18.781 (Introduction to the Theory of Numbers)

This is an introductory course in Number Theory, covering divisibility, primes, common divisors, congruences, quadratic reciprocity, arithmetic functions, Diophantine equations, Farey fractions, continued fractions, RSA cryptography, and more.

Vital stats

Meeting times: 2:30-4, Tuesdays and Thursdays, Room 4-163.

Instructor: Ben Webster
Syllabus: here
My office: 2-380
My office hours: Tuesday 11-12, Wednesday 12:30-2:30 (or by appointment)
My email: bwebster@math.mit.edu

Stellar site: stellar.mit.edu/S/course/18/fa09/18.781
Wiki site: wikis.mit.edu/confluence/display/18DOT781fa09/Home

Problem Sets

Homework is due Thursdays at the beginning of class (it can also be slipped under the door of my office before class or submitted electronically via Stellar) and will not be accepted late barring exceptional circumstances.

Homeworks will be posted here at least a week before they are due. The first assignment will be due September 17th.

As usual, it's fine to discuss assignments with other students and work together, but you should do your write up separately.

  1. HW for September 17th
  2. HW for September 24th
  3. HW for October 1st
  4. HW for October 15th
  5. HW for October 22th
  6. HW for October 29th
  7. HW for November 5th
  8. HW for November 19th

Exams

Exams will be in class, open note and open book.

Midterm 1 Oct. 8
Solutions

Midterm 2 Nov. 12
Solutions

Final Exam Dec. 15th, 1:30pm - 4:30pm, Room 56-154

Class Schedule

Date          
Topics
 Reading
Sep. 10 (R)
Intro; Pythagorean triples
 Niven 5.3, 1.2
Sep.  15 (T) Division algorithm;
gcd's & Euclidean algorithm;
Fundamental theorem of arithmetic;
Niven 1.2, 1.3
Sep. 17 (R)
 congruences; Dirichlet's theorem;
Solving linear congruence equations;
Niven 2.1, 2.2
Sep. 22 (T)
 Fermat's little theorem; polynomial congruences;
Chinese remainder theorem
Niven 2.2, 2.3
Sep. 24 (R)
 Euler's totient function; Arithmetic functions;
Mobius inversion;
Niven 4.2,4.3
Sep. 29 (T)
 Computational techniques; primality testing;
 Niven 2.4
Oct. 1 (R)
 Multiplicative structure mod primes;
RSA cryptography
 Niven 2.5, 2.7
Oct. 6 (T)
Hensel's lemma; primitive roots Niven 2.6, 2.8
Oct. 8 (R)
 Midterm Exam I
Oct. 13 (T)
 No class; Monday schedule

Oct. 15 (R)
 Primitive roots modulo primes and composites; quadratic residues
 Niven 2.8, 3.1
Oct. 20 (T)
 Legendre symbols; quadratic reciprocity
Niven 3.2
Oct. 22 (R)
 Jacobi symbols; quadratic forms
 Niven 3.3, 3.4
Oct. 27 (T)
 Continued fractions
 Niven 7.1,7.2,7.3
Oct. 29 (R)
 Farey sequences, approximations
 Niven 6.1,6.2,7.5
Nov. 3 (T)
 Periodic continued fractions, square roots.
Niven 7.7
Nov. 5 (R)
 Pell's equation;
Niven 7.8