Prime numbers are those that have only two factors: 1 and the number itself. Consider the number 5, which has only two elements (1 and 5). This indicates that it is a prime number. Consider the number 6, which has more than two factors, namely 1, 2, 3, and 6. As a result, 6 is not a prime number. If we consider the number 1, we can see that it has only one element. As a result, it cannot be a prime number because a prime number must have exactly two elements. This indicates that 1 is neither a prime nor a composite number; it is singular.

A prime number is any whole number bigger than one that is divisible exclusively by one and itself.

Definition of a Prime Number

## What Are Prime Numbers?

- A prime number (or prime) is a natural number higher than one that cannot be divided by two lesser natural numbers. A composite number is a natural number greater than one that is not prime.
- 5 is prime, for example, because the only methods to write it as a product, 1 5 or 5 1, involve 5 itself. However, 4 is composite since it is a product of two numbers that are both less than four.
- Because of the fundamental theorem of arithmetic, every natural number bigger than one is either a prime or can be factorised as a product of primes that is unique up to their order.
- A prime number is a number bigger than one with exactly two elements, i.e. 1 and the number itself. In other words, a prime number is one that cannot be divided into equal groups.
- Only if a number can be factorised as a product of two numbers can we split it into groups with equal numbers of items/elements.
- For example, 7 cannot be divided into equal-number groups. This is due to the fact that 7 can only be factorised as follows:
- 7 × 1 = 7

- 1 × 7 = 7

- This means that the sole elements of 7 are 1 and 7. Because it cannot be divided into groups of equal numbers, 7 is a prime number.

- As Euclid established circa 300 BC, there exist an endless number of primes. There is no known straightforward formula for distinguishing prime numbers from composite numbers.
- The distribution of primes within natural numbers in the big, on the other hand, can be statistically modelled.
- The first result in that direction is the prime number theorem, which states that the probability of a randomly chosen large number being prime is inversely proportional to its number of digits, or logarithm.

## Property of Primarily:

- The property of being prime is called primality.
- Faster algorithms include the Miller–Rabin primality test, which is fast but has a small chance of error, and the AKS primality test, which always produces the correct answer in polynomial time but is too slow to be practical.
- Particularly fast methods are available for numbers of special forms, such as Mersenne numbers.

## History of Prime Numbers

- Since ancient times, humans have been fascinated by prime numbers. Mathematicians are still looking for prime numbers with mystical qualities.
- Euclid proposed the prime number theorem, which states that there are an endless number of prime numbers.
- Eratosthenes, a famous scientist who lived a few decades after Euclid, devised a clever method for determining all prime numbers up to a certain integer.
- The Sieve of Eratosthenes is the name given to this procedure. If you need to locate prime numbers up to n, we will generate a list of all numbers ranging from 2 to n.
- Starting with the smallest prime number, p = 2, we will remove from the list all multiples of 2 except 2. Likewise, assign the next p value that is a prime number bigger than 2.

## List of Prime Numbers

There are 25 prime numbers from 1 to 100. The complete list of prime numbers from 1 to 100 is given below:

List of Numbers | Prime Numbers |

Between 1 and 10 | 2, 3, 5, 7 |

Between 11 and 20 | 11, 13, 17, 19 |

Between 21 and 30 | 23, 29 |

Between 31 and 40 | 31, 37 |

Between 41 and 50 | 41, 43, 47 |

Between 51 and 100 | 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 |

## Properties of Prime Numbers

• The following are some of the most essential qualities of prime numbers:

o A prime number is one that is a whole number greater than one.

o It has only two factors: 1 and the number itself.

o There is only one even prime number, 2;

o Any two prime numbers are always co-prime numbers.

o Every number can be expressed as a prime number product.

## Prime Numbers vs Composite Numbers

- A prime number is one that has exactly two factors and is greater than one, whereas a composite number contains more than two elements.
- For example, 5 can only be factorised in one way: 1 5 (OR) 5 1. It only contains two elements, which are 1 and 5. As a result, 5 is a prime number.
- A composite number is one that is higher than one and has more than two elements. For example, the number four can be factorised in a variety of ways.
- As a result, the factors of 4 are 1, 2, and 4. It contains more than two elements. As a result, 4 is a composite number.
- Let us examine the distinction between prime numbers and composite numbers using the table below:

Prime Numbers | Composite Numbers |

Numbers, greater than 1, having only two factors, 1 and the number itself | Numbers greater than 1 having at least three factors |

2 is the smallest and the only even prime number | 4 is the smallest composite number |

Examples of prime numbers are 2, 3, 5, 7, 11, 13, etc | Examples of composite numbers are 4, 6, 8, 9, 10, etc. |

## Prime Numbers and Co-prime Numbers

- There is a distinction between prime and co-prime numbers. Co-prime numbers are always paired, whereas a single number can be read as a prime number.
- If a pair of numbers has no common factor other than 1, they are referred to as co-prime numbers.
- The only requirement for co-prime numbers to be prime or composite is that their GCF is always 1.
**Examples of co-prime numbers:**- 5 and 9 are co-primes.

- 6 and 11 are co-primes.

- 18 and 35 are co-primes.

- Co-prime numbers need not necessarily be prime numbers.

## Easy Way to Find Prime Numbers

- There are several methods for determining prime numbers. Let us look at two of these ways.
**Method 1:**In the formula ‘n2 + n + 41,’ substitute whole numbers for n. This formula returns all prime numbers bigger than 40.

Let’s try a couple whole numbers and see what happens. - 0
^{2}+ 0 + 41 = 0 + 41 = 41 - 1
^{2}+ 1 + 41 = 2 + 41 = 43 - 2
^{2}+ 2 + 41 = 6 + 41 = 47

Continuing in this manner, you can compute any prime integers greater than 40.**Method 2:**Except for 2 and 3, any prime number can be written as ‘6n + 1 or 6n – 1’. So, if you have a number other than 2 or 3, you may determine if it is prime or not by attempting to express it in the form of 6n + 1 or 6n – 1.

6(1) – 1 = 5

6(1) + 1 = 7

6(2) – 1 = 11

6(2) + 1 = 13

Now, we know that the numbers 5, 7, 11, and 13 are prime.

**List of Odd Prime Numbers**

A prime number chart is a diagram that displays a list of prime numbers in a logical order. The prime number chart from 1 to 100, which illustrates the list of odd prime numbers, is shown below.

## Is There a Pattern in Prime Numbers?

A pattern can be used to find the set of prime numbers between any two numbers. The picture below depicts a few prime integers surrounded and all the numbers divisible by these prime numbers struck out. This sequence can be repeated until you reach the square root of the larger number, which is 100.

## Co-Prime Numbers

When two numbers have only one common factor, they are said to be co-prime. It is not required that these numbers be prime numbers. For example, the numbers 9 and 10 are co-primes. Let’s double-check

## Some Facts about Prime Numbers

- The smallest prime number is two.
- The only even prime number is 2.
- The only consecutive prime numbers are 2 and 3.
- A whole number, with the exception of 0 and 1, is either a prime number or a composite number.
- Not all odd integers are prime numbers. For example, 21, 39, and so on.
- No prime number bigger than 5 has an ending of 5.
- The Sieve of Eratosthenes was one of the first methods for determining prime numbers.
- As the quantity grows larger, prime numbers become more scarce.
- There is no such thing as the greatest prime number. As of September 2021, the largest known prime number is 282,589,933 1, which contains 24,862,048 digits when expressed in base 10.

**Examples of Prime Numbers**

Example 1: Which of the two numbers is a prime number, 13 or 15?

Solution

The number 15 has more than two factors: 1, 3, 5, and 15. Hence, it is a composite number. On the other hand, 13 has just two factors: 1 and 13. Hence, it is a prime number. Therefore, 13 is a prime number.

**Frequently Asked Questions**

Q1 Is the number one a prime number?

**Answer:** No, one is not a prime nor a composite number.

Q2 Is it possible for a prime number to be negative?**Answer:**Prime numbers can never be negative. The set of natural numbers includes prime numbers.

Q3 Why is the number 2 the only even prime number?

**Answer:** Every even number greater than 2 is a multiple of 2. As a result, 2 is the only even prime number.

Q4 What is the distinction between a prime number and a co-prime number?

**Answer:** A prime number has two factors: one and the number itself. The common factor of co-prime numbers is 1.

Q5 Which Is the largest known prime number?

**Answer:** The largest known prime number (as of September 2021) is 282,589,933 − 1, a number that has 24,862,048 digits. By the time you read this, it may be even larger than this.

Q6. Why is the number 2 the only even prime number?

**Answer:** All even numbers greater than two are multiples of two. As a result, 2 is the only even prime number.

Q7 What are the rules for determining whether a number is prime?

**Answer:** A prime number is one that contains only two factors: itself and one. In other words, it can only be split evenly by itself and 1.

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