Simple numbers: the routine of an unsolved mystery

Simple numbers are one of the most interesting mathematical phenomena, which attracts the attention of scientists and ordinary citizens for more than two millennia. Despite the fact that now we live in the age of computers and the most up-to-date information programs, many riddles of prime numbers have not been solved so far, there are even those to which scientists do not know how to approach.

Simple numbers are, as is known even from the course of elementary arithmetic, those natural numbers that divide without a remainder only by one and itself. By the way, if a natural number is divided, besides the above, into another number, then it is called a composite number. One of the most famous theorems states that any composite number can be represented as the only possible product of prime numbers.

A few interesting facts. First, the unit is unique in that it does not, in fact, belong to either simple or composite numbers. At the same time in the scientific community it is still customary to refer it to the first group, since formally it fully meets its requirements.

Secondly, the only even number that is woven into the group "prime numbers" is, of course, two. Any other even number can not get here, because by definition, apart from itself and a unit, it is divided into two.

Simple numbers, the list of which, as was indicated above, can be started from unity, represent an infinite series, as infinite as a series of natural numbers. Relying on the basic theorem of arithmetic, one can come to the conclusion that prime numbers are never interrupted and never end, since otherwise a number of natural numbers would inevitably be interrupted.

Simple numbers do not appear in a natural series randomly, as it may seem at first glance. After carefully analyzing them, you can immediately notice several features, the most interesting of which are related to the so-called "twin" numbers. They are called so because in some incomprehensible way they were in the neighborhood with each other, separated only by an even delimiter (five and seven, seventeen and nineteen).

If you look closely at them, you can see that the sum of these numbers is always a multiple of three. Moreover, when dividing a left-wing colleague into a trio, the remainder always has a deuce, and the right-hand one has one. In addition, the very distribution of these numbers along the natural series can be predicted if we represent the entire series in the form of oscillatory sinusoids, the main points of which are formed by dividing the numbers by three and two.

Simple numbers are not only an object of close scrutiny by mathematicians around the world, but have long and successfully been used in the compilation of various series of numbers, which is the basis, including for ciphering. At the same time, it should be recognized that a huge number of puzzles related to these remarkable elements are still waiting for their clues, many questions have not only a philosophical but also a practical meaning.