Number Theory
M405, Spring 2018

Professor: Dr. Rachel Pries
e-mail: pries ATSYMBOL math DOT colostate DOT edu
web page: www.math.colostate.edu/~pries
office: Weber 205A, office hours: Wed 10-11 am and Thurs 1-2 pm.

Lecture: MWF 2:00-3:00, Engineering E105.

Course description:
The study of number theory originated in several ancient civilizations including China and India. Many famous results in number theory were proved in Europe in the 17th and 18th centuries. It is a core subject in mathematics and has important connections to algebra, geometry, combinatorics, and analysis. One motivation for studying number theory is that it provides many applications to cryptography and error-correcting codes.

Number theory is known for having problems that are easy to state yet which can only be solved using complicated structures. For example, it took 300 years to find a complete proof of Fermat's Last Theorem. Number theory is a vast subject; we will focus on the topics in the course description including distribution of primes; Diophantine equations; multiplicative functions; finite fields; quadratic reciprocity; quadratic number fields. If time permits, we will study elliptic curves and some applications to cryptography.

Prerequisite: M360 or M366 or equivalent experience.

Texts: We will use the following free on-line textbook:
Elementary Number Theory: Primes, Congruences, and Secrets, by William Stein.

We will use sections of the following books, which are freely available to CSU students through SpringerLink:

step 1: go to the CSU library page CSU library page or (on-campus only?) link.springer.com .
step 2: in the almost everything box, type in keywords (e.g. Everest and Ward and Introduction and Number) and click the search icon.
step 3: click on the item, look for full text available, and download.

primary (easier): Stillwell, Elements of Number Theory
primary (harder) : Ireland and Rosen, A Classical Introduction to Modern Number Theory, second edition
later: Everest and Ward, An Introduction to Number Theory
later: Silverman and Tate, Rational Points on Elliptic Curves
This textbook might be useful for review and details: Elementary Number Theory, by Clark.
Eisenstein article by Cox.

30% homework, computer labs, group project, and reading summaries; 15% final poster; 15% midterm one; 15% midterm two; 25% final.
Borderline grades will be decided on the basis of class participation.

Homework: Due every week. Doing homework problems is crucial for doing well in this class. Doing homework problems helps you develop skills in computation and logical reasoning. The process of doing homework will help you solve unfamiliar problems on the tests. Homework must be neat, legible, and stapled. I encourage you to brainstorm the problems in groups and write up your solutions independently.

Computer labs: We will sometimes use the computer program SAGE to solve some problems in number theory. SAGE is a free on-line math program which is helpful to solve complicated numerical problems. It can also be used to collect data and develop greater understanding of topics in number theory. Don't worry if you haven't used math software before - we will go over all the basics together. There will be several homework assignments using SAGE. Quick guide to SAGE:

Reading summaries: Each student will give a 1-2 minute summary of several pages of the reading several times during the semester.

Poster project: This class will participate in the spring 2018 math department poster competition. The assignment is to make a poster, covering some new material or application in number theory. This gives us a chance to have an overview of many fantastic topics in number theory that we wouldn't see otherwise.

Classroom policies:
Yes! Respect: we all have different backgrounds, skills, and goals.
Yes! Reciprocity: active and engaged learning.
Yes! Focus: deep connections and high level critical thinking.
Yes! Integrity: do your best, take responsibility, build a true education.
No :( disrespect, disengagement, distraction, despair. Please turn off your cell phone.

Important Dates: Midterm 1 is Friday 2/23; Midterm 2 is Friday April 6.
The group project will be done in class during the week of April 9-13.
Poster session: tentatively, Thursday May 3, 9 am to noon. The final exam is Tuesday May 8, 7:30-9:30 am.
There are no makeups for missed exams, regardless of the reason for absence. You must take the final examination at this time scheduled by the university; no final exams will be given earlier. Examinations will not be rescheduled because of travel arrangements. It is your responsibility to schedule travel appropriately.

Help: Help is always available if you have trouble with homework or lecture material. If your classmates can't answer your question, come ask me! Office hours are Mon 2-3 pm and Thurs 1-2 pm in Weber 118.

CSU Honor Pledge: Academic integrity is important to me. Paraphrasing the words of Greg Dickinson, Director of Graduate Studies; Professor, Dept. of Communication Studies:
Plagiarism is the unauthorized or unacknowledged use of a person's academic or scholarly work. Regardless of how it occurs, plagiarism is a theft of intellectual property.
Academic integrity means having a true educational experience. It involves doing your own reading and studying. It includes regular class attendance, careful consideration of all class materials, and engagement with the class and other students. Academic integrity lies at the core of our common goal: to create an intellectually honest and rigorous community.
Because academic integrity is so central to our mission as students, teachers, scholars, and citizens, we will ask you (but not require you) to sign the CSU Honor Pledge when completing all major assignments.
"I have not given, received, or used any unauthorized assistance."
If you plagiarize in your work you could lose credit for the plagiarized work, fail the assignment, or fail the course. Each instance of plagiarism, classroom cheating, and other types of academic dishonesty will be addressed according to the principles published in the CSU General Catalog (see page seven, column two: http://www.catalog.colostate.edu/FrontPDF/1.6POLICIES1112f.pdf).