THE **number theory** It is the study of the natural numbers or positive integers 1, 2, 3, 4,… and their properties. Mathematician Leopold Kronecker once remarked that when it comes to mathematics, God created natural numbers, and the rest is the work of mankind. However, positive integers undoubtedly represent the first human mathematical creation, and it is difficult to imagine humanity without the ability to count.

Although natural numbers constitute, in a sense, the most elementary mathematical system, the study of their properties has exerted great fascination in the human mind since the earliest times of antiquity, challenging countless generations of mathematicians and lay people who appreciate their utterances. simple and intriguing, whose demonstrations are beyond simplicity.

Among the treasures of ancient Egypt is Papyrus Rhind describing the mathematics practiced in Egypt about 2000 years BC… Historical records show that the Sumerians developed some form of arithmetic because, around 3500 BC, they had a calendar, and around 2500 BC, developed a number system using number 60 as a base. The Babylonians followed this tradition and became calculating experts; Babylonian clay tablets dating from 2000 BC have been found with elaborate mathematical tables. By the end of the third millennium BC, cuneiform tables of Mesopotamia showed that arithmetic was already quite sophisticated.

The **numbers** they were used in business transactions for over 2000 years until one thought to study them systematically. The first scientific approach to the study of **whole numbers**, that is, the true origin of **number theory**, is generally attributed to the Greeks. Around 600 BC Pythagoras and his disciples did several interesting studies. They were the first to classify the **integers** in many ways: **pair numbers**, **odd**, **cousins**, **etc… **

In fact, it's not exactly the **natural numbers** that exert aesthetic, mystical and practical fascination, but the relationships they establish with each other. It is within these deep and subtle relationships that one finds the beauty, charm, and fascination that numbers exert across generations.

THE **number theory** It is the area of mathematics whose aim is to discover and establish the deep and subtle relationships that numbers of different kinds keep to each other. For example, consider the **squares** of natural numbers **1, 4, 9, 16, 25**,… If we take the **sum of two squares**eventually we will get another square as a result. The most famous example is: , but there are other examples: , , and many others. Such suits, (3, 4, 5), (5, 12, 13), (20, 21, 29), are called **Pythagorean suits**. On the other hand It is not a square. So follow questions like “There are endless **Pythagorean suits**? ”And“ If the answer is positive, can we find a formula that fully describes them? ”. These are some of the types of **questions** that **number theory** investigate.

THE **number theory** it is populated by a huge variety of objects: prime, square, odd and perfect numbers; sets of rational, algebraic, and transcendent numbers, some **analytical functions** quite specific such as Dirichlet series and modular forms; equations such as that of **Fermat** it's from **Pell**, elliptic curves, codes, some **geometric objects** such as lattices, Z-beams, and many others we will encounter on our journey through number theory.

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