Understanding and Calculating Prime Numbers

What is the easiest way to find prime numbers? 

As defined by the definition of a prime number, it only contains two factors. The first is 1, and the second is its original value. We, therefore, need to find the numbers consisting only of two factors. 

By using a simple method called prime factorisation, this can be accomplished. This article explains how we can find prime numbers (small and large) using the factorisation algorithm.

First of all, start by understanding what prime numbers are. It is easy to find the primes when dealing with smaller numbers, but we have to find another way for larger numbers. Therefore, we have illustrated how the prime numbers are evaluated, not just for small numbers but also for bigger ones. 

A shortcut method for finding prime numbers between 1 and 100 will also be discussed here.

So, let’s first understand, “What are prime numbers?”

Definition of prime numbers

Prime numbers are natural numbers greater than 1, which are divisible only by themselves and by 1. Here are the first few prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23.

If you divide a prime number by any other number, you will end up with a remainder. As an example, 13 is a prime number. Only 1 and 13 can divide it. As a result of dividing a prime number by another number, there will be remainders, e.g., 13 * 6 = 2 remainders. 15 can be divided by both 5 and 3 as well as by itself and 1. This does not qualify it as a prime number.

Prime Numbers: 1-100 

To make it easier for you to understand, create a table by writing 2-100 in 10 rows. Then follow the simple steps to learn about the prime number from 2- 100 because 1 cannot be a prime number (it has only one factor).

• Let's begin with 2. 2 is a prime number, but since it is divisible by 2, all its multiples will be composite numbers. In this case, we cross every multiple of 2 out of the table.
• The next prime number is 3, so all multiples of 3 are composite numbers to remove them. 
• The next prime number is 5, cross out all multiples of 5 like 10, 15, 20 and keep going till 100. 
• Now we have the prime number 7, and we cross out all of the multiples of 7.
• Our next prime number is 11, so we cross out all multiples of 11, 22, 33, 44, 55, 66, 77, 88, and 99. 

Having already crossed out all of these numbers, we have completed our chart by crossing out all composite numbers. Now you can look at your table and read aloud which numbers you have found as a prime number from 2-100. 

Smallest prime number

In the prime number series, 2 is the smallest. In addition, it is the only even prime number - all other even numbers can be divided by themselves at least one and two times, so they have at least three factors in common.

Largest prime number

The famous mathematician Euclid proved that there was no largest prime number in the classical era. However, the Great Internet Mersenne Prime Search is still searching for it, and many scientists and mathematicians still hope to find it.

As of November 2020, the largest prime number is 2^(82,589,933)^-1, which has 24,862,048 digits when written in base 10.

How to easily find prime numbers

Prime numbers can be determined in a variety of ways. The factorisation is the most effective way to determine prime numbers. By factorising a number, the factors are obtained, making it easy to determine if a number is prime.

# Method 1

The Factorisation Method for Finding Prime Numbers

A prime number can be found by factorisation. To use this method, follow these steps:

• Step 1: Find the factors of the number in the first place
• Step 2: Determine how many factors there are in that number
• Step 3: The number does not qualify as a prime if more than two factors are involved.

Example: 

Suppose you take 36 as an example.

36 can now be expressed as 2 x 3 x 2 x 3. Therefore, the factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36. Considering 36 has more factors than 2, it is a composite number rather than a prime.

Let's take the example of 17. Factorising 17 is 1 x 17. This means that 17 is composed of two factors. This makes it a prime number.

# Method 2

Using the Formula 6n ± 1

• Step 1: Divide the given number by 6.
• Step 2: If the remainder is 1, then the given number is a prime number; otherwise, it is a composite number.

Example:

Let's take the example of 37.

If we divide 36 by 6, we get 1 as a remainder. We can express the number 37 using the formula 6 x 6 1 = 36 1 = 37. Thus making 37 a prime number. 

Rules of the Prime Numbers Formula

In dealing with prime numbers and their formulas, keeping a few important details in mind is essential. 

• It is impossible for any even number placed in the unit's place to be prime.
• There is only ever one even prime number, and that is 2.
• In the case of large numbers, if the sum of all the digits is divisible by 3, it isn't a prime number.
• All other numbers, except numbers 2 and 3, can be expressed as 6n ± 1 (prime number formula), where n = natural number. 

Prime number examples

# Example 1

Check whether 57 is a prime number.

Solution: According to the division method, 1, 3, 19, and 57 divide 57 completely and leave no remainder.

57 consists of 4 factors. Due to the fact that it has more than two factors, 57 is not a prime number.

The number 57 could not be expressed as 6n ± 1 with 'n' equal to 1,2,3...

The number 57 is therefore not a prime number.

# Example 2

Find out if 79 is a prime number. 

Solution: Using the division method, 1 and 79 divide 79 completely, leaving no remainder.

79 cannot be divided entirely by any other number. Therefore, 79 only has two factors.

Dividing 79 by 6 is another way to calculate this. We get the remainder 1. 

As a result, we can represent it as 6n 1 : 6 × 13 1 = 78 1 = 79

As a result, 79 is a prime number.

# Example 3

Use the prime numbers formula to determine if 19 is a prime number. 

Solution: 19 has two factors, 19 and 1.

We can check if 19 is a prime number using this formula: 6n ± 1

19 divided by 6 gives you 1 as the remainder. 

Therefore, 19 can be represented as 6 × 3 1 = 18 1 = 19

The number 19 is, therefore, a prime number. 

Conclusion

Cyber security - making the internet safer as information is shared over the internet - is one of the most important uses for prime numbers. So it is important to understand the concept of prime numbers right from the basic. Once you understand what prime numbers are and how to find them, you will never forget this. Keep learning!