Introduction


Welcome to the fascinating world of number theory! Number theory is a branch of pure mathematics that deals with the properties and relationships of numbers. It's one of the oldest branches of mathematics, dating back to ancient times, and it continues to be a vibrant and active field of research today.

At its core, number theory focuses on the study of integers and their properties. Integers are whole numbers, including positive numbers, negative numbers, and zero. Number theorists explore various aspects of these integers, seeking patterns, relationships, and structures within the numerical realm.

One of the central themes in number theory is prime numbers. Prime numbers are integers greater than 1 that have no positive divisors other than 1 and themselves. They play a fundamental role in number theory and are often considered the building blocks of the integers. The distribution and properties of prime numbers have intrigued mathematicians for centuries and continue to be a subject of active research.

Another key concept in number theory is modular arithmetic. This involves studying arithmetic operations like addition, subtraction, multiplication, and division, but within a fixed modulus (or remainder). Modular arithmetic finds applications in cryptography, coding theory, and various other areas of computer science and engineering.

Number theory also delves into topics such as Diophantine equations, which involve finding integer solutions to polynomial equations, and algebraic number theory, which extends the study of numbers to more general algebraic structures.

Throughout history, number theory has played a crucial role in the development of mathematics and its applications. From ancient Greek mathematicians like Euclid to modern-day researchers, mathematicians have been captivated by the beauty and depth of number theory.

In our exploration of number theory, we'll uncover the mysteries of prime numbers, unravel the secrets of modular arithmetic, and embark on a journey through the rich landscape of mathematical structures and relationships that characterize this captivating branch of mathematics. Get ready for an intellectually stimulating adventure into the realm of numbers!

Notations
Numbers
We use standard notation for various sets of numbers:
$\mathbb{N}$:= set of natural numbers $\{1,2,\ldots\}$.
$\mathbb{Z}$ := the set of integers = $\{. . . ,-2,-1, 0, 1, 2, . . .\}$,
$\mathbb{Q}$ := the set of rational numbers = $\{\frac{a}{b} : a, b \in \mathbb{Z}, b \ne  0\}$
$\mathbb{R}$ := the set of real numbers,
$\mathbb{C}$ := the set of complex numbers.

$\mathbb{N} ⊂ \mathbb{Z} ⊂ \mathbb{Q} ⊂ \mathbb{R} ⊂ \mathbb{C}$

$\#S$: the number of elements is a set $S$.
$Re(C)$: the real part of a complex number $C$.
$|x|$ is the absolute value of $x$.

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