Number Theory
M405, Spring 2012

Professor: Dr. Rachel Pries
e-mail: pries ATSYMBOL math DOT colostate DOT edu
web page: www.math.colostate.edu/~pries
office: Weber 118, office hours: Mon 2-3 pm and Thurs 1-2 pm.

Lecture: MWF 11:00-12:00, Engineering E104 (code 14826).

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 in its own right and has important connections to algebra, geometry, combinatorics, and analysis. One motivation for studying number theory is that it provides many applications to coding theory and cryptography.

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 Diophantine equations; distribution of primes; multiplicative functions; finite fields; quadratic reciprocity; quadratic number fields. In addition, we will study elliptic curves and some applications to cryptography. We will use the computer program SAGE to solve many problems in number theory.

Prerequisite: M360 or M366 or equivalent experience.

Text: The textbook we will use is legally available for free from the author here: Elementary Number Theory: Primes, Congruences, and Secrets, by William Stein.
The next textbook might be useful for review and to get more details: Elementary Number Theory, by Clark.

20% homework; 20% midterm; 30% projects and presentations; 30% 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. The process of doing homework will help you solve unfamiliar problems on the tests. The homework problems will help you develop skills in computation and logical reasoning. Homework must be neat, legible, and stapled. I encourage you to brainstorm the problems in groups and write up your solutions independently.

Project: In this class, we will use the computer program SAGE, 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 two group projects using SAGE and one group presentation. This gives us a chance to have an overview of many fantastic topics in number theory that we wouldn't see otherwise.

Important Dates: The midterm is in class on Friday March 2.
Group project 1 will be done in class during the week of March 5-9.
Group presentations will be the week of March 19-23.
Project 2 will be done in class during the week of March 26-30.
The final exam is Monday May 7, 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.