Math 319 — Number Theory — Spring 2015

Here is a shortcut to the course schedule/homework page.

Here is a shortcut to the summary table below of components of the grades for this course.

**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:

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 1.9Divisibility, prime numbers, linear congruences, multiplicative functions, cryptology, primitive roots, and quadratic residues.

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):

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.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.

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

- 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:

- Keep an eye on the course schedule page, at least as frequently as we have class.
- 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.

- First, just to get an idea of the new terms which are used and the
over-all shape of the reading —
- 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&Q**s. 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. - 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. - 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. - 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. - 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:- problem entirely missing (or wrong problem done!)
- some work present, but also several errors and/or important missing parts;
- 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);
- all content is present, all notation is defined, all steps are explained and justified.
- 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.

- 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.

- 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. - 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 27 ^{th}, 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 % | |
---|---|---|---|---|---|

T&Qs: | 1 | ≈36 | 6 | 0% | 5% |

Miniquizzes: | 2 | ≈24 | 4 | 50% | 10% |

Maxiquizzes: | 5 | ≈12 | 2 | 75% | 10% |

Homework: | 3/prob | ≈45 probs | 5 probs | 75% | 25% |

Midterms: | >100 | 2 | 0 | 33.3% | 25% |

Final Exam: | >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 (

**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.

From | and | to | and | . |

Jonathan Poritz (jonathan.poritz@gmail.com) |
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