Math 419 — Number Theory — Spring 2012

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

**Lectures:** MWF 11-11:50am in PM 116
**Office Hours:** MWF10-10:50am, T10-11:50am, 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: A Friendly Introduction to Number Theory
(3^{rd} edition)**, by Joseph H. Silverman.

**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 are
comfortable reading and writing proofs, which will be a huge part of this
course.

**Course Content/Objective:** The

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):

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 corresponds roughly to chapters 1–15 & 20–23 of our textbook. 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

- quadratic residues and quadratic reciprocity
- cryptology
- Diophantine approximation
- continued fractions
*p*-adic numbers

**Class organization:** The textbook we are using is, as it says, quite
**friendly**. [Almost *too* friendly.] We will center our activity
in the course around the book, covering a chapter approximately every two
classes, in the following way:

- Keep an eye on the course schedule page, at least as frequently as we have class.
- Read the chapter we will be discussing
**before**the class in which we will discuss it (and probably again after the first class discussion of that material). **Every student**must submit to the course web site thoughts and/or questions that have come up during their reading and thinking about the chapter,**at least an hour before each class**.*[We'll call these your***T&Q**s.]- The course schedule page will also name two students for each chapter
as
who will be responsible for taking and writing up class notes. These students should work together and make absolutely sure they have clear, complete, formal versions of all the definitions, statements (lemmata, propositions, theorems, corollaries,*student chapter leaders [SCLs]**etc.*), and proofs. - The SCLs will get their material into our electronic
. This text will be available at all times to all students, and should provide a great help by filling in all the*Class Collaborative Textbook [CCT]**unfriendly*details the paper textbook is missing. - Along with each class discussion of a chapter, we will be discussing
several of the chapter's problems. One such should also go in the
, as part of the SCLs' write-up.*CCT* - Students who are not SCLs for a given chapter will have 2–4
**homework problems**to do and to write up from each chapter. These must be turned in at the next class after we are done with that chapter. -
You may turn in the HW on paper or electronically. If you use paper, you
will be given
**1 point**of*green points***extra credit**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. If you do electronic HW submission, you will automatically get**5 green points**for every set.

**Revision of work on homework, CCT work, 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, CCT sections,
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 revised work 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
**Friday, May 4th from 10:30am-12:50pm in our usual classroom**.

**Grades:** On exam days or days you are a SCL, attendance is required
— if you miss such, you will get a **zero** as score; you will be
able to replace that zero only if you are regularly attending class and have
informed me, **in advance**, of your valid reason for missing that day.

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.]*

# of such | # dropped | revision % | course % | |
---|---|---|---|---|

T&Qs: | ≈45 | 5 | 0% | 10% |

CCT §§: | ≈4 | 0 | 75% | 15% |

Homework: | ≈20 | 2 | 75% | 25% |

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

Final Exam: | 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, 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).

**Contact inside class:** Here are some useful hand gestures which can
be used during class discussions (or lectures) for everyone to participate
without the room becoming too cacophonous:

**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. Also, if you are writing
me for help on a particular problem, please do not assume I have my book, it
is often not available to me when I am answering e-mail; therefore, you should
give me enough information about the problem so that I can actually help you
solve it (*i.e.,* "How do you do problem number *n* on page *p*"
is often not a question I will be able to answer).

**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 this assistance, you must be registered with and provide
documentation of your disability to the Disability Resource Office, which is
located in the Psychology Building, Suite 232.

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