This course offers a fascinating and in-depth journey into the heart of number theory. Throughout the course, we will delve into the depths of natural and integer numbers, uncovering surprising properties and complex relationships between them.

We will begin with a precise definition of the concept of division, explore its intriguing properties, and prove the fundamental theorem of arithmetic, which serves as a cornerstone in number theory. From there, we will transition to the mysterious world of prime numbers, define them formally, and reveal central theorems that illuminate their unique nature.

We will learn and practice advanced algorithms for finding prime numbers and sophisticated methods for determining primality, while understanding their importance in modern cryptography and information security.

In the final part of the course, we will delve into modular arithmetic, which acts as a kind of "arithmetic of remainders". We will explore how this calculation is applied in everyday life, from determining the days of the week for future events to planning advanced communication systems. We will conclude by presenting Fermat's famous theorem and its proof, emphasizing its broad implications in modern number theory.

Throughout the course, we will develop advanced mathematical thinking, practice formal proof techniques, and be exposed to practical applications of number theory in diverse fields such as computer science, encryption, and signal processing.

This course is designed not only for budding mathematicians but for anyone interested in deeply understanding the mathematical structures underlying modern technology and developing advanced analytical thinking.

* The course material will be tailored to the age of the learners.