# Number theory I.c - Combinatorial number theory

## Summary

This is an introductory course to combinatorial number theory. The main objective of this course is to learn how to use combinatorial, topological, and analytic methods to solve problems in number theory.

## Content

Combinatoics is the study of discrete structures, and number theory the study of arithmetic. At the interface of these domains we encounter the fascinating field of combinatorial number theory (sometimes also called arithmetic combinatorics), which concentrates on the study of arithmetic structures. Two of the key areas that we will focus on during this course are Ramsey theory, encompassing Schur's Theorem, van der Waerden's Theorem, and the Erdos-Szekeres Theorem, and additive combinatorics, featuring Hindman's Theorem and Roth's Theorem. This course will help foster both your combinatorial and analytic intuition in mathematics and will allow you to visualize the natural numbers in new and complex ways. We will also discover connections to subjects that you have seen before, such as number theory, analysis, group theory and set theory.

## Keywords

Combinatorial number theory, additive combinatorics, arithmetic combinatorics, additive number theory, Ramsey theory, discrete mathematics

## Learning Prerequisites

## Required courses

First year math courses

## Learning Outcomes

By the end of the course, the student must be able to:

- Apply tools from combinatorics, probability theory, and discrete harmonic analysis to solve problems in number theory
- Prove results in additive combinatorics and Ramsey theory
- Transpose ideas from analysis and number theory

## Transversal skills

- Use a work methodology appropriate to the task.
- Demonstrate a capacity for creativity.
- Demonstrate the capacity for critical thinking
- Continue to work through difficulties or initial failure to find optimal solutions.

## Teaching methods

Weekly leactures, weekly exercises classes, weekly homework assignments

## Expected student activities

Paritcipation in lectures and exercise calsses

## Assessment methods

25% written homework assignments, 75% written final exam

## Supervision

Office hours | No |

Assistants | Yes |

Forum | Yes |

## In the programs

**Semester:**Fall**Exam form:**Written (winter session)**Subject examined:**Number theory I.c - Combinatorial number theory**Lecture:**2 Hour(s) per week x 14 weeks**Exercises:**2 Hour(s) per week x 14 weeks**Type:**optional