Department of Mathematics & Statistics
189-346A / 377B
(The chapter numbers refer to Leveque, "Fundamentals of Number Theory").
- Jan 3-Jan 5: (Chapters 1,2). Overview of the course. Basic
properties of the integers. The GCD and the Euclidean algorithm.
Proof of the fundamental theorem of arithmetic. Application to
Jan 8-Jan 12: (Chapter 2).
Modular arithmetic. Wilson's Theorem.
Fermat's Little Theorem.
Primality testing and factorisation.
- Jan 15-Jan 19: (Sections 3.1, 3.2, 4.1, 4.2).
The structure of (Z/nZ)x. The Euler phi-function.
Application to primality testing and cryptography.
Congruence equations. The Chinese remainder theorem.
Structure of the polynomial ring $Z/pZ[x]$.
- Jan 22-Jan 26:
Congruence equations, continued.
- Jan 29- Feb 2 : (Chapter 4,5).
Discrete logarithms. The Diffie-Hellman key exchange.
- Feb 5 - Feb 9:
Midterm exam on Monday.
The p-adic logarithm, and some review.
- Feb 12 - Feb 16: (Chapter 6).
The law of quadratic reciprocity.
- Feb 19- Feb 23:
Study break. A good time to work seriously on
your project! In particular, your topic should have been chosen by then.
- Feb 26 - March 2 :
(Chapter 6). Introduction to analytic number theory.
Euler's proof of the infinitude of primes.
The sieve of Eratosthenes.
Dirichlet's theorem on primes in arithmetic progressions.
- March 5 - March 9: (Chapter 6).
Dirichlet's Theorem, continued.
- March 12 - March 16:
(Sec. 2.2 and Chapter 8).
Quadratic equations and
quadratic fields: Legendre's Theorem.
- March 19 - March 23:
(Chapter 7) Sums of squares.
- March 26 - March 30: (Chapter 8, 9).
Diophantine approximation and continued fractions.
- April 2 - April 4: Continued fractions and Pell's equation
- April 11: Review.