Prime numbers are positive integers **(natural numbers)** greater than one, which is divisible by only two numbers, the same and one without the remainder; the name is indivisible,

and Primary numbers are infinitesimal Infinite numbers are infinite numbers, and positive integers greater than one, and the number of denominators more than two are called non-prime numbers

or complex numbers, which can be broken down, while the two numbers (0.1) are excluded from the list of prime numbers and complex numbers.

Scientists have been interested in the preliminary numbers since ancient times, and is still to the attention of mathematicians;

history records indicate the ancient Egyptians know the concept of a prime number, but the ancient Greeks are still the first to conduct serious studies on this subject,

for example, Euclid in 300 BC In his books the prime numbers are endless. There has been much controversy over the truth of the number 1;

whether it is a prime number or non-prime, and from the definition of Primary numbers are numbers that have two natural denominators,

but the number one has only one denominator and is the same only, and may not repeat similar factors, and to clarify (1 × 1 = 1); so its factors are only one set without repeating the same denominators.

## The importance of prime numbers

Data is based on its security on several concepts, including Primary numbers; it is one of the most important tools used in the encryption of electronic data, banking transactions,

and logging into social networking sites, and the principle of these numbers lies in the encryption of information initially, and the transfer of the message to

a large number produces This number is called the open number, which is the secret number, and can only be breached if the primary factors used for this complex process are known, which is very difficult.

Examples of non Primary numbers The number 4 is not a prime number because it has three divisors: 1, 4, 2, and 15 is not a prime number,

because it has four divisors: 1, 15, 3, 5, and 24 is not prime because it has six divisors, These are: 24, 1, 4, 6, 8, 3. It is worth noting that even numbers cannot be an absolute prime number except two.

Using prime numbers The Primary numbers are used in many areas of information technology, including encryption by means of the declared key, and this technology depends mainly on specific characteristics.

## Examples of prime and complex numbers

**Example:** Why are the numbers (29,13,7,5) prime numbers? Solution: The number 5 is prime; the reason is that it is divisible by itself and the number is only one;

The number 7 is a prime number; Number 13 is a prime number, and number 29 is a prime number because of both divide themselves and only one.

**Example:** What are the prime numbers smaller than 100? Solution:

**The prime numbers smaller than 100 are: **

**2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, and 97.**

**Example:** Are the numbers **(2.5,28.8)** prime or complex? Solution: The number 8 is a non-prime number; it is a complex number because the factors of the number eight are the

following numbers: **(8,4,2,1)**; that is, it has more than two (4), so it is excluded from the list of prime numbers. Also, the number 28 is not prime, because it accepts division by more than two factors.

The number 2.5 is a non-prime number because one of the conditions for prime numbers is to be positive integers, not fractions, fractional numbers, or negative numbers

### Characteristics of prime numbers

The prime numbers have several characteristics: Primary numbersare distributed irregularly, and the main reason is that scientists do not understand the method of distribution of prime numbers, unlike odd or even numbers.

All Primary numbers except (5,2) end in one of the numbers **(9,7,3,1)**; Which ends with (5.0) multiples of number five, which is not primary. (If (a, b) are integers, and (c) is a third number, where (c) is a prime, and the product of

multiples of (an x b) is divisible by c, then a or b is divisible by Number C, this is called Euclid’s theorem.

The number 2 is the smallest number in the Primary numbers list and is the only even number in it.

### Discover the largest prime number

The largest preliminary number was discovered by Dr Curtis Cooper at the University of Missouri in America by computer.

The 49th ranking in the Mersey preliminary series; these very large initial figures are useful in future computing processes because the current numbers are in the hundreds and have not yet reached millions of numbers.

### Eratosthenes sieve

Eratosthenes sieve is one way of knowing Primary numbers discovered by a Greek scientist called Eratosthenes.

The Eratosthenes sieve method, assuming that (b = 2), the smallest prime number less than 100, is omitted (b) and all its multiples (8,6,4,2 …) and so up to a hundred,

the first number remains and has not yet been It is a Primary number (3), and the step is repeated by deleting the number 3 and all its multiples from the list until the last

A number follows the number 3 that has not been omitted (5), so it is prime, and the steps are repeated several times to get all Primary numbers resulting from this screening.

**Test a number **

To determine the prime number of a number, several tests are used to determine the Primary numbers of any number.

**Mercenary test **

In 1644, the scholar Mersini developed a formula as follows: (L = 2 L -1; where (L) prime number, and then found that M = 23 × 89 complex number),

and Mersini formula was used to determine the largest prime number, was determined in 1984 There is no formula for determining prime numbers,

which is found in the study of lists of these numbers as dispersed and not distributed in a certain and regular pattern,

and that as the values of Primary numbers increase, the spacing between them will increase.

#### Kaus test

In 1793, the world introduced the so-called theorem of the Primary numbers. It was only in 1896 that it was proved by CJVPoussin and J.

Hadamard separately, and this formula was proved by several proofs, but it lacked ease and simplicity until the Norwegian presented Silberk in 1949 a proof in

which he resorted to the use of the concepts of the theory of numbers, and took it in his proof, which was characterized by simplicity and lacked The complexity.

** Finding Primary numbers**