Prime numbers. How to appoint them?
Prime numbers are natural numbers greater than 1 that have exactly two natural divisors: 1 and itself. The set of all prime numbers is marked with the symbol ℙ. Which numbers are prime numbers? What are them quickly set?
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1. When did natural numbers appear?
It is difficult to clearly define when the natural numbers first appeared. Presumably simple numerical operations have always been with people. Already prehistoric hunters distinguished between concepts like zero, one, two and many, and then used these skills to describe the number of animals hunted.
In addition, it has been proven that many animal species readily recognize the numbers of small harvests, so this is not only the domain of mankind, although in humans these skills have developed on a very large scale.
Man naturally developed with increasing numeracy because he needed it for everyday life. On the one hand, because the tribes grew, but also developed in techniques of breeding, hunting and trading, which required precise manipulation of ever greater numbers. Greater natural numbers were also useful to man in determining the size of hostile tribes and the resources that could be taken from them.
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Very often, the trophies obtained were a common good, as independent hunting was a thing of the past, and groups of looters and hunters worked together to provide food and well-being for families. So it became natural to ask how to fairly divide the acquired resources.
It turned out that some sets can be divided into equal parts so that each set contains an element of each kind. According to historical evidence, small numbers describing the size of such sets (which we now call prime numbers) were known to the people of today's Congo even 20,000 years ago.
2. Which numbers are prime?
Prime numbers divide by 1 and by themselves. In the range from 1 - 100 we can list the following numbers:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.
The list of these numbers does not include the number 4, for example, because it has 3 divisors: 1, 2 and 4. Similarly, the number 6 has 4 and these divisors are: 1,2,3 and 6.
Natural numbers that are greater than 1 and not prime are called composite numbers, so 4 and 6 are composite numbers.
It is worth adding that the numbers 0 and 1 are neither complex nor prime.
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3. Twin numbers in prime numbers
Among the prime numbers we can distinguish the so-called twin numbers - two prime numbers whose difference is 2. For example, they are:
- 3 and 5;
- 11 and 13;
- 59 and 61.
4. Properties of prime numbers
- The smallest natural divisor of a natural number greater than one, other than 1, is a prime number;
- A never-complete set does not contain all prime numbers, as Euclid had already proved.
- All natural numbers greater than 1 can be uniquely written as the product of a finite nondecreasing sequence of some prime numbers. The aforementioned Euclid was able to prove this theorem, he even created the tools necessary for this, but only Gauss did it in the end. This theorem compares prime numbers to the atoms out of which the remaining numbers are constructed by multiplication.
5. How to determine prime numbers?
Prime numbers are used quite commonly in mathematics, mainly in fields related to algebra, number theory, algorithmics, and information processing. Many mathematicians are more interested in finding them than the prime numbers themselves.
To determine them, one can use an algorithm known as Erastothenes' sieve, which is one of the oldest methods of searching for prime numbers. The oldest known reference to this topic appears in the work of Nikomachos of Gerasa from the period around 60-120 CE.
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This method allows you to quickly find all prime numbers that are less than a given limit. It consists in systematically removing from the list of potential primes those values which, after checking, are not prime.
Erastothenes' sieve is the fastest search algorithm for all primes less than or equal to a given value. However, it is not suitable for searching for large primes. This is because building a sequence of prime numbers incrementally is not appropriate when we are only interested in the largest of them.
Many mathematicians spend a lot of their time looking for a pattern that could organize the set of prime numbers. It seems, however, that although the distribution of these numbers is fairly even, it is by no means predictable.