What are the prime numbers ?

What are the prime numbers?, prime numbers, Mathematics, How to determine prime numbers, Use factorization method, Use the calculator, properties

 

Mathematics

Mathematics involves the study of many subjects including quantity such as number theory, structure such as algebra , space such as geometry and change such as mathematical analysis;  Mathematics is present in everything surrounding man and in everything he does as well, as it is the cornerstone of everything in daily life, including mobile devices, engineering in general and architecture in particular, ancient or modern, art, money and sports, The need for mathematics was based on the needs of society; The more sophisticated and complex society is, the more its needs for mathematics increase than others. For example, the needs of primitive tribes were much less than the ability to count, but they relied on mathematics to calculate the position of the sun. In this article, the question will be answered, "What are the prime numbers?" .

What are the prime numbers?

Many students at all levels of study face the question “What are the prime numbers?”, as these numbers are one of the most important concepts in mathematics that every student must understand and study  , which are defined as integers greater than 1 i.e. numbers Positive, which is divisible only by the number 1 and the number itself, and to answer briefly the question "What are the prime numbers?" It is possible to say that they are numbers that have only a pair of factors, namely: the number 1 and the number itself,  For example, the number 2 can be obtained only when the number 1 is multiplied with the number 2, so it has only a pair of factors, which are 1,2 as follows: 1 * 2 = 2, that is, there are no other two factors that can be multiplied together to get the number 2, Also, one of the most important examples of prime numbers is the number 3, which can only be obtained if the numbers 1 and 3 are multiplied with each other, that is, it also has a pair of factors, which are the number 3,1 as follows:

1*3 = 3 

The same applies to each of the following numbers: 5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83 89, 97, 101, 103, 107, 109, 113 and many more. It is worth noting that the study of prime numbers continued for thousands of years, as the scientist Euclid proved in his book The Elements, which was published approximately in 300 BC, many results regarding the prime numbers. According to several studies, the largest prime number discovered to date is 1 - 57,885,161^2, which consists of 17,425,170 numbers.

How to determine prime numbers

To answer the question "What are prime numbers?" In a practical and clear way, it is necessary to use many methods that guarantee a complete and definite answer, and here are some of the ways in which it is possible to determine whether the number is prime or not: 

Use factorization method

Factoring is one of the fastest methods used by mathematicians to determine whether a number is prime or not. Before using this method, it is necessary to know that the factors of a number are any two numbers that can be multiplied with each other to get this number, for example the factors of a number 10 is 2 and 5 because these integers can be multiplied together to get 10, in addition to that, 1 and 10 are also factors of 10 because their product is also 10, that is, the number 10 did not meet the conditions of the prime number because it has A pair of factors, and is also divisible by two numbers other than 1 and 10, that is, when answering the question "What are the prime numbers?" All numbers with more than one factor pair are always excluded.

The number can also be factored in the tree method; For example, to test the number 30, the factors of the number 30 are first determined, which are either the numbers 10 * 3 or 15 * 2, that is, there are two cases and each of them has the same result, as the analysis of the first case, which is 10 * 3, is as follows: 

First, the number 10 is factored into 2 * 5, and the number 3 has no factors, so the answer is 2 * 5 * 3 and the result is 30. 

As for the analysis of the second case, which is 15 * 2, it is as follows: the number 15 is first analyzed into 5 * 3, and the number 2 has no factors, so that the answer becomes 5 * 3 * 2, and the result is also 30.

Use the calculator

A calculator can also be used given the concept of divisibility; This is to be able to determine whether the number is prime or not. The number is entered into the calculator and divided, taking into account that the result is an integer. For example, to test the number 57, it is entered into the calculator and divided, for example, by the number 2; The result is 27.5, which is not an integer, so it is not acceptable. Again, for example, 57 is divided by 3, and the result will be an integer 19, so both 3 and 19 are factors of 57, so 57 is not prime. Because it does not meet the conditions of prime numbers.

properties of prime numbers

After answering the question "What are prime numbers?" It has become necessary to talk about the properties of these numbers; Which are the building blocks of integers, where the ability to get a distinct integer as a product of prime numbers is the main reason behind the entire number theory and behind the interesting results in this theory, as many important theories, applications and guesses have been formulated based on the properties of prime numbers , and the following are some of these characteristics:

• There are an infinite number of prime numbers, as there are 10 different ways to prove this fact . 
• If p is a prime number and a is an integer then the product of this equation "a ^ p - a" is divisible by p . 
• p is a prime number if and only if 1x2x3x.....x(p - 1) +1 is divisible by p . 
• The only even prime number is 2, and all remaining even numbers are not primes; Because they are all divisible by 2 .
•  There is no prime number greater than 5 that ends in 5, because any number ending in 5 is divisible by 5; Therefore, it is not a prime number .

The importance of prime numbers

Many students come to mind after they get the answer to the question "What are the prime numbers?" Lots of questions about the importance of knowing and studying prime numbers; So it was necessary to talk about its importance; These numbers are used in many fields, such as in data encryption, by multiplying two prime numbers to get a new number, i.e. a complex number, which is called the public key, and because of the largeness of these numbers, it becomes very difficult to analyze this number into its prime factors again, which It is called the private key, and it is worth mentioning here that these resulting numbers are very large numbers and not small numbers that can be handled in a simple way and re-parsed; This is the feature that many scientists have resorted to exploiting in the encryption process, which means that it is very difficult to access the private key easily by anyone .

Prime numbers are also widely used in the mathematical theory of music, electronics , computers , information processing, in addition to that, prime numbers have applications to many common processes in signal processing, and applications to quantum mechanics and physics. There is no aspect of life that is not affected by it.