Prime Number Formula

Prime Numbers vs Composite Numbers

From the above text, we see that 6 has factors of 1, 2, 3, and 6. The factors are in addition to 1 and 6. If this case occurs, we say that 6 is a composite number.


However, a number “N” having only two factors, i.e., 1 and N itself is a prime number.


We do have the formula to find prime numbers; also, a way of how to calculate the prime number;  this page will discuss the same in detail.


Prime Numbers

Let’s suppose that we have a set of numbers: 1, 3, 5, 7, 11,........., and so on. Additionally, we can determine the prime number or not by looking at its factors. 


For instance, 

6 = 1 x 6 = 1 x 2 x 3

6 = 2 x 3


So, 6 has 4 factors: 1, 2, 3, and 6


However, 11 = 11 x 1; it has only two factors, 11 and 1 itself. 


Properties of Prime Numbers

  • A number greater than 1 is divisible by at least one prime number.

  • We can express an even positive integer greater than 2 as the sum of two primes.

  • Other than 2, other prime numbers are odd. Thus, we can say that 2 is the only even prime number.

  • Two prime numbers are coprime to each other.

Formula to Find Prime Numbers

The prime number definition says that it is a natural number greater than 1 and that is not a product of two smaller natural numbers. 


In simple words, it is a natural number that has only two distinct natural number divisors: 1 and itself. 

For instance, 7 is prime because the only ways of writing it as a product are 1 × 7 or 7 × 1, involving 7 itself. 

Point To Note:

2 is the only even prime number.


So, there are two methods in a prime number formula:


Method 1: 


A prime number is written in the form of 6n - 1 or 6n + 1. Here, n can be any number except the multiple of prime numbers, i.e., 2, 3, 5, 7, 11,......, etc.


Let’s suppose that n = 1. Now put this value in 6n - 1:


=> 6 x 1 - 1 = 5


∴  5 is a prime number


Another Example,


Putting n = 3 in the equation 6n + 1, we get:


=> 6 x 3 + 1 

=> 18 + 1

=> 19


We get that 19 is a prime number.


Conversely, put n = 4 in the above formula:


=>   6 x 4 - 2  = 24 - 2 = 20


Here, 20 has five factors, i.e., 1, 2, 4, 5, and 20. This is not possible because a prime number can have only two factors, i.e., 1 and the number 20 itself. 


Method 2: How to Calculate the Prime Number?


To determine the list of prime numbers greater than 40, the below prime number formula can be used:


=> n2 + n + 41….(1)


Here, we will put whole numbers one by one from 0, 1, 2, 3,....., till 39 (the number should be less than 40).


Let’s start with 0 and put the same in the equation (1):


=> 02 + 0 + 41



=> 41 


You get a prime number, which means the formula you have used here is correct.


Let’s verify with another number, say 14, and put the same in equation (1):


=> 142 + 14 + 41


=> 196 + 14 + 41


=> 196 + 55


Solving this equation, we get:


=> 251


∴ On putting n = 14, we get another prime number, i.e., 251. 


The factors of 251 are 1 and 251.


We have the formula to find the sum of prime numbers; let’s understand it with the help of illustrative examples:

Formula for Sum of Prime Numbers

Let’s suppose that you went to a grocery shop to purchase some items and you find that all of these are arranged in an odd-even sequence. The sequence resembles the ODD-EVEN formula of Delhi used by CM Arvind Kejriwal to prevent pollution in the city. 


Considering only the items in the prime number list, we find the sum of prime numbers from 1 …..n prime number items by using the following formula:


=> 2, 5, 7, 17, 19, 23, 37


=> 2 + 5 + 7 + 17  + 19 + 23 + 27


The sum of prime numbers is:


=> 100

Average of Prime Numbers Formula

You know that the average of a number is given by the formula:


= Sum of observations/number of observations


Let’s consider a set of prime numbers: 3, 5, 7, 13, and 17. The average of these numbers is:

 

=> (3 + 5 + 7 + 13 + 17) 5


We get the average as;


=> 45/5 = 9


Let’s take another set of prime numbers:


Set = 3, 5, 7, 13, 17, 23, and 29


Now, finding the average:


=> (3 + 5 + 7 + 13 + 17 + 23 + 29)/7


The average of prime numbers is:


=> 97/7 


=> 13.87

Conclusion

The numbers that are divisible by 1 and themselves are called prime numbers. Also, such numbers are positive integers and they have only two factors, i.e., 1 and the integer themselves.


Now, let’s have a look at some FAQs on Prime Number by using various formulas:

FAQs (Frequently Asked Questions)

Q1: Determine Whether the Following Numbers are Prime or Composite: 54, 53, 61, 88

Ans: 

1. 54

The prime factors of 54: 1, 2, 3, 6, 9, 18, 27, and 54. Here, 54 has many factors, which shows that 54 is a composite number, not prime.


2. 53

The prime factors of 53 are 1 and 53, which means 53 is a prime number.


3. 61

The prime factors of 61 are 1 and 61, which means 61 is a prime number.


4. 88

The prime factors of 88 are 2, 4, 8, 11, and 88. This shows that 88 is a composite number.

Q2: Write the First 100 Prime Numbers.

Ans: The list of prime numbers from 1 to 100 is: 


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.

Q3: Is 1 a Prime Number?

Ans: As per the prime number definition, a prime number always has two factors; however, 1 carries only one factor. It means 1 is not a prime number.