Ma 007/107:  Number Theory for Beginners (Spring 2016-17)

ANNOUNCEMENTS

Registration for Spring term opens Thursday, February 23, 2016.

This page was last updated:  .


COURSE DESCRIPTION

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.


SCHEDULE

MWF 13:00-13:55, 159 Sloan.


INSTRUCTORS

Zavosh Amir Khosravi
OLGA TAUSSKY AND JOHN TODD INSTRUCTOR IN MATHEMATICS
258 Sloan
626-395-4339


TA's

Serin Hong: shong2 at caltech dot edu (website)


OFFICE HOURS

Serin: Mondays 3-4pm, Sloan 382

Zavosh: Fridays 3-4pm (or by appointment), Sloan 258


POLICIES

Grades

The course grade will be based on the assignments and a final collaborative project.

Homework Policy

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.


TEXTBOOKS

The Higher Arithmetic, H. Davenport (Eighth Edition), ISBN: 0521722365


NOTES

Date Description
  Lecture Notes
  Notes from Prof. Ramakrishnan's course
  Introduction to Sage, Sage Quick Reference

Topics

Date Description
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


HOMEWORK

Due Date Homework Solutions
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  
     
     
     

EXAMS

 


READING