### What are Prime Numbers?

Prime numbers are natural numbers greater than one. They only have two factors: 1 and the number itself. This means that these numbers cannot be divided by any number other than 1 and the number itself without leaving a remainder. Some prime numbers are 2, 3, 5, 7, 11, 13, 17, 19 and so on. Except for 2, all prime numbers are odd.** **Let’s understand what prime numbers are by examples.

### Understanding prime numbers through illustrations

Understanding 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 that 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 |

Thus, between 1 and 100, there are 25 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. All these numbers are divisible only by one and by the number itself. As a result, these are known as prime numbers. These are also the first 25 prime numbers.** **

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

### Properties of Prime Numbers

- Every number greater than one has at least one prime number that can be divided by it.
- The sum of two primes can be expressed as any even positive integer greater than 2.
- With the exception of 2, all other prime numbers are odd.
- Co-prime numbers are always two 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 the 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 listed 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.

### Examples of Prime Numbers

**Example 1. **Between 40 and 50, what prime numbers are there?

**Ans:** The Prime numbers are 43 and 47

**Example 2. **Write 40** **as the sum of two odd** **primes**.**

**Ans:** 40 = 37 + 3

**Example 3.** What is the biggest prime number between 1 and 20?

**Ans:** The prime numbers are 2, 3, 5, 7, 11, 13, 17 and 19

** **The biggest are between 1 and 20 is 19

**Example 4. **What is the smallest prime number ranging from 1 to 10?

**Ans: **The prime numbers are 2, 3, 5 and 7

** ** 2 is the smallest between 1 and 10.

**Example 5.** Find out if 79 is a prime number. ** **

**Ans: **The only factors of 79 are 1 and 79.

#### Frequently Asked Questions?

#### 1. Is 1 a Prime Number?

The definition of a prime number is that it has exactly two factors: 1 and itself. As such, when we look at the factors of 1, we see that there are only two possibilities: one and itself. So while 1 isn’t exactly an odd number, it does have exactly two factors: 1 and itself.

#### 2. How Do You Calculate Prime Numbers?

To calculate prime numbers, you can use the sieve of Eratosthenes. This technique will only work for numbers up to a certain limit (the size of your computer’s memory). For larger numbers, you’ll need to use a more advanced method.

#### 3. What Is the Average of Prime Numbers Between 1 and 100?

There are 8 tween primes between 1 and 100. The sum of prime numbers between 1 and 100 is 1060. The average of prime numbers between 1 and 100 is 42.4. The product of prime numbers between 1 and 100 is 2.3056 × 1036.

#### 4. What Is the Highest and Smallest Prime Number Between 1 and 100?

The highest prime number between 1 to 100 is 97. And the smallest between 1 to 100 is 2.

#### 5. How Many Even Prime Numbers Can Be Divided by 2?

The answer is zero. All even prime numbers are even, and all even numbers can be divided by 2.

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