### Prime Numbers

##### Prime Numbers

Prime numbers are whole numbers greater than 1 that have only two factors—1 and the number itself. For example, 2, 3, 5, 7, and 11 are all prime numbers.

Prime numbers are essential in number theory and cryptography. They are also used in computer science, such as in the design of error-correcting codes and public-key cryptography.

The distribution of prime numbers is not uniform. There are infinitely many prime numbers, but they become increasingly rare as the numbers get larger.

The largest known prime number is 2^82,589,933 - 1, which has over 24 million digits. It was discovered by Patrick Laroche in December 2018.

Prime numbers continue to fascinate mathematicians and computer scientists alike, and they remain an active area of research.

##### What are Prime Numbers?

**Prime Numbers**

A prime number is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number.

For example, 5 is a prime number because it cannot be made by multiplying two smaller natural numbers. 10 is a composite number because it can be made by multiplying 2 and 5.

The first few prime numbers are:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, …

There are infinitely many prime numbers. This was proven by the Greek mathematician Euclid in the 3rd century BC.

**Properties of Prime Numbers**

Prime numbers have a number of interesting properties. Some of these properties include:

- Every prime number greater than 3 can be written in the form 6n ± 1, where n is a natural number.
- The only even prime number is 2.
- The sum of two consecutive prime numbers is always odd.
- The product of two consecutive prime numbers is always greater than the sum of the two numbers.
- There are infinitely many twin primes, which are prime numbers that differ by 2.

**Applications of Prime Numbers**

Prime numbers have a number of applications in mathematics and computer science. Some of these applications include:

- Prime numbers are used in cryptography to encrypt and decrypt messages.
- Prime numbers are used in number theory to study the properties of numbers.
- Prime numbers are used in computer science to design efficient algorithms.

**Conclusion**

Prime numbers are a fascinating and important part of mathematics. They have a number of interesting properties and applications. The study of prime numbers has been going on for centuries, and it is still an active area of research today.

##### Download PDF – Prime Numbers

**Prime Numbers**

A prime number is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number.

For example, 5 is a prime number because it cannot be made by multiplying two smaller natural numbers. 10 is a composite number because it can be made by multiplying 2 and 5.

The first few prime numbers are:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, …

There are infinitely many prime numbers. This was proven by Euclid in the 3rd century BC.

**Properties of Prime Numbers**

Prime numbers have a number of interesting properties. Some of these properties include:

- Every prime number greater than 3 can be written in the form 6n ± 1, where n is a natural number.
- The only even prime number is 2.
- The sum of two consecutive prime numbers is always odd.
- The product of two consecutive prime numbers is always greater than the sum of the two numbers.
- There are infinitely many twin primes. Twin primes are two prime numbers that differ by 2.

**Applications of Prime Numbers**

Prime numbers have a number of applications in mathematics and computer science. Some of these applications include:

- Prime numbers are used in cryptography to encrypt and decrypt messages.
- Prime numbers are used in number theory to study the properties of numbers.
- Prime numbers are used in computer science to design efficient algorithms.

**Conclusion**

Prime numbers are a fascinating and important part of mathematics. They have a number of interesting properties and applications. Prime numbers are still being studied today, and new discoveries are being made all the time.

##### Download PDF

**Downloading PDF**

Downloading a PDF file from the internet is a simple process that can be done with any web browser. Here are the steps involved:

**Find the PDF file you want to download.**You can do this by searching for the file name or by browsing through a website’s directory.**Click on the link to the PDF file.**This will open the PDF file in your browser’s PDF viewer.**Click on the “Download” button.**This will save the PDF file to your computer.

**Here are some examples of how you might download a PDF file:**

- You might download a PDF file of a research paper from a university website.
- You might download a PDF file of a product manual from a manufacturer’s website.
- You might download a PDF file of a travel brochure from a tourism website.

**Here are some tips for downloading PDF files:**

- Make sure you have a PDF reader installed on your computer.
- If you are downloading a PDF file from a website that you do not trust, be sure to scan the file for viruses before opening it.
- Save the PDF file to a location on your computer where you can easily find it.

**Downloading PDF files is a simple and convenient way to save information for later use. By following these steps, you can easily download any PDF file that you need.**

##### Properties of Prime Numbers

**Properties of Prime Numbers**

Prime numbers are whole numbers greater than 1 that have only two factors—1 and the number itself. For example, 2, 3, 5, 7, and 11 are all prime numbers.

There are an infinite number of prime numbers. This was proven by the Greek mathematician Euclid in the 3rd century BC.

The distribution of prime numbers is not uniform. There are more small prime numbers than large prime numbers. For example, there are 25 prime numbers between 1 and 100, but only 10 prime numbers between 100 and 200.

**Some important properties of prime numbers include:**

- Every prime number greater than 3 can be written in the form 6n ± 1, where n is a natural number.
- The only even prime number is 2.
- The sum of two consecutive prime numbers is always odd.
- The product of two consecutive prime numbers is always greater than the square of either prime number.
- Every prime number greater than 3 can be written as the sum of two squares. For example, 5 = 1^2 + 2^2, 13 = 2^2 + 3^2, and 17 = 1^2 + 4^2.
- Every prime number greater than 5 can be written as the difference of two squares. For example, 7 = 4^2 - 1^2, 11 = 5^2 - 2^2, and 13 = 7^2 - 2^2.

**Applications of Prime Numbers**

Prime numbers have a variety of applications in mathematics, computer science, and cryptography.

- In mathematics, prime numbers are used to study number theory, which is the branch of mathematics that deals with the properties of numbers.
- In computer science, prime numbers are used in cryptography, which is the science of encrypting and decrypting data.
- In cryptography, prime numbers are used to create public-key cryptography, which is a method of encrypting data that allows anyone to encrypt a message, but only the intended recipient can decrypt it.

**Conclusion**

Prime numbers are a fascinating and important part of mathematics. They have a variety of properties that make them useful in a variety of applications.

##### Prime Numbers Chart

**Prime Numbers Chart**

A prime number is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number.

For example, 5 is a prime number because it cannot be made by multiplying two smaller natural numbers. 10 is a composite number because it can be made by multiplying 2 and 5.

The first few prime numbers are:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, …

There are infinitely many prime numbers. This was proven by Euclid in the 3rd century BC.

**Prime Numbers Chart**

A prime numbers chart is a table that lists prime numbers in order. The following is a prime numbers chart up to 100:

Number | Prime |
---|---|

2 | Yes |

3 | Yes |

4 | No |

5 | Yes |

6 | No |

7 | Yes |

8 | No |

9 | No |

10 | No |

11 | Yes |

12 | No |

13 | Yes |

14 | No |

15 | No |

16 | No |

17 | Yes |

18 | No |

19 | Yes |

20 | No |

21 | No |

22 | No |

23 | Yes |

24 | No |

25 | No |

26 | No |

27 | No |

28 | No |

29 | Yes |

30 | No |

31 | Yes |

32 | No |

33 | No |

34 | No |

35 | No |

36 | No |

37 | Yes |

38 | No |

39 | No |

40 | No |

41 | Yes |

42 | No |

43 | Yes |

44 | No |

45 | No |

46 | No |

47 | Yes |

48 | No |

49 | No |

50 | No |

51 | No |

52 | No |

53 | Yes |

54 | No |

55 | No |

56 | No |

57 | No |

58 | No |

59 | Yes |

60 | No |

61 | Yes |

62 | No |

63 | No |

64 | No |

65 | No |

66 | No |

67 | Yes |

68 | No |

69 | No |

70 | No |

71 | Yes |

72 | No |

73 | Yes |

74 | No |

75 | No |

76 | No |

77 | Yes |

78 | No |

79 | Yes |

80 | No |

81 | No |

82 | No |

83 | Yes |

84 | No |

85 | No |

86 | No |

87 | No |

88 | No |

89 | Yes |

90 | No |

91 | No |

92 | No |

93 | No |

94 | No |

95 | No |

96 | No |

97 | Yes |

98 | No |

99 | No |

100 | No |

**Examples of Using a Prime Numbers Chart**

- You can use a prime numbers chart to find the prime factors of a number. For example, the prime factors of 12 are 2 and 3.
- You can use a prime numbers chart to determine if a number is prime or composite. For example, 17 is a prime number because it is not divisible by any other number except 1 and itself.
- You can use a prime numbers chart to generate a list of prime numbers. For example, the following is a list of the first 100 prime 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

**Conclusion**

Prime numbers are an important part of mathematics. They have many applications, including finding the prime factors of a number, determining if a number is prime or composite, and generating a list of prime numbers.

##### List of Prime Numbers 1 to 100

**Prime Numbers 1 to 100**

A prime number is a natural number greater than 1 that is not a product of two smaller natural numbers. In other words, a prime number can only be divided by 1 and itself without a remainder.

The first 25 prime numbers 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, 97
```

**How to Find Prime Numbers**

There are a few different ways to find prime numbers. One simple way is to use the sieve of Eratosthenes. This method works by starting with a list of all the numbers from 2 to the number you are looking for. Then, you cross out all the multiples of 2, starting with 4. Next, you cross out all the multiples of 3, starting with 9. You continue this process, crossing out all the multiples of each prime number, until you reach the number you are looking for. The numbers that are not crossed out are the prime numbers.

For example, to find all the prime numbers up to 100, you would start with the following list:

```
2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100
```

Next, you would cross out all the multiples of 2, starting with 4:

```
2, 3, ~4~, 5, ~6~, 7, ~8~, 9, ~10~, 11, ~12~, 13, ~14~, 15, ~16~, 17, ~18~, 19, ~20~, 21, ~22~, 23, ~24~, 25, ~26~, 27, ~28~, 29, ~30~, 31, ~32~, 33, ~34~, 35, ~36~, 37, ~38~, 39, ~40~, 41, ~42~, 43, ~44~, 45, ~46~, 47, ~48~, 49, ~50~, 51, ~52~, 53, ~54~, 55, ~56~, 57, ~58~, 59, ~60~, 61, ~62~, 63, ~64~, 65, ~66~, 67, ~68~, 69, ~70~, 71, ~72~, 73, ~74~, 75, ~76~, 77, ~78~, 79, ~80~, 81, ~82~, 83, ~84~, 85, ~86~, 87, ~88~, 89, ~90~, 91, ~92~, 93, ~94~, 95, ~96~, 97, ~98~, 99, ~100~
```

Then, you would cross out all the multiples of 3, starting with 9:

```
2, 3, ~4~, 5, ~6~, 7, ~8~, ~9~, ~10~, 11, ~12~, 13, ~14~, 15, ~16~, 17, ~18~, 19, ~20~, 21, ~22~, 23, ~24~, 25, ~26~, 27, ~28~, 29, ~30~, 31, ~32~, 33, ~34~, 35, ~36~, 37, ~38~, 39, ~40~, 41, ~42~, 43, ~44~, 45, ~46~, 47, ~48~, 49, ~50~, 51, ~52~, 53, ~54~, 55, ~56~, 57, ~58~, 59, ~60~, 61, ~62~, 63, ~64~, 65, ~66~, 67, ~68~, 69, ~70~, 71, ~72~, 73, ~74~, 75, ~76~, 77, ~78~, 79, ~80~, 81, ~82~, 83, ~84~, 85, ~86~, 87, ~88~, 89, ~90~, 91, ~92~, 93, ~94~, 95, ~96~, 97, ~98~, 99, ~100~
```

You would continue this process, crossing out all the multiples of each prime number, until you reach the number 100. The numbers that are not crossed out are the prime 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
```

**Applications of Prime Numbers**

Prime numbers have a number of applications in mathematics and computer science. For example, prime numbers are used in:

- Cryptography: Prime numbers are used to create secure encryption algorithms.
- Number theory: Prime numbers are used to study the properties of numbers.
- Computer science: Prime numbers are used in a variety of algorithms, such as the Fast Fourier Transform (FFT) and the RSA algorithm.

Prime numbers are also used in a variety of other fields, such as physics, chemistry, and biology.

##### How to Find Prime Numbers?

**How to Find Prime Numbers**

A prime number is a natural number greater than 1 that is not a product of two smaller natural numbers. In other words, a prime number can only be divided by itself and 1 without leaving a remainder.

The first few prime numbers are:

```
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ...
```

There are an infinite number of prime numbers, but they become increasingly rare as the numbers get larger.

**Methods for Finding Prime Numbers**

There are a number of different methods for finding prime numbers. Some of the most common methods include:

**Trial division:**This is the most straightforward method for finding prime numbers. It involves dividing a number by all of the numbers less than or equal to its square root. If the number is not divisible by any of these numbers, then it is prime.**The Sieve of Eratosthenes:**This is a more efficient method for finding prime numbers. It involves creating a list of all of the numbers from 2 to a given number. Then, for each number in the list, all of its multiples are crossed out. The numbers that are not crossed out are the prime numbers.**The AKS primality test:**This is a deterministic primality test that can be used to determine whether a given number is prime in polynomial time. However, the AKS primality test is not practical for large numbers.

**Examples**

Here are some examples of how to find prime numbers using the trial division method and the Sieve of Eratosthenes:

**Trial division:**

To find all of the prime numbers less than or equal to 100, we can use the trial division method. We start by dividing 2 by all of the numbers less than or equal to its square root (which is 1). Since 2 is not divisible by any of these numbers, it is prime.

We then move on to 3 and repeat the process. We find that 3 is not divisible by any of the numbers less than or equal to its square root, so it is also prime.

We continue this process for all of the numbers up to 100. The prime numbers that we find 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, 97
```

**Sieve of Eratosthenes:**

To find all of the prime numbers less than or equal to 100 using the Sieve of Eratosthenes, we start by creating a list of all of the numbers from 2 to 100.

```
2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100
```

We then cross out all of the multiples of 2, starting with 4.

```
2, 3, ~4~, 5, ~6~, 7, ~8~, 9, ~10~, 11, ~12~, 13, ~14~, 15, ~16~, 17, ~18~, 19, ~20~, 21, ~22~, 23, ~24~, 25, ~26~, 27, ~28~, 29, ~30~, 31, ~32~, 33, ~34~, 35, ~36~, 37, ~38~, 39, ~40~, 41, ~42~, 43, ~44~, 45, ~46~, 47, ~48~, 49, ~50~, 51, ~52~, 53, ~54~, 55, ~56~, 57, ~58~, 59, ~60~, 61, ~62~, 63, ~64~, 65, ~66~, 67, ~68~, 69, ~70~, 71, ~72~, 73, ~74~, 75, ~76~, 77, ~78~, 79, ~80~, 81, ~82~, 83, ~84~, 85, ~86~, 87, ~88~, 89, ~90~, 91, ~92~, 93, ~94~, 95, ~96~, 97, ~98~, 99, ~100~
```

We then cross out all of the multiples of 3, starting with 6.

```
2, 3, ~4~, 5, ~6~, 7, ~8~, 9, ~10~, 11, ~12~, 13, ~14~, 15, ~16~, 17, ~18~, 19, ~20~, 21, ~22~, 23, ~24~, 25, ~26~, 27, ~28~, 29, ~30~, 31, ~32~, 33, ~34~, 35, ~36~, 37, ~38~, 39, ~40~, 41, ~42~, 43, ~44~, 45, ~46~, 47, ~48~, 49, ~50~, 51, ~52~, 53, ~54~, 55, ~56~, 57, ~58~, 59, ~60~, 61, ~62~, 63, ~64~, 65, ~66~, 67, ~68~, 69, ~70~, 71, ~72~, 73, ~74~, 75, ~76~, 77, ~78~, 79, ~80~, 81, ~82~, 83, ~84~, 85, ~86~, 87, ~88~, 89, ~90~, 91, ~92~, 93, ~94~, 95, ~96~, 97, ~98~, 99, ~100~
```

We continue this process for all of the numbers up to the square root of 100 (which is 10). The numbers that are not crossed out are the prime 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
```

**Applications of Prime Numbers**

Prime

##### Prime Numbers vs Composite Numbers

**Prime Numbers**

The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29.

**Composite Numbers**

A composite number is a natural number greater than 1 that can be made by multiplying two smaller natural numbers.

For example, 10 is a composite number because it can be made by multiplying 2 and 5. 12 is a composite number because it can be made by multiplying 2, 2, and 3.

The first few composite numbers are 4, 6, 8, 9, 10, 12, 14, 15, 16, and 18.

**Properties of Prime Numbers**

- There are an infinite number of prime numbers.
- The only even prime number is 2.
- Every prime number greater than 3 can be written in the form 6n ± 1, where n is a natural number.
- The sum of two prime numbers is always odd.
- The product of two prime numbers is always odd.

**Properties of Composite Numbers**

- Every composite number can be written as a product of prime numbers.
- The sum of two composite numbers is not always even.
- The product of two composite numbers is not always even.

**Examples**

- 7 is a prime number because it cannot be made by multiplying two smaller natural numbers.
- 10 is a composite number because it can be made by multiplying 2 and 5.
- 12 is a composite number because it can be made by multiplying 2, 2, and 3.
- 15 is a composite number because it can be made by multiplying 3 and 5.
- 21 is a composite number because it can be made by multiplying 3 and 7.

**Applications**

Prime numbers have many applications in mathematics, computer science, and cryptography.

- In mathematics, prime numbers are used to study number theory.
- In computer science, prime numbers are used to generate random numbers and to encrypt data.
- In cryptography, prime numbers are used to create public-key encryption systems.

##### Solved Examples on Prime Numbers

**Example 1: Is 7 a prime number?**

To determine if 7 is a prime number, we need to check if it is divisible by any number other than 1 and itself. We can start by checking if it is divisible by 2, 3, 4, 5, and 6.

- 7 is not divisible by 2 because 7 is an odd number.
- 7 is not divisible by 3 because the sum of its digits (7) is not divisible by 3.
- 7 is not divisible by 4 because the last two digits (07) are not divisible by 4.
- 7 is not divisible by 5 because the last digit (7) is not 0 or 5.
- 7 is not divisible by 6 because it is not divisible by both 2 and 3.

Since 7 is not divisible by any number other than 1 and itself, it is a prime number.

**Example 2: Find all the prime numbers between 1 and 100.**

To find all the prime numbers between 1 and 100, we can use the Sieve of Eratosthenes. This method involves creating a list of all the numbers from 2 to 100 and then marking off all the multiples of each number. The numbers that are not marked off are the prime numbers.

Here is the Sieve of Eratosthenes for the numbers between 1 and 100:

```
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 numbers that are not marked off are the prime numbers between 1 and 100.

**Example 3: Find the largest prime factor of 100.**

To find the largest prime factor of 100, we can first find all the prime factors of 100. The prime factors of 100 are 2, 2, 5, and 5. The largest prime factor of 100 is 5.

##### Frequently Asked Questions on Prime Numbers

##### What are Prime Numbers in Maths?

**Prime Numbers**

In mathematics, a prime number is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number.

For example, 5 is a prime number because it cannot be made by multiplying two smaller natural numbers. However, 6 is a composite number because it can be made by multiplying 2 and 3.

The first few prime numbers are:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, …

There are infinitely many prime numbers. This was proven by the Greek mathematician Euclid in the 3rd century BC.

**Properties of Prime Numbers**

Prime numbers have a number of interesting properties. For example:

- Every prime number greater than 3 can be written in the form 6n ± 1, where n is a natural number.
- The only even prime number is 2.
- The sum of two consecutive prime numbers is always odd.
- The product of two consecutive prime numbers is always greater than the square of either prime number.

**Applications of Prime Numbers**

Prime numbers have a number of applications in mathematics and computer science. For example:

- Prime numbers are used in cryptography to encrypt and decrypt messages.
- Prime numbers are used in number theory to study the distribution of numbers.
- Prime numbers are used in computer science to design efficient algorithms.

**Conclusion**

Prime numbers are a fascinating and important part of mathematics. They have a number of interesting properties and applications, and they continue to be studied by mathematicians today.

##### How to find prime numbers?

**Prime Numbers**

A prime number is a whole number greater than 1 whose only factors are 1 and itself. For example, 2, 3, 5, 7, and 11 are all prime numbers.

There are an infinite number of prime numbers, but they become increasingly rare as the numbers get larger. For example, there are 25 prime numbers between 1 and 100, but only 168 prime numbers between 100 and 1,000.

**How to Find Prime Numbers**

There are a few different ways to find prime numbers. One simple method is called the **sieve of Eratosthenes**. This method works by starting with a list of all the numbers from 2 to n, where n is the largest number you want to check. Then, you cross out all the multiples of 2, starting with 4. Next, you cross out all the multiples of 3, starting with 9. You continue this process, crossing out all the multiples of each prime number, until you reach the square root of n. The numbers that are left uncrossed are all prime numbers.

For example, here is how you would use the sieve of Eratosthenes to find all the prime numbers between 1 and 100:

- Start with a list of all the numbers from 2 to 100:

```
2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100
```

- Cross out all the multiples of 2, starting with 4:

```
2, 3, ~4~, 5, ~6~, 7, ~8~, 9, ~10~, 11, ~12~, 13, ~14~, 15, ~16~, 17, ~18~, 19, ~20~, 21, ~22~, 23, ~24~, 25, ~26~, 27, ~28~, 29, ~30~, 31, ~32~, 33, ~34~, 35, ~36~, 37, ~38~, 39, ~40~, 41, ~42~, 43, ~44~, 45, ~46~, 47, ~48~, 49, ~50~, 51, ~52~, 53, ~54~, 55, ~56~, 57, ~58~, 59, ~60~, 61, ~62~, 63, ~64~, 65, ~66~, 67, ~68~, 69, ~70~, 71, ~72~, 73, ~74~, 75, ~76~, 77, ~78~, 79, ~80~, 81, ~82~, 83, ~84~, 85, ~86~, 87, ~88~, 89, ~90~, 91, ~92~, 93, ~94~, 95, ~96~, 97, ~98~, 99, ~100~
```

- Cross out all the multiples of 3, starting with 9:

```
2, 3, ~4~, 5, ~6~, 7, ~8~, ~9~, ~10~, 11, ~12~, 13, ~14~, 15, ~16~, 17, ~18~, 19, ~20~, 21, ~22~, 23, ~24~, 25, ~26~, 27, ~28~, 29, ~30~, 31, ~32~, 33, ~34~, 35, ~36~, 37, ~38~, 39, ~40~, 41, ~42~, 43, ~44~, 45, ~46~, 47, ~48~, 49, ~50~, 51, ~52~, 53, ~54~, 55, ~56~, 57, ~58~, 59, ~60~, 61, ~62~, 63, ~64~, 65, ~66~, 67, ~68~, 69, ~70~, 71, ~72~, 73, ~74~, 75, ~76~, 77, ~78~, 79, ~80~, 81, ~82~, 83, ~84~, 85, ~86~, 87, ~88~, 89, ~90~, 91, ~92~, 93, ~94~, 95, ~96~, 97, ~98~, 99, ~100~
```

- Continue this process, crossing out all the multiples of each prime number, until you reach the square root of 100, which is 10:

```
2, 3, ~4~, 5, ~6~, 7, ~8~, ~9~, ~10~, 11, ~12~, 13, ~14~, 15, ~16~, 17, ~18~, 19, ~20~, 21, ~22~, 23, ~24~, 25, ~26~, 27, ~28~, 29, ~30~, 31, ~32~, 33, ~34~, 35, ~36~, 37, ~38~, 39, ~40~, 41, ~42~, 43, ~44~, 45, ~46~, 47, ~48~, 49, ~50~, 51, ~52~, 53, ~54~, 55, ~56~, 57, ~58~, 59, ~60~, 61, ~62~, 63, ~64~, 65, ~66~, 67, ~68~, 69, ~70~, 71, ~72~, 73, ~74~, 75, ~76~, 77, ~78~, 79, ~80~, 81, ~82~, 83, ~84~, 85, ~86~, 87, ~88~, 89, ~90~, 91, ~92~, 93, ~94~, 95, ~96~, 97, ~98~, 99, ~100~
```

- The numbers that are left uncrossed are all prime numbers:

```
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 4
##### What are the examples of prime numbers?
Prime numbers are whole numbers greater than 1 that have only two factors – 1 and the number itself. For example, 2, 3, 5, 7, 11, and 13 are all prime numbers.
Here are some additional examples of prime numbers:
* 17
* 19
* 23
* 29
* 31
* 37
* 41
* 43
* 47
* 53
* 59
* 61
* 67
* 71
* 73
* 79
* 83
* 89
* 97
There are an infinite number of prime numbers, and they become increasingly rare as the numbers get larger. For example, there are only 25 prime numbers between 1 and 100, but there are over 10,000 prime numbers between 1 and 10,000.
Prime numbers have many interesting properties. For example, every even number greater than 2 is composite (not prime). Also, the sum of two prime numbers is always odd.
Prime numbers are used in many areas of mathematics and computer science. For example, they are used in cryptography, which is the study of how to encode and decode messages. Prime numbers are also used in number theory, which is the study of the properties of numbers.
##### What is the smallest prime number?
The smallest prime number is 2. A prime number is a natural number greater than 1 that is not a product of two smaller natural numbers. In other words, a prime number can only be divided by 1 and itself without a remainder.
Here are some examples of prime numbers:
* 2
* 3
* 5
* 7
* 11
* 13
* 17
* 19
* 23
* 29
The sequence of prime numbers is infinite, meaning that there are an infinite number of prime numbers. However, the distribution of prime numbers becomes less dense as the numbers get larger. For example, there are 25 prime numbers between 1 and 100, but only 168 prime numbers between 100 and 1000.
Prime numbers have many important applications in mathematics and computer science. For example, they are used in cryptography, which is the study of how to encode and decode messages so that they cannot be read by unauthorized people. Prime numbers are also used in number theory, which is the study of the properties of numbers.
The smallest prime number, 2, is a very special number. It is the only even prime number, and it is the only prime number that is also a Mersenne prime. A Mersenne prime is a prime number that can be expressed in the form 2^p - 1, where p is a prime number. The largest known Mersenne prime is 2^82,589,933 - 1, which has over 24 million digits.
##### What is the largest prime number so far?
The largest known prime number as of December 2022 is 2^(82,589,933) - 1, a Mersenne prime discovered by Patrick Laroche in December 2018. It has 24,862,048 digits.
Mersenne primes are prime numbers of the form 2^p - 1, where p is a prime number. They are named after Marin Mersenne, a French mathematician who studied them in the 17th century.
The search for Mersenne primes is a challenging and ongoing endeavor. It requires extensive computational resources and specialized algorithms. The discovery of each new Mersenne prime is a significant achievement in the field of number theory.
Here are some interesting facts about the largest known prime number:
- It is so large that it cannot be written out in full using standard notation.
- It would take approximately 2,700 years to write out the digits of the number by hand.
- The number is so large that it cannot be stored in the memory of a typical computer.
- It would take a supercomputer several months to verify the primality of the number.
The search for larger prime numbers continues, and it is possible that an even larger prime number will be discovered in the future.
##### Which is the largest 4 digit prime number?
The largest 4-digit prime number is 9973.
A prime number is a natural number greater than 1 that is not a product of two smaller natural numbers. In other words, a prime number can only be divided by 1 and itself without a remainder.
The 4-digit prime numbers are:
1009
1013
1019
1021
1031
1033
1039
1049
1051
1061
1063
1069
1087
1091
1093
1097
1103
1109
1117
1123
1129
1151
1153
1163
1171
1181
1187
1193
1201
1213
1217
1223
1229
1231
1237
1249
1259
1277
1279
1283
1289
1291
1297
1301
1303
1307
1319
1321
1327
1361
1367
1373
1381
1399
1409
1423
1427
1429
1433
1439
1447
1451
1453
1459
1471
1481
1483
1487
1489
1493
1499
1511
1523
1531
1543
1549
1553
1559
1567
1571
1579
1583
1597
1601
1607
1609
1613
1619
1621
1627
1637
1657
1663
1667
1669
1693
1697
1699
1709
1721
1723
1733
1741
1747
1753
1759
1777
1783
1787
1789
1801
1811
1823
1831
1847
1861
1867
1871
1873
1877
1879
1889
1901
1907
1913
1931
1933
1949
1951
1973
1979
1987
1993
1997
1999
2003
2011
2017
2027
2029
2039
2053
2063
2069
2081
2083
2087
2089
2099
2111
2113
2129
2131
2137
2141
2143
2153
2161
2179
2203
2207
2213
2221
2237
2239
2243
2251
2267
2269
2273
2281
2287
2293
2297
2309
2311
2333
2339
2341
2347
2351
2357
2371
2377
2381
2383
2389
2393
2399
2411
2417
2423
2437
2441
2447
2459
2467
2473
2477
2503
2521
2531
2539
2543
2549
2551
2557
2579
2591
2593
2609
2617
2621
2633
2647
2657
2659
2663
2671
2677
2683
2687
2689
2693
2699
2707
2711
2713
2719
2729
2731
2741
2749
2753
2767
2777
2789
2791
2797
2801
2803
2819
2833
2837
2843
2851
2857
2861
2879
2887
2897
2903
2909
2917
2927
2939
2953
2957
2963
2969
2971
2999
3001
3011
3019
3023
3037
3041
3049
3061
3067
3079
3083
3089
3109
3119
3121
3137
3163
3167
3169
3181
3187
3191
3203
3209
3217
3221
3229
3251
3253
3257
3259
3271
3299
3301
3307
3313
3319
3323
3329
3331
3343
3347
3359
3361
3371
3373
3389
3391
3407
3413
3433
3449
3457
3461
3463
3467
3469
3491
3499
3511
3517
3527
3529
3533
3539
3541
3547
3557
3559
3571
3581
3583
3593
3607
3613
3617
3623
3631
3637
3643
3659
3671
3673
3677
3691
3697
3701
3709
3719
3727
3733
3739
3761
3767
3769
3779
3793
3797
3803
3821
3823
3833
3841
3851
3853
3863
3877
3881
3889
3893
3907
3911
3917
3919
3923
3929
3931
3943
3947
3967
3989
4001
4003
4007
4
##### What are prime numbers between 1 and 50?
**Prime Numbers Between 1 and 50**
A prime number is a natural number greater than 1 that is not a product of two smaller natural numbers. In other words, a prime number can only be divided by itself and 1 without a remainder.
The prime numbers between 1 and 50 are:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47
**Examples of Prime Numbers**
* 2 is a prime number because it can only be divided by itself and 1 without a remainder.
* 3 is a prime number because it can only be divided by itself and 1 without a remainder.
* 5 is a prime number because it can only be divided by itself and 1 without a remainder.
* 7 is a prime number because it can only be divided by itself and 1 without a remainder.
* 11 is a prime number because it can only be divided by itself and 1 without a remainder.
**Non-Examples of Prime Numbers**
* 4 is not a prime number because it can be divided by 2 without a remainder.
* 6 is not a prime number because it can be divided by 2 and 3 without a remainder.
* 8 is not a prime number because it can be divided by 2 and 4 without a remainder.
* 9 is not a prime number because it can be divided by 3 without a remainder.
* 10 is not a prime number because it can be divided by 2 and 5 without a remainder.
**Properties of Prime Numbers**
* There are an infinite number of prime numbers.
* The only even prime number is 2.
* Every prime number greater than 3 can be written in the form 6n ± 1, where n is a natural number.
* The sum of two consecutive prime numbers is always odd.
* The product of two consecutive prime numbers is always even.
**Applications of Prime Numbers**
* Prime numbers are used in cryptography to encrypt and decrypt messages.
* Prime numbers are used in computer science to generate random numbers.
* Prime numbers are used in mathematics to study number theory.
##### Why 1 is not a prime number?
**Why 1 is not a prime number**
A prime number is a natural number greater than 1 that is not a product of two smaller natural numbers. In other words, a prime number can only be divided by itself and 1 without leaving a remainder.
1 is not a prime number because it can be divided by itself and 1 without leaving a remainder. Therefore, 1 is not a prime number.
**Examples of prime numbers**
Some examples of prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29.
**Why 1 is not a composite number**
A composite number is a natural number greater than 1 that can be written as a product of two smaller natural numbers. In other words, a composite number is not a prime number.
1 is not a composite number because it cannot be written as a product of two smaller natural numbers. Therefore, 1 is not a composite number.
**1 is a unit**
1 is a special number that is neither prime nor composite. It is known as a unit. Units are important in mathematics because they can be used to simplify expressions and equations.
**Conclusion**
1 is not a prime number because it can be divided by itself and 1 without leaving a remainder. 1 is also not a composite number because it cannot be written as a product of two smaller natural numbers. 1 is a unit.
```