Registration for Spring term opens Thursday, February 23, 2016.
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This course is an introduction to number theory. No background in the topic is required, and all are welcome.
Number theory is the study of the natural numbers. Although its ultimate scope is vast, in this course we will start at the very beginning. The topics covered will include:
Primes and divisibility, Diophantine equations and Pythagorean triples, modular arithmetic, Euler's phi-function, solutions of congruence equations, Fermat and Mersenne primes, RSA encryption, Quadratic reciprocity, sums of squares, Gaussian integers, diophantine approximations, continued fractions, and binary quadratic forms.
The final project will be a programming assignment. Students are encouraged to learn to use PARI, which is designed for number theoretic calculations. It is now incorporated into Sage, which is also very useful. See the notes section below for links to introductory guides.
Zavosh Amir Khosravi
OLGA TAUSSKY AND JOHN TODD INSTRUCTOR IN MATHEMATICS
Serin Hong: shong2 at caltech dot edu (website)
The course grade will be based on the assignments and a final collaborative project.
Students may collaborate with classmates on assignments, but may not consult anyone else other than the TA or the instructor. They also may not consult internet resources such as online forums. If using a theorem not proved in class, the student must supply their own proof.
|Notes from Prof. Ramakrishnan's course|
|Introduction to Sage, Sage Quick Reference|
|04/03||GCD, Euclidean Algorithm, solvability of ax+by=c|
|04/05||The general solution of ax+by =c, Fundamental Theorem of Arithmetic|
|04/07||Unique prime factorization, Application to gcd and lcm|
|04/10||Diophantine equations, Infinitude of primes|
|04/12||Congruences, Divisibility Tests, Wilson's Theorem|
|04/14||Fermat's Little Theorem, Euler's φ-function, Euler's Theorem|
|04/17||Chinese Remainder Theorem, Multiplicative Property of φ|
|04/19||Lifting solutions mod pⁿ, Hensel's Lemma, p-adic integers|
|04/21||Polynomial equations, Roots of polynomials modulo p|
|04/24||Multiplicative orders, Mersenne primes, Fermat primes|
|04/26||Fermat numbers, Primitive roots, Existence modulo pⁿ|
|April 18||Problem Set 1||Problem Set 1 Solutions|
|April 25||Problem Set 2||Problem Set 2 Solutions|
|May 2||Problem Set 3||Problem Set 3 Solutions|
|May 9||Problem Set 4||Problem Set 4 Solutions|
|May 16||Problem Set 5||Problem Set 5 Solutions|
|May 23||Problem Set 6||Problem Set 6 Solutions|
|June 16 (June 9 for seniors)||Final Project|