Prime numbers are one of the most interesting mathematical phenomena that has attracted the attention of scientists and ordinary citizens for more than two millennia. Despite the fact that we now live in the age of computers and the most modern information programs, many mysteries of prime numbers have not yet been solved, there are even those that scientists do not know how to approach.

Prime numbers are, as is known from the course of elementary arithmetic, those natural numbers that are divisible without remainder only by one and itself. By the way, if a natural number is divisible, in addition to those listed above, by another number, then it is called composite. One of the most famous theorems states that any composite number can be represented as the only possible product of prime numbers.

Some interesting facts. First, the unit is unique in the sense that, in fact, it does not belong to either prime or composite numbers. At thatAt the same time, in the scientific community it is still customary to attribute it to the first group, since formally it fully satisfies its requirements.

Secondly, the only even number in the "prime numbers" group is, of course, two. Any other even number simply cannot get here, since by definition, in addition to itself and one, it is also divisible by two.

Prime numbers, the list of which, as mentioned above, can begin with one, are an infinite series, as infinite as the series of natural numbers. Based on the fundamental theorem of arithmetic, one can come to the conclusion that prime numbers are never interrupted and never end, since otherwise the series of natural numbers would inevitably be interrupted.

Prime numbers do not appear randomly in natural numbers, as it might seem at first glance. After carefully analyzing them, you can immediately notice several features, the most curious of which are associated with the so-called "twin" numbers. They are called that because, in some incomprehensible way, they ended up next to each other, separated only by an even delimiter (five and seven, seventeen and nineteen).

If you look closely at them, you will notice that the sum of these numbers is always a multiple of three. Moreover, when dividing by three, the left brother always has a remainder of two, and the right brother always has a remainder of one. In addition, the very distribution of these numbers over the natural series can bepredict if we represent this entire series in the form of oscillatory sinusoids, the main points of which are formed by dividing numbers by three and two.

Prime numbers are not only an object of close scrutiny by mathematicians around the world, but have long been successfully used in compiling various series of numbers, which is the basis, including for ciphergraphy. At the same time, it should be recognized that a huge number of mysteries associated with these wonderful elements are still waiting to be solved, many questions have not only philosophical, but also practical significance.