MAU23101 - Introduction to number theory

Course information

Suggested textbooks

• A course in computational number theory, by D. Bressoud and S. Wagon
• The higher arithmetic, by H. Davenport (up to chapter VI)
• A classical introduction to modern number theory, by K. Ireland and M. Rosen (up to chapter 5)
• Primes of the form x²+ny², by D. Cox (up to section 2.B)

Recorded lectures

• Chapter 0: Introduction - Diophantine equations
  Video - Notes (Complete)


• Chapter 1: Divisibility and factorisation
  Notes (Complete)
  Videos:
  • Euclidean division
  • Divisibility, gcd, lcm
  • The fundamental theorem of arithmetic
  • Practical factoring
  • Valuations
  • Multiplicative functions
  • The Diophantine equation ax+by=c
  • Complements on primes


• Chapter 2: Congruences and Z/nZ
  Notes (Complete)
  Videos:
  • The set Z/nZ
  • The ring Z/nZ
  • Application to Diophantine equations
  • Invertible elements
  • Chinese remainders
  • Additive and mutiplicative order
  • Primitive roots


• Chapter 3: Power residues, Legendre symbols, and quadratic reciprocity
  Notes (Complete)
  Videos:
  • Powers in Z/pZ
  • The Legendre symbol
  • Proofs part 1/3
  • Proofs part 2/3
  • Proofs part 3/3: quadratic reciprocity


• Chapter 4: Sums of squares
  Notes (Complete)
  Videos:
  • Introduction
  • The Gaussian integers
  • Arithmetic with the Gaussian integers
  • Classification of the Gaussian primes
  • Conclusion


• Chapter 5: Binary quadratic forms
  Notes (Complete)
  Videos:
  • Introduction, representation by a form
  • Equivalent forms, discriminant
  • Equivalence vs. representability
  • Reduced forms
  • The class number
  • The class number one case


• Chapter 6: Continued fractions
  Notes (Complete)
  Videos:
  • Concept and notations
  • Continued fraction expansion of a real number
  • Convergents
  • Convergence of continued fractions
  • Diophantine approximation
  • Continued fractions attached to quadratic irrationals
  • The Pell-Fermat equation

Homework

Exam