This module is an elementary introduction to number theory, whose aim is to familiarize the audience with the classical ideas and methods of the field, as well as some of its more recent applications especially in cryptography and related technologies. Topics covered in this module include prime numbers and their distribution, the fundamental theorem of arithmetic, modular arithmetic, primitive roots, finite fields, applications to modern cryptography (e.g. RSA cryptographic platform), discrete logarithm problem, applications to error correcting codes, and quadratic reciprocity. The second part of the module is more topical and deals with more advances topics such as Riemann Zeta function, primes in arithmetic progressions, continued fractions and diophantine approximations, Pell’s equation, Minkowski’s Geometry of numbers, the Gauss circle problem and related lattice point counting problems.

Here is a rough outline of the course, subject to minor changes.

1 Divisibility in integers:

2 The distribution of prime numbers

3 The greatest common divisor

4 Primitive roots, applications to cryptographic platforms

5 Varieties over finite fields

6 Fourier analysis on finite abelian groups

7 The quadratic reciprocity

8 Arithmetic of Gaussian integers

9 Primes in arithmetic progressions

10 Geometry of Numbers

11 Diophantine approximation

12 Arithmetic functions

Class hours:

The class will meet on in Lecture Hall of Research I.

Please familiarize yourself with new Covid-19 related regulations regarding class attendance.

Tutorials:

Office hours:

Office hours for this course will be online on Zoom. Time will be announced later.

Lecture notes

Week 1 ( 1 September 2020)

Week 2-3

Section 5

Section 6

Section 7

Complete lecture notes (Note: section 12 is optional reading)

Problem Sets

Problem set 1

Problem Set 2

Problem Set 3

Problem Set 4

Problem Set 5