Sophie Marques

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MATH-UA 248.001:
Number theory

Contents

  • Divisibility theory
    • Interlude on natural numbers, induction and well ordering
    • Divisibility
    • Euclidean division, algorithm
    • The fundamental theorem of arithmetic
  • Arithmetic functions
    • Definitions, examples
    • Euler function
    • Convolution, Möbius inversion
  • Congruences
    • Motivation
    • (Z/nZ , + , .)
    • Congruences and polynomials
    • Linear congruence
    • Group of units
    • Quadratic congruences
  • Continued fraction
    • Generality
    • Continued fractions for quadratic irrationals
    • Pell's equation
  • Gaussian integers
    • Basic properties
    • Fermat's two square theorem
    • Pythagorean triples
    • Primes of the form 4n+1
  • Other diophantine equation
    • Fermat's equation
    • Mordell's equation
    • The 'abc'-conjecture
    • Mordell's conjecture
    Syllabus

Schedule:

MW 3:30PM- 4:45PM

Location:

CIWW 201

Office hours:

Wednesday 2:00-3:00pm or by appointment.

Grading:

  • Homework: 10%
  • Quizes: 10%
  • Midterm : 30%
  • Final : 50%

Recommended book:

I.N. Herstein- Topics in Algebra, 2nd edition, 1975


Notes of the course

Notes.pdf


Exam basics and advice

Help.pdf

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Quizzes 2014

Midterm 2014