We know that a positive integer is prime if it is divisible only by itself beyond 1. Prime numbers play a fundamental role in arithmetic, analogous to the role of atoms in the structure of matter, that is, integers that are not numbers. primes may be expressed as the product of prime numbers. Therefore, any integer greater than 1 is either a prime number or is expressed as a product of prime numbers.

Although the notion of prime number in the above sense seems obvious, in general, questions involving prime numbers are not easy to answer at the current stage of mathematics. For example, every odd number is expressed in form 4.*x* + 1 or 4*x* + 3; so we ask which are the cousins of form 4*x* + 1 and what are the cousins of form 4*x* + 3. *If we generate the numerical sequences of the form above, replacing x with positive integers, the resulting sequences will have an infinite number of prime numbers.*?

Euclid of Alexandria (about 300 BC) gave a very ingenious demonstration that there is an infinite number of prime numbers. The same argument given by Euclid can be used to demonstrate the infinity of form 4 cousins.*x* + 3. Since 2 is the only even prime, the set of odd prime numbers is divided into two families:

i) 5, 13.17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137, 149, 157, 173…;

ii) 3, 7, 11, 19, 23, 31, 43, 47, 59, 67, 71, 79, 83, 103, 107, 127, 131 139, 151,…

where the first sequence of numbers refers to the cousins of form 4*x* + 1 and the second to form 4 cousins*x* + 3. Let's demonstrate that there are infinite type 4 cousins*x* + 3 using Euclid's method which demonstrates the existence of infinite cousins.

In fact, suppose there were a finite number of prime numbers of the form 4*x* + 3; let's name them *what*_{1}, *what*_{2}, *what*_{3},… , *what*_{no}. Consider the positive integer:

N = 4* what*_{1}.*what*_{2}.*what*_{3}… *what*_{no} - 1 = 4 *what*_{1}.*what*_{2}.*what*_{3}… *what*_{no} - 4 + 3 = 4 ( *what*_{1}.*what*_{2}.*what*_{3}… *what*_{no}- 1) + 3

and let N = *r*_{1}.*r*_{2}.*r*_{3}… *r*_{M} its decomposition into prime numbers. Since N is an odd integer, it follows that *r*_{k} is different from 2 for all *k*, and each *r*_{k} it is therefore of the form 4*x* +1 or 4*x* + 3. However, the product of two or more integers of form 4*x* +1 also results in an integer like this, that is,

(4*m* + 1).(4*no* + 1) = 16*mn* + 4*m* + 4*no* + 1 = 4(*mn* + *m* + *no*) + 1 = 4*z* + 1.

Thus, it follows that N has at least one prime factor of form 4*x* + 3, say *r*_{i }= 4*x* + 3.

Now we claim that *r*_{i} is not an element of our original finite list of prime numbers: *what*_{1}, *what*_{2}, *what*_{3},… , *what*_{no}. In fact, otherwise we would have *r*_{i} =* what*_{j}for some cousin *what*_{j }from our original list of cousins and then* r*_{i} would divide the product *what*_{1}.*what*_{2}.*what*_{3}… *what*_{no}. On the other hand, being *r*_{i} a factor of N, *r*_{i} divide N - 4* what*_{1}.*what*_{2}.*what*_{3}… *what*_{no} = -1. Soon, *r*_{i} divide -1. Therefore, we conclude that there are an infinite number of cousins of form 4*x* + 3 therefore assume that there is a finite number of prime of the form 4*x* + 3 leads us to a contradiction.

The next question would be: *there is an infinite number of cousins of the form *4*x + *1? The answer is yes, but we must use another argument. A similar situation arises with respect to the number sequences of form 6.*x* +1 and 6*x* + 5.

Note that if we generate the number sequence of form 4*x* + 3:

3, 7, 11, 15, 19, 23, 27, 31, 35, 39, 43, 47, 51, 55, 59, 63, 67, 71, 75, 79, 83, 87,… ,

The difference between a term in the sequence and its predecessor is always equal to 4.

The same is true of form 4 sequences*x* + 1, 6*x* + 1 or 6*x* + 5. In fact, we have the following definition: “a **Arithmetic Progression** is a sequence of integers in which the difference between a term (from 2^{O}.) and the antecedent term is always the same ”.

*Could the fact that there are infinite cousins in some arithmetic progressions, such as those mentioned above, be generalized*?

Note that the progressions cited above are as follows. *B* + *ax* Where *The* and *B* are fixed and *x* = 0, 1, 2, 3, 4, 5,…, that is, they are of the form

*B*, *B* + *The*, *B* + 2*The*, *B* + 3*The*, *B* + 4*The*,…

If *The* and *B* have a common factor, so arithmetic progression does not contain prime numbers, because every element of progression has that factor. For example, consider the arithmetic progression given by 6 + 2*x*, this is,

6, 8, 10, 12, 14, 16, 18, 20, 22, 24,…

Note that 2 is a common factor of 2 and 6, and every progression term has the number 2 as a factor. This fact suggests that we should consider progressions *B* + *ax* on what *The* and *B* be prime to each other to get an infinite number of prime as specified *B* + *ax*. It seems that the mathematician Legendre was the first to realize the importance of this question, and in 1808 published the following conjecture:*If a ≥ *2* and b *0* are positive integers and prime to each other, so there is a plethora of prime numbers in arithmetic progression*

*B*, *B* + a, *B* + 2a, *B* + 3*The*,… ”

This conjecture became a major theorem and was demonstrated by Dirichlet in 1837. This result was monumental for a number of reasons. Dirichlet relied on Euler's original idea to demonstrate the infinity of cousins. Revolutionary analytical methods such as infinite series, series convergence, boundaries, logarithms, etc., and many other concepts hitherto foreign to whole number theory were used. Dirichlet's demonstration is regarded as one of the first important applications of analytical methods in number theory and provided new lines of development. The ideas underlying Dirichlet's arguments are very general in character and were instrumental in developing the subsequent work of applying analytical methods in number theory.

In 1949 the mathematician Atle Selberg gave an elementary demonstration of Dirichlet's theorem, analogous to his earlier demonstration of the prime number theorem.

Dirichlet also demonstrated that any quadratic form in two variables, that is, any form of the type *ax ^{2 }*+

*bxy*+

*cy*Where

^{2 }*The*,

*B*, c, are cousins to each other, generate an infinity of cousins. Not much is known about other ways that generate infinite prime numbers.

On the other hand, we can demonstrate that there is no arithmetic progression in which all terms are prime numbers. Until the last century, an old open problem was to determine an arbitrarily long but finite arithmetic progression in which all terms were prime numbers.

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