McGill University

Department of Mathematics & Statistics

Number Theory

189-346A / 377B

Detailed Syllabus

(The chapter numbers refer to Leveque, "Fundamentals of Number Theory").
  1. 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 Diophantine equations.

  2. Jan 8-Jan 12: (Chapter 2). Modular arithmetic. Wilson's Theorem. Fermat's Little Theorem. Primality testing and factorisation.

  3. 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 $Z/pZ[x]$.

  4. Jan 22-Jan 26: Congruence equations, continued. p-adic numbers.

  5. Jan 29- Feb 2 : (Chapter 4,5). Discrete logarithms. The Diffie-Hellman key exchange. Power residues.

  6. Feb 5 - Feb 9: Midterm exam on Monday. The p-adic logarithm, and some review.

  7. Feb 12 - Feb 16: (Chapter 6). The law of quadratic reciprocity.

  8. 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.

  9. 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.

  10. March 5 - March 9: (Chapter 6). Dirichlet's Theorem, continued.

  11. March 12 - March 16: (Sec. 2.2 and Chapter 8). Quadratic equations and quadratic fields: Legendre's Theorem.

  12. March 19 - March 23: (Chapter 7) Sums of squares.

  13. March 26 - March 30: (Chapter 8, 9). Diophantine approximation and continued fractions.

  14. April 2 - April 4: Continued fractions and Pell's equation (Chapter 9)

  15. April 11: Review.