Have you wondered why some numbers are divisible by numbers smaller than themselves and some are not divisible at all? If not, then it’s time to think about it now. If yes, then why are some numbers not divisible at all? They may be divided by other numbers; you will get answers in decimals or fractions when they do. So, why does it happen so? You are going to learn in this article.

## What are prime numbers?

Prime numbers are not divisible by any other number than themselves. In the paragraph mentioned above, prime numbers are the ones that do not get divisible by the smaller number. In fact, they are not divisible by any other number. In other terms, prime numbers are the ones who have only two factors – 1 and itself. All prime numbers will have two factors only. Let’s understand what prime numbers are by examples.

### Examples of Prime Numbers

Take a number, for instance, 16. You can divide 16 by 2. 16 is also divisible by 4, 8, and 16. After dividing 16 from 2, you will get 8. After dividing 16 from 4 and 8, you will get 4 and 2, respectively. In the same way, when 16 gets divided by 16, you will get 1.

Now, let us take another number, say 31. Is 31 divisible by any other number? 31 is not divisible by 2, 3, 4, or any other number. 31 is only divisible by 31, which results in 1. 31 is also divisible by 1, which results in 31. Hence, 31 is a prime number having only two factors, 1 and 31, i.e., 1 and the number itself.

#### Understanding prime numbers through illustrations

Understanding prime numbers is easier through illustrations. Take a number, say 6 and 11. If you can draw the given numbers in equal rows or columns, then it means it is divisible. Firstly, take 6. You can write 6 in two rows of 3 and 2, as shown in the image. This means 6 is divisible by 3 and 2.

Let us take another number, 11. You cannot draw 11 in equal rows or columns. The possible arrangements can be either in a single row or column or haphazardly, as shown in the picture below. This shows 11 is a prime number as it is not divisible by any other number.

This method is convenient for smaller numbers. For bigger numeric values, this method is not appropriate for finding which numbers are prime and non-prime.

## Prime Numbers List

Here is the list of prime numbers from 1 to 100 given below in the table:

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 |

You can memorize these prime numbers up to 100 for ease of calculations in your exams. This list of prime numbers to 100 can be memorized by understanding how to find prime numbers.

### How to find prime numbers

It is important to understand how to find prime numbers if you need to find all prime numbers from a given set of numbers. There are several methods to find prime numbers. The most commonly used are the given two methods:

**Method 1:**If you need to find all prime numbers up to 100, this formula can come in handy – n^{2}+ n + 41. However, this formula will give prime numbers greater than 40 only. For prime numbers below 40, you have to memorize the table given above. From the formula, replace n with the number starting from 0. For example, let us take 0. If you replace n with 0, the formula will give the value – 0^{2}+ 0 + 41 = 41. Similarly, for other numbers greater than 0, the prime numbers will be:

1^{2 }+ 1 + 41 = 43

2^{2 }+ 2 + 41 = 47

3^{2 }+ 3 + 41 = 53

4^{2 }+ 4 + 41 = 61… and so on.

**Method 2:**Apart from numbers 2 and 3, every prime number can be written in the form of 6n – 1 or 6n + 1. If you cannot write the number in either of these forms, then it means the number is not prime. Let us understand this concept by an example.

Take a number, say 16. We cannot write this number in either of these forms; neither 6n – 1 nor 6n + 1. Take another number, say 13. You can write number 13 in the form of 6n + 1 as 6 (2) + 1 = 13. Therefore, 13 is a prime number. This method is useful in finding various prime numbers because you need to check the divisibility by 6 in the first place and then check whether it fits in this formula or not.

Using method 2, you can find all prime numbers list greater than 100. The table below shows prime numbers list up to 500:

2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281 283 293 307 311 313 317 331 337 347 349 353 359 367 373 379 383 389 397 401 409 419 421 431 433 439 443 449 457 461 463 467 479 487 491 499 |

### Types of Prime Numbers

Have you ever heard of types of prime numbers? Yes, types of prime numbers. Well, there are various types of prime numbers. Some commonly known types of prime numbers are:

- Balanced primes have an equal size gap between the former and later prime numbers so that they are equal to the arithmetic mean of the nearest prime number. Example – 5, 53, 157, 173, 211, 257, 263, 373…
- Gaussian primes follow the form p
_{n}^{2}> p_{n-i}p_{n+i}for all 1 ≤ i ≤ n – 1, where p_{n}is the n^{th}prime number. Example – 5, 11, 17, 29, 37, 41, 53, 59, 67, 71, 97, 101, 127, 149, 179… - Circular primes remain prime on any cyclic rotation of their digits. Example – 2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, 97, 113…
- Factorial primes are in the form n! – 1 or n! + 1. Example – 2, 3, 5, 7, 23, 719, 5039, 39916801, 479001599…
- Euler irregular primes divide Euler number E
_{2n}for some 0 ≤ 2n ≤ p – 3. Example – 19, 31, 43, 47, 61, 67, 71, 79, 101, 137, 139, 149… - Harmonic primes have no solutions to H
_{k}≡ 0 and H_{k}≡ -⍵_{p}for 1 ≤ k ≤ p – 2, where H_{k}denotes the k^{th}harmonic number, and ⍵_{p}denotes the Wolstenholme quotient. Example – 5, 13, 17, 23, 41, 67, 73, 79, 107, 113, 139, 149, 157, 179… - Pythagorean primes follow the form 4n + 1. Example – 5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137, 149, 157, 173, 181…
- Ramanujam primes are the smallest integers to give at least n prime numbers from x/2 to x for all x ≥ R
_{n}. Example – 2, 11, 17, 29, 41, 47, 59, 67, 71, 97, 101, 107, 127, 149, 151, 167… - Eisenstein primes are irreducible real numbers that follow the form 3n – 1. Example – 2, 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101, 107, 113, 131, 137, 149, 167, 173…
- Fibonacci primes are the primes that follow the Fibonacci sequence. Example – 2, 3, 5, 13, 89, 233, 1597, 28657, 514229, 433494437, 2971215073, 99194853094755497…

#### Facts about Prime Numbers

To summarize prime numbers, the numbers having only two factors – 1 and itself – are known as prime numbers. The smallest even prime number is 2. If the prime numbers occur just after one another, they are known as twin prime numbers. For example, 41 and 43 are the two odd numbers that come just after another. Hence, they are twin prime numbers. Obviously, no even number is a prime number as every even number is divisible by 2.

Prime numbers play a crucial role in everyday life. From making computers to keeping them secure, prime numbers are used abundantly as they cannot be figured out easily. Living beings also use prime numbers. Cicadas’ life cycle revolves around prime numbers; modern screens are made using prime numbers, manufacturers use prime numbers to balance their products, and much more. Thus, it is important to understand and learn prime numbers to locate fascinating facts in nature.