Prime number is called the natural number which is divided only into unit and into itself. All other numbers, besides unit, are composite. Properties of prime numbers are studied by science under the name the theory of numbers.
Instruction
1. According to the main theorem of arithmetics, any natural number which is more than unit can be spread out to the work of prime numbers. Proceeding from it, it is possible to draw a conclusion that prime numbers represent certain "blocks" for natural numbers.
2. Operation of representation of natural number in the form of the work simple is called factorization or prime factorization. Polynomial algorithms of decomposition of numbers are unknown, but also there are no proofs that in the nature they do not exist.
3. Some cryptosystems are based on complexity of the calculations connected with factorization of numbers, for example, one of known is RSA. For quantum computers there is an algorithm Shora which allows to carry out factorization of numbers with polynomial complexity.
4. There are algorithms by means of which it is possible to carry out search and also to distinguish prime numbers. The simplest of them are Eratosthenes's sieve, Atkin's sieve, Sundaram's sieve. In practice often there is a problem not of receiving prime numbers, and checks of number on whether it is simple. The algorithms designed to solve similar problems are called as tests of simplicity.
5. Still Euclid proved the fact that prime numbers exists infinitely much. The essence of its evidence produced in the book of "Beginning" consists in the following. Let there is a final number of prime numbers. Let's multiply them then we will add to them unit. The number which turned out cannot be divided on any prime number from final set without the rest (it will be equal to 1). In that case this number is divided into a prime number which is not a part of the presented final set. In addition, exist as well other mathematical proofs of infinity of prime numbers.