This page exists to give some information about Math 214, which will be offered in Fall 2004.
Math 214 is a course in classical number theory, which will both introduce students to the language and mores of higher mathematics and lay out the basics of number theory. Number theory has been called the "queen of mathematics"; the subject begins with the Greeks and has been an active research topic more or less continuously for the past eight hundred years.
Who should take this class? Anyone who is contemplating being a math major and wants to know what higher mathematics is like. Also, anyone who isn't contemplating being a math major and wants to know what higher mathematics is like.
Should I take Math 214 or Math 215? Either is an excellent entry into the math department. Students who take Math 214 will probably be more prepared for Math 217 than Math 215 students, and less prepared for Math 218. Students could take a sequence of 214-203-217 and then move on to 300-level courses. An ambitious student could also try 214-217-218, with the understanding that a certain amount of self-study will be necessary in order to catch up with the analysis taught in 218.
This course will be taught with the understanding that students have not studied proof-based mathematics before. There are no prerequisites. (The people who invented number theory did not know calculus; you will not need it either.) There will be a final paper, through which you can investigate more deeply a number-theoretic topic that particuarly interests you.
The textbook for the course will be Niven, Zuckerman, and Montgomery's An Introduction to the Theory of Numbers.
Syllabus The course will cover the foundations of classical number theory, including: properties of integers and prime factorizations. Infinitude of prime numbers of various types. Which numbers are the sums of two squares? Properties of greatest common divisor. Fermat's little theorem. Euler's phi function. Chinese Remainder Theorem. The Fundamental Theorem of Algebra. Solutions of diophantine equations. Primitive roots. Quadratic residues, Legendre symbols, and quadratic reciprocity. Applications of number theory to cryptography (RSA algorithm.) We will make a special effort to emphasize the similarities and differences between different number systems (complex numbers, real numbers, rational numbers, integers, mod p numbers, and more.) Here is the Fall 2004 syllabus.