## Prime Numbers

Prime numbers are a special category of natural numbers (or positive integers) which are exactly divisible by 1 and by itself. So, a prime number has exactly two divisors. Prime number is a number which is greater than 1 and which can be efficiently divided by 1 and by itself, not by any other number. It is also a whole number.

An Integer P > 1 is called a **prime number** when its only divisors are 1 and P. Any number m > 1 which is not a prime is called a composite.

**Simple properties of primes:**

- A prime ‘p’ is either relatively prime to a number ‘n’ or divide it.
- A product is divisible by a prime ‘p’ only when ‘p’ divides one of the factors.
- Every n > 1 is divisible by some prime.

## Table of Contents

## What is a Prime Number?

**Definition:** Prime numbers are a special category of natural numbers or positive integers which are exactly divisible by 1 and the number itself. So, a prime number has exactly two divisors.

- Largest Prime Number: The largest known prime number is 2
^{82,589,933}− 1 as of June 2021. It is said to have 24,862,048 digits. - Smallest Prime Number: As we know, 2 is the smallest prime number.

### Is 1 a Prime Number?

According to the prime number definition, we can say that 1 is not a prime number. The definition says that a prime number should have exactly two divisors. But 1 has only one divisor. So, 1 is not a prime number.

### Is 2 a Prime Number?

Yes, 2 is a prime number because it is divisible by itself and 1.

### Are all Prime Numbers Odd?

Yes, all prime numbers are odd, but only even prime number is 2. Since every other even number can be divide by a 2.

## Types of Prime Numbers

1. Twin prime numbers

Example: (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43)

2. Cousin prime numbers

Examples: (3, 7), (7, 11), (13, 17), (19, 23), (37, 41)

3. Balanced prime numbers

Examples: 5, 53, 157, 173, 211, 257, 263

4. Palindromic prime numbers

Example: 2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919

5. Reversible prime numbers

Example: 13, 17, 31, 37, 71, 73, 79, 97, 107, 113, 149, 157

6. Pythagorean prime numbers

Example: 5, 13, 17, 29, 37, 41, 53, 61, 73

7. Permutable prime numbers

Example: 2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79

8. Mersenne prime numbers

Example: 3, 7 , 31, 127, 511, 2047

## List of Prime Numbers

Till now, we have explained the concept of prime numbers and given below is list of prime numbers for your better knowledge. The list of prime numbers from 1 to 1000 are as follows:

Numbers | Number of prime numbers | List of prime numbers |

1 to 100 | 25 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 |

101-200 | 21 numbers | 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199 |

201-300 | 16 numbers | 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293 |

301-400 | 16 numbers | 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397 |

401-500 | 17 numbers | 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499 |

501-600 | 14 numbers | 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599 |

601-700 | 16 numbers | 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691 |

701-800 | 14 numbers | 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797 |

801-900 | 15 numbers | 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887 |

901-1000 | 14 numbers | 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997 |

Total number of prime numbers from 1 to 1000 = 168 |

## Prime Numbers up to 100

The following are the list of prime numbers up to 100. So students, go ahead with us and learn about prime numbers.

### Prime Numbers up to 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.

Numbers are found everywhere in mathematics as well as in day to day life. There are different types of numbers in mathematics. Prime numbers are one of the important ones. A prime number is a position integer that is divisible by only 1 and itself. i.e. there is no number other than 1 and itself that divides a prime number. There are various properties that prime numbers possess. These properties are listed below :

- Prime numbers are positive numbers greater than 1 .
- For a number to be a prime number, it must be non-zero whole number.
- Prime numbers are the numbers that cannot be divided by any number except themselves and one.
- Prime numbers have only two factors.
- The two factors of prime numbers are one and the number itself.
- The way of finding the prime numbers is called integer factorization or prime factorization.

## How to Find Prime Numbers

The Greek Eratosthenes formed a technique to find out these prime numbers, although it only worked over a restricted range:

1 | 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 |

- Write out the numbers from 1 to 100 as shown above.
- Cross out the number 1, because all primes are greater than 1.
- Number 2 is a prime, so we can keep 2 as it is, and we cross out all the numbers that are multiplies of 2. (like 4,6,8,10 ….)
- Number 3 is also a prime number, so we keep 3 as it is and we cross out all the numbers that are the multiples of 3.( like 3,6,9,…….)
- The next number left is 5, so we keep 5 as it is and we cross out all the numbers that are the multiples of 5 (like 10 ,15, 20, 25…..)
- At last, the number left in the first row is number 7, now we keep 7 as it is and we cross out all the numbers that are the multiples of 7 (like 14, 21, 28 ……)
- Now, all the remaining numbers in the table are prime numbers.

**Let’s identify whether the following numbers are prime or not**

The number 2 is exactly divisible by 2 and 1.

The number 3 is exactly divisible by 3 and 1.

The number 4 is exactly divisible by 4, 2 and 1

The number 5 is exactly divisible by 5 and 1

The number 6 is exactly divisible by 6, 3, 2 and 1

The number 7 is exactly divisible by 7 and 1

The number 8 is exactly divisible by 8, 4, 2 and 1

The number 9 is exactly divisible by 9,3 and 1

The number 10 is exactly divisible by 10, 5, 2 and 1

The numbers 4, 6, 8, 9, 10 are not prime numbers since it has more than 2 factors.

The numbers 2, 3, 5, 7 are prime numbers since it has exactly two factors, 1 and the number itself. You can learn more about prime numbers online with the help of our tutorials.

## Twin Prime Numbers

A twin prime numbers are defined as a pair of prime numbers having a difference of two. In other words, A pair of prime numbers which have a gap of 2 are called twin prime numbers. It means that twin primes are consecutive odd numbers which are both prime.

For example: (3, 5), (5, 7), (11,13), (17, 19), (41, 43) etc. Whether this list ends or not, it is unknown.

Also, (2, 3) are not categorized as twin prime, since they do not have a difference of two.

## Prime Number Theorem

Prime Number Theorem precisely gives us the value to which the density of prime numbers less than a given number approaches as the numbers grow larger and larger i.e. tend to infinity. There are many proofs easily accessible on the internet or otherwise. Hence, only its statement is being given here.

**Statement of Prime Number Theorem:**

Let P(x) be the number of prime numbers less than x, where x > 1 and x is a real number. Then the prime number theorem states

This asymptotic formula precisely gives the value to which prime number density approaches as the numbers grow larger and larger. It has been termed as one of the most beautiful and important theorems in the history of mathematics and number theory.

## Prime Number Calculator

Is it a prime number? Find out and generate the list of prime numbers using our prime numbers calculator.

### Prime Number Calculator

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## Prime Number FAQs

**is 1 a prime number?**

No, 1 is not prime number, since it has only one divisor, namely 1.

**is 2 a prime number?**

Yes, 2 is a prime number because it has two divisors, namely 1 and itself.

**is 3 a prime number?**

Yes, 3 is a prime number, since it divisible by 1 and 3.

**What is largest prime number?**

The largest prime number is 2^{82,589,933} − 1 and it have 24,862,048 digits.

**What is smallest prime number?**

The smallest prime number is 2.