Internet, to the point where, for example, the US National Security Agency is the world's largest employer of Ph.D. mathematicians, many of them number theorists.

During the first, foundational part of the course we will cover a portion of the basics which are needed in any more specialized topic of number theory. This should take us approximately 3/4 of the term. A sightly more indicative list (than the catalog description) of topics in this first part would include

• linear and quadratic Diophantine equations ... hints of higher degrees
• divisibility, the greatest common divisor
• primes
• the Fundamental Theorem of Arithmetic
• congruences
• the Chinese Remainder Theorem
• Wilson's Theorem and Fermat's Little Theorem
• arithmetic functions, Euler's Φ function
• primitive roots
On top of these foundations, we will spend approximately a quarter of the semester on a major modern application of number theory:
• cryptology

Another objective of this course is to help with the transition from earlier stages of the mathematics curriculum, which is very often centered in some large part on calculation, to the rest of the mathematical world, which revolves much more around ideas, words and explanations. (There is in fact an argument, which I find very persuasive, that calculation is given a greatly exaggerated importance even in the earlier stages of mathematics....) What this means is that we will investigate during class time, as you will on your own (e.g., for homework), these ideas and words using a variety of mental tools including clear, formal definitions and proofs — so there will be much more reading and writing (of words) in this class than you may have seen in previous math classes!

Class [dis]organization: Daily procedures:

1. Keep an eye on the course schedule page, at least as frequently as we have class.
2. Read the assigned material we will be discussing before the class in which we will discuss it (and probably again after the first class discussion of that material). (In fact, one way to read hard texts about mathematics (and probably other subjects) is to do it three times:
• First, just to get an idea of the new terms which are used and the over-all shape of the reading — i.e., is it a long proof of a difficult theorem, or a definition and then several carefully worked-out examples, or whatever, and is there an important diagram after the discussion, or is there an informal proof followed by a statement "What we have just seen is: Theorem: ....".
• Second, read it through very carefully, following all the details of logic and calculation with a pen and paper in your hand to check them and see that you understand every single step in its full glory.
• Third, skim a last time now that you understand the details, to reassess the over-all structure and even to think about connections you can make on your own to other things you know, other examples you make on your own of the definitions or theorems under discussion, and how everything would change (if it would!) if some part were changed or omitted.
3. Every student must e-mail me (at jonathan.poritz@gmail.com thoughts and/or questions that have come up during their reading and thinking about the assigned material, at least an hour before each class, and no earlier than 5pm of the previous business day. We'll call these your T&Qs. They are graded 0 or 1: almost anything which shows serious engagement with the material of the day is worth a 1. Your six lowest such scores will be dropped.
4. Most Monday and Wednesday classes will end with a 5-10 minute miniquiz, almost always on vocabulary — if you can give a formal definition of the terminology we are using in a particular class, you should have absolutely no trouble with the miniquiz. These are graded out of 2 points, and your four lowest miniquiz scores will be dropped.
5. Most Friday classes will end with a 10-20 minute maxiquiz, which will be a bit more significant than the miniquizzes, perhaps along the lines of a (rather easy) homework problem. These are graded out of 5 points, and your lowest two maxiquiz scores will be dropped.
6. There will be regular homework assignments, roughly once a week, which will consist each of only a few problems — but they will be fairly challenging problems! One part of this HW which be new to many students is the importance of the quality of exposition: we will be working on clarity of explanation as much as anything else in this class! A good way to approach this is to write a rough draft of your work, doodling with ideas and computations and logical strategies before putting them together in the work to be handed in.
7. Some specifics about the homework:
• Homework is due either in class or at my office, before the end of assigned due date.
• Homework is assigned in sets but graded by problem. Each problem will typically be worth 3 points, meaning:
1. problem entirely missing (or wrong problem done!)
2. some work present, but also several errors and/or important missing parts;
3. most of the correct content is present, but there is at least one key idea or step which is missing, and/or there is a significant flaw in exposition (a variable used without definition, that kind of thing);
4. all content is present, all notation is defined, all steps are explained and justified.
5. For the homework pages you turn in, you will be given 1 point of extra credit, called a Green Point [GP] for every page you reuse: take a page from some other source (handout you are finished with from another class, draft page of a paper you are writing, whatever), put a big X through the written part and do your math work on the (previously blank) back of the paper.
Note that is is far better to be clear about a gap in your work than to pretend it doesn't exist: if you say "I think this next step is true but I cannot fully justify it" that will be worth much more than just stating the step as if it were obvious.
• Homework problems will appear on the homework web page on a regular basis. Please get used to going to that page frequently — at least every other day, and certainly before starting your work on a homework set.
• Late homework will count, but at a reduced value — generally, the score will be reduced by around a point for each day late.
• Your lowest five HW problem scores will be dropped.
8. Regular attendance in class is a key to success. But more than merely attending, you are also expected to be engaged with the material in the class. In order for this to be possible, it is necessary to be current with required outside activities such as reading textbook sections, thinking about problems, doing the small writing assignments and larger problem sets. You are expected to spend 2-3 hours per hour of class on this outside work — this is not an exaggeration (or a joke!), in fact it is closer to a legal requirement. Note that engagement with class and the outside materials should make the miniquizzes almost entirely trivial; if they are not, that is a strong sign you need to try to get more in step with the class, and perhaps come see me in office hours.
9. If you absolutely have to miss a class, please inform me in advance and I will video the class and post the video on the 'net. You can then watch the class you missed in the comfort of you home and (hopefully) not fall behind. Classes I have videoed will have the icon next to that day's entry on the schedule/homework page to remind you of the available video. (But you must e-mail me for a link to the video, you will not be able to search for it.)

Revision of work on homework, quizzes, and tests: A great learning opportunity is often missed by students who get back a piece of work graded by their instructor and simply shrug their shoulders and move on. In fact, painful though it may be, looking over the mistakes on those returned papers is often the best way to figure out exactly where you tend to make mistakes. If you correct that work, taking the time to make sure you really understand completely what was missing or incorrect, you will often truly master the technique in question, and never again make any similar mistake.

In order to encourage students to go through this learning experience, I will allow students to hand in revised solutions to all homeworks, mini- and maxiquizzes, and midterms. There will be an expectation of slightly higher quality of exposition (more clear and complete explanations, all details shown, all theorems or results that you use carefully cited, etc.), but you will be able to earn a percentage of the points you originally lost, so long as you hand in the original, graded work and a new, better version at the very next class meeting. The percentage you can earn back is given in the "revision %" column of the table in the Grades section, below.

Exams: We will have two midterm exams on dates to be determined (and announced at least a week in advance). These may have a take-home portion in addition to the in-class part. Our final exam is scheduled for Monday, April 27th, 1-3:20pm, in our usual classroom.

Grades: In each grading category, the lowest n scores of that type will be dropped, where n is the value in the "# dropped" column. The total remaining points will be multiplied by a normalizing factor so as to make the maximum possible be 100. Then the different categories will be combined, each weighted by the "course %" from the following table, to compute your total course points out of 100. Your letter grade will then be computed in a manner not more strict than the traditional "90-100% is an A, 80-90% a B, etc." method. [Note that the math department does not give "+"s or "-"s.]

pts each # of such # dropped revision % course % 1 ≈36 6 0% 5% 2 ≈24 4 50% 10% 5 ≈12 2 75% 10% 3/prob ≈45 probs 5 probs 75% 25% >100 2 0 33.3% 25% >200 1 0 0% 25%

Nota bene: Most rules on due dates, admissibility of make-up work, etc., will be interpreted with great flexibility for students who are otherwise in good standing (i.e., regular classroom attendance, homework (nearly) all turned in on time, no missing quizzes and tests, etc.) when they experience temporary emergency situations. Please speak to me — the earlier the better — in person should this be necessary for you.

Contact outside class: Over the years I have been teaching, I have noticed that the students who come to see me outside class are very often the ones who do well in my classes. Now correlation is not causation, but why not put yourself in the right statistical group and drop in sometime? I am always in my office, PM 248, during official office hours. If you want to talk to me privately and/or cannot make those times, please mention it to me in class or by e-mail, and we can find another time. Please feel free to contact me for help also by e-mail at jonathan.poritz@gmail.com,d to which I will try to respond quite quickly (usually within the day, often much more quickly); be aware, however, that it is hard to do complex mathematics by e-mail, so if the issue you raise in an e-mail is too hard for me to answer in that form, it may well be better if we meet before the next class, or even talk on the telephone (in which case, include in your e-mail a number where I can reach you).

A request about e-mail: E-mail is a great way to keep in touch with me, but since I tell all my students that, I get a lot of e-mail. So to help me stay organized, please put your full name and the course name or number in the subject line of all messages to me.

Academic integrity: Mathematics is more effectively and easily learned — and more fun — when you work in groups. However, all work you turn in must be your own, and any form of cheating is grounds for an immediate F in the course for all involved parties. In particular, some assignments, such as take-home portions of tests, will have very specific instructions about the kinds of help you may seek or resources you may use, and violations of of these instructions will not be tolerated.

Students with disabilities: The University abides by the Americans with Disabilities Act and Section 504 of the Rehabilitation Act of 1973, which stipulate that no student shall be denied the benefits of education "solely by reason of a handicap." If you have a documented disability that may impact your work in this class for which you may require accommodations, please see the Disability Resource Coordinator as soon as possible to arrange accommodations. In order to receive accommodations, you must be registered with and provide documentation of your disability to: the Disability Resource Office, which is located in the Library and Academic Resources Center, Suite 169.

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