Prime Numbers

Prime numbers are natural numbers greater than 1, having exactly two factors, 1 and the number itself. It can also be defined as a whole number that can’t be exactly divided by anything except 1 and itself. Thus negative numbers cannot be prime.

They are important in mathematics because they are the building blocks of the natural numbers. 2, 3, 5, 7, 11, 13, 17, 19 are a few examples of prime numbers.

Natural numbers greater than 1 that are not prime are known as composite numbers.

0 is not a prime number, as it has infinite factors. Similarly, 1 is also not a prime number as it has only one factor (the number 1 itself).

Difference between Prime and Composite Number

Prime Number	Composite Number
It has exactly two factors.	It has more than two factors.
It can be divided evenly only by 1 and the number itself. For example, 2 is divisible only by 1 and 2.	It can be divided evenly by all its factors. For example, 4 is divisible by 1, 2, and 4
Examples: 11, 13, 19, 37, 41 etc.	Examples: 10, 18, 24, 45, 49 etc.

List

A list of prime numbers from 1 to 100 is provided below for reference:

Between 1 and 10	2, 3, 5, 7
Between 11 and 20	11, 13, 17, 19
Between 21 and 30	23, 29
Between 31 and 40	31, 37
Between 41 and 50	41, 43, 47
Between 51 and 60	53, 59
Between 61 and 70	61, 67
Between 71 and 80	71, 73, 79
Between 81 and 90	83, 89
Between 91 and 10	97

Thus there is a total of 25 prime numbers between 1 and 100. 2 is the only even prime number, and the rest are all odd. Again, 2 and 3 are the only two consecutive prime numbers.

Chart

Eratosthenes’s method of finding prime numbers resulted in the formation of charts. Even today, they are helpful for our direct applications and mathematical calculations.

How to Find Prime Numbers

There are different ways of finding prime numbers. They are:

Method 1: The Sieve of Eratosthenes

Before the discovery of calculators and computers, this method was the most widely used for finding prime numbers. It was named after its discoverer Eratosthenes. 

Here the steps show how to identify prime numbers between 1 and 100.

1. All the numbers from 1 to 100 are listed. 1 is left out as it is neither prime nor a composite number.
2. We start at the first prime number, 2, and encircle it. All the multiples of 2 except 2 are crossed out as those are not prime numbers. For example, 4, 6, 8, 10, 12, and so on, up to 100 are not prime.
3. The first uncrossed number next to 2, i.e., 3, is encircled. All the multiples of 3, except 3, are crossed out. As few multiples like 6, 12, 18, 24, 30 … are already crossed out, the remaining ones, like 9, 15, 21, 27, 33, and so on, up to 100, are crossed out now.

The process repeats until all the numbers on the list are either encircled or crossed out.

Thus, we obtain a list with all the encircled 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 and 97. We can also verify the list from the list of prime numbers above.

Method 2

It is used to find all prime numbers except 2 and 3. Each prime number except the only consecutive prime numbers 2 and 3 can be represented as (6n + 1) or (6n – 1); here, n is a natural number.

For example, 11 can be written as (6 × 2) – 1, (here n = 1). So, 11 is a prime number.

Similarly,

31 = (6 ×5) + 1

37 = (6 ×6) + 1

53 = (6 ×9) – 1,

61 = (6 × 10) + 1  and so on.

Thus 11, 31, 37, 53, and 61 are prime numbers.

Method 3

It is used to find prime numbers greater than 40. If a number greater than 40 can be expressed as (m² + m +41), m = 0, 1, 2, … it is a prime number.

For example,

41 = 0² + 0 + 41

43 = 1² + 1 + 41

47 = 2² + 2 + 41

Thus, 41, 43, and 47 are prime numbers.

Other Related Terms

Smallest Prime Number	The smallest prime number is 2, which is the only even prime number. 3 is the smallest odd prime number.
Largest Prime Number	There exists no largest prime number as the natural numbers do not end. The largest known prime number (as of April 2023) is 282,589,933 − 1, which has 24,862,048 digits when written in base 10.
Co-prime Numbers	A pair of numbers are co-prime to each other if their Greatest Common Factor (GCF) is 1. Co-prime numbers can be either prime or composite numbers. 4 and 13 are co-prime numbers as their GCF is 1. A pair of two prime numbers are always co-prime to each other as they have 1 as their only common factor. (3, 5), (5, 7), and (7, 11) are some pairs of co-prime numbers that are also prime.
Twin Prime Numbers	Two prime numbers are twin primes if only one composite number exists between them. They always differ by 2. (3, 5) as there is only one composite number, 4, between them. (11, 13), (17, 19), and (29, 31) are some other examples.

Odd Numbers and Prime Numbers

Though all the prime numbers except 2 are odd, the converse is false.

For example, 5, 7, 11, and 13 are prime and odd numbers.

But if we consider 15, 21, and 27, all of them are odd numbers.

15 has 4 factors:  1, 3, 5 and 15

21 has 4 factors: 1, 3, 7, and 21

27 also has 4 factors 1, 3, 9 and 27

So these are not prime numbers.

Solved Examples

E.g.3.