# Prime Numbers: Definition | Examples | Types of Prime Number

Prime numbers are the positive integers that have only two factors, 1 and the integer itself. It means that any number that is divisible by itself and One. Let’s take an example of the number 3, it can be factored as 3 and 1. So, 3 is a Prime number.

Also, learn about the below classifications of numbers:

## Definition of Prime Numbers

It is a natural number greater than 1 and has exactly two factors. One is the number itself and the other is 1. Numbers that are not Prime numbers are Composite numbers.

From the above definition, it’s clear that 1 is not a Prime number. As the definition says the number is greater than one.

## First ten Prime Numbers

Below are the first ten Prime numbers in the number series:

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

## What are Co-prime Numbers?

Co-prime numbers are those numbers that have their Highest Common Factor (HCF) as 1. Or in other words, we can define two numbers are said to be co-prime if they have no common factor other than 1.

These are considered in pairs. Individually they can be prime or not but together if they’ve the Highest Common Factor of 1 then that number pair is said to be Co-prime Numbers.

Example: 18 and 35. The factors of 18 are 1, 2, 3, 6, 9, and 18 while the factors of 35 are 1, 5, 7, and 35. From the factors list, we can see that the Highest Common Factor is 1.

## How to find Prime Numbers?

### Small Prime Numbers

Factorization is the best and shortest way to find Prime Numbers that are smaller. The following steps are used in Factorization:

• First, find the factors of a number.
• Check the number of factors.
• If the number of factors is greater than 2 then that number is not Prime.

### Large Prime Numbers

#### Simple Checks method for Prime Numbers

To find whether any given large number is a prime number or not, we follow the simple checks method as per the below steps:

1. The given number should not end with 0,2,4,6, or 8.
2. Find the sum of all digits of the given number and check if it’s divisible by 3 or not. If it’s divisible then that number is not Prime.
3. Find the square root of the given number.
4. If the square root is a perfect square then retake the square root.
5. Divide the given number by all the prime numbers below its square root.
6. If the given number is divisible by any of the prime numbers less than its square root, then that number is not a prime number, else it’s a prime number.

For Example: let’s test for the number 104729. Checks as per the above simple checks method:

1. The given number ends with 9. Proceed to the next step.
2. The Sum of all digits of the number (104729) is 23. 23 is not divisible by 3. Let’s proceed to the next step.
3. The square root of the given number (104729) is 323.61…
4. This step is not required as the square root is not a perfect square.
5. Divide the given number (104729) by all the prime numbers less than 323.
6. Final Result: 104729 is Prime Number.

### Formula to Find Prime Numbers

#### Formula 1

Every prime number which is greater than 11 and is not multiples of prime numbers below 13 (for example 2,3,5,7, and 11) can be written in the form of 6n + 1 or 6n – 1, where n is a natural number.

For Example:
For n= 1 : 6(1) – 1 = 5
For n= 1 : 6(1) + 1 = 7
For n= 2 : 6(2) – 1 = 11
For n= 2 : 6(2) + 1 = 13
For n= 3 : 6(3) – 1 = 17
For n= 3 : 6(3) + 1 = 19
For n= 4 : 6(4) – 1 = 23
For n= 4 : 6(4) + 1 = 25 (multiple of 5) we’ll exclude this number.

#### Formula 2

This formula is applicable when we try to find out the prime numbers that are greater than 40. n2 + n + 41.

For Example:
For n= 0 : (0)2 + 0 + 0 = 41
For n= 1 : (1)2 + 1 + 41 = 43
For n= 2 : (2)2 + 2 + 41 = 47
For n= 3 : (3)2 + 3 + 41 = 53

## Prime Numbers between 1 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, and 97

## Prime Numbers between 100 to 200

101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, and 199

## Prime Numbers between 200 to 500

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, and 499

## Prime Numbers between 500 to 1000

503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, and 997

## FAQs

### What are Prime Numbers?

Prime numbers are the positive integers that have only two factors, 1 and the integer itself.

### Is 1 Prime Number?

No, as the definition of Prime Numbers says any number greater than one.

### First ten Prime Numbers?

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

### A negative number can be Prime Numbers?

No, as the definition of Prime Numbers says any number greater than one.

### Smallest Prime Number?

The smallest prime number is 2.

### Largest Prime Number?

Great Internet Mersenne Prime Search (GIMPS) was able to successfully find the largest prime number as 282,589,933 − 1, a number with 24,862,048 digits in December 2018.

## Interesting facts on Prime Numbers

• ‘1’ is not a prime number.
• ‘2’ is the smallest prime number.
• Every prime number is co-prime to each other.
• Any two successive numbers or integers are always co-prime.
• The sum of any two co-prime numbers is always co-prime with their product.
• ‘2’ is the only prime number that is even, the rest of the other primes are odd numbers.
• ‘2’ and ‘3’ are the only two consecutive prime numbers.
• Largest prime number as 282,589,933 − 1, a number with 24,862,048 digits found in December 2018.