We will now make a foray into one of the oldest and most interesting areas of Number Theory: *the distribution of prime numbers*. This inquiry has fascinated the minds of men since classical antiquity: How do prime numbers distribute themselves over whole numbers?

Studying the distribution of prime numbers developed the theory of functions of a complex variable, in particular the theory of integer functions. During the development of this investigation, deep methods of algebra and analysis have emerged, but they do not always produce the expected success. On the other hand, some of the most important results can be obtained through surprisingly simple but ingenious reasoning, such as Euclid's demonstration of the infinity of the set of prime numbers.

First we need to know what a prime number is, in all respects of number theory, the essential notion. Given two integers, their sum, their difference, and your product are also whole numbers. However, the quotient of dividing one integer by another may not result in an integer, for example, the result of dividing integer 5 by integer 3 is not an integer. Efforts to construct sets that would yield results from the desired operations led mathematicians to successive generalizations of the concept of number. For example, if we consider the set of rational numbers, that is, the fractions *a / b*, Where *The* and *B* are integers, and *B* ¹ 0, so the division quotient is always defined, ie (*a / b*) ç*/ d*) = (*ab*)*/*(*CD*). However, data **the integers The and B, if there is an integer what such that a = bq we say that The is divisible by B, or that B divide The**. The number

*B*is a number divisor

*The*is the number

*The*is a multiple of the number

*B*. We often indicate the fact that

*B*divide

*The*in the following way:

*B*½

*The*. For example, 2½4 (read 2 splits 4), 4 is a multiple of two, and 2 is a divisor of 4.

Every positive integer *The* which is greater than 1 has two obvious dividers, 1 and itself *The*. If beyond these dividers the integer *The* own another divider let's say *B*, 1< *B* < *The*, then *The* It is called a compound number. Otherwise the integer *The* is called *a prime number*, or simply *a cousin*. For example, 2, 3, 5, 7, 11, 13, 17 are prime numbers because they have exactly two dividers. The number 6 has as dividers 1, 2, 3 and 6 and is therefore called *a compound number*. Therefore, except for 1, the natural numbers regarding their divisibility behavior are divided into two numerical sets:* Prime numbers* and the *compound numbers*.

When we multiply prime numbers, we get a compound number and, conversely, when we isolate prime dividers from a number *The*we represent *The* as a product of prime factors, i.e. For example, the number 90 is divisible by 2, and so we get: 90 = 2 x 45. In turn 45 is divisible by 3 and then 45 = 3 x 15. If we continue this process, we get: 90 = 2 x 3 x 3 x 5.

An interesting question is whether this decomposition is unique. The answer is yes, that is, any positive integer can be represented as a product of prime numbers, and this representation is unique unless of the order of factors. For this reason prime numbers are often called integer building blocks.

We observe that there are many sets with addition and multiplication operations where the decomposition into prime factors is not unique, we will return to this issue in other columns.

The representation of integers as a product of cousins has long been seen as an obvious fact, but mathematician Gauss has demonstrated this statement in his famous work. *Disquisitiones Arithmeticae* 1801. This statement is known as the fundamental arithmetic theorem (APT), or *unique factorization*.

This theorem shows that prime numbers form a multiplicative basis. Knowing some of the properties of this base is very important, because this is equivalent to knowing some properties of prime numbers. The first question that arises is related to the infinity of prime numbers, ie is there an infinite number of prime numbers? The answer is yes and this theorem was demonstrated by Euclid:

“*Suppose that the set of prime numbers, P, be finite. Let r be the exact number of prime numbers, ie the cardinality of the set P. In this case_{ } , … up until which is the largest (and last) prime number. Thus we emphasize that the set P contains all existing prime numbers. Consider now a new integer n = TFA states that it cannot be prime factored, n =_{ }where the cousins are elements of the set P and k> *1

*. Follow that*½

_{ }*where is a cousin of the set*

**P**. Therefore for some j where 1 £ j £ r. Consequently*½*

*. Thus*

*½*

*huh*

*½*

*; soon*

*½*

*n -*1

_{ }. On the other hand, n - =*and this way,*

*½*

*no*

*- =*1

*, that is,*

*½1*

*, contrary to the definition of prime number. This contradiction shows that no finite set*

**P**can contain all prime numbers.Another very interesting issue related to prime numbers concerns the frequency of occurrence of prime numbers in their natural order of appearance in the set of numbers. *natural. *In other words, how many primes are there between the natural numbers 1, 2,…, *X* When *X* is it a big number? This number, which generally depends on *X*, is denoted by p (*X*), ie p (*X*) is the number of cousins less than or equal to *X*. For example, p (4) = 2, p (7) = 4.

The first conjecture about the magnitude of p (*X*) as a function of *X *was made by mathematicians Gauss and Legendre independently at the end of the 18th century. Based on extensive calculations, Gauss and Legendre made the conjecture that

P (*X*) ~ *X */ log *X*,

ie p (*X*) is approximately *X* / log *X* When *X* It is a very large natural number. This conjecture suggests that the quotient of p (*X*) per *X* / log *X* tends to limit 1 when *X* tends to infinity. This formulation is known as the Prime Number Theorem and was independently demonstrated by de la Vallée - Poussin and Hadamard in 1896 using powerful new analytical methods of complex variable theory. In 1948 Atle Selberg and Paul Erdös gave another demonstration without the use of complex variable theory. Many mathematicians contributed to the demonstration of the Prime Number Theorem: Riemann, Mertens, von Mangoldt, Hadamard, de la Vallée-Poussin, Tchebychev, etc. This was one of the greatest achievements of nineteenth-century mathematics and gave rise to the Analytical Number Theory.

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