Number theory

Number theory is a branch of mathematics that studies integers. Number theory is a fascinating field, because many questions involving integers are very difficult to solve even if they seem simple at first glance.

As an example, consider the following equation: \[x^3 + y^3 + z^3 = 33\] It is easy to find three real numbers $x$, $y$ and $z$ that satisfy the equation. For example, we can choose \[ \begin{array}{lcl} x = 3, \\ y = \sqrt[3]{3}, \\ z = \sqrt[3]{3}.\\ \end{array} \] However, it is an open problem in number theory if there are any three integers $x$, $y$ and $z$ that would satisfy the equation [6].

In this chapter, we will focus on basic concepts and algorithms in number theory. Throughout the chapter, we will assume that all numbers are integers, if not otherwise stated.