Math 40520 Theory of Numbers (Fall 2015)
Undergraduate number theory

 

The official syllabus I distributed during the first lecture can be found here. Below you may find a continuously updated list of topics covered/to be covered in the course, together with dates when the topics were covered.

Topics
  1. The Euclidean algorithm
    1. Two variable linear and quadratic diophantine equations using initial guesses.
    2. The Euclidean algorithm
    3. Bezout's identity
    4. Linear diophantine equations
  2. Congruences
    1. Basics
    2. Applications to diophantine equations
    3. Fermat's little Theorem
      1. Euler's theorem and Fermat's little theorem
      2. Polynomials mod p and Wilson's theorem
    4. Primitive roots
      1. Multiplicative orders mod n
      2. Primitive roots mod p and applications
      3. Primitive roots mod p^n
    5. Polynomials and congruences
      1. Polynomials mod p and Fermat's little theorem
      2. Roots of polynomials mod p^n and Hensel's lemma
    6. Congruences mod n and simultaneous congruences
      1. The Chinese Remainder Theorem
      2. Roots of polynomials mod n
      3. A formula for Euler's function
    7. Quadratic residues
      1. The Legendre symbol (October 5-7)
      2. Special examples and Gauss' lemma (October 9)
      3. Sums of Legendre symbols (October 12-16))
      4. Quadratic forms mod p (October 26)
      5. Quadratic reciprocity (October 28)
      6. Public key cryptography: Rabin, RSA and ElGamal (October 30)
      7. Zero knowledge proofs (November 2)
  3. Primes and combinatorics
    1. Overview (November 4th)
    2. Integer primes
      1. Infinitely many primes (November 6th)
      2. Sum of reciprocals of primes (November 6th)
      3. Special types of primes (November 6th)
      4. Primality testing (November 9th and 11th)
      5. Quadratic expressions for primes (November 11th and 13th)
      6. Combinatorial expressions and primes (November 13th and 16th)
    3. Irreducible polynomials mod p
      1. Testing irreducibility of a polynomial mod p (November 18th)
      2. Counting irreducible polynomials mod p (November 18th)
      3. Dirichlet series and the Mobius inversion formula (November 18th)
      4. Many examples of Dirichlet series (November 23rd)
  4. Gaussian integers and other Euclidean domains
    1. Euclidean domains
      1. Euclidean functions (November 30thrd)
      2. Quadratic rings of integers (December 2nd)
      3. Unique factorization in the Gaussian integers (December 4th)
    2. How many ways to write an integer as a sum of two squares (December 7th)
    3. Pell's equation and descent (December 9th)

The design of this webpage is based on the MIT course web page template.