Prime Number Formula

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How to Find Prime Numbers?

Introduction to Prime Numbers

If a number is given and you are asked whether it is a prime number or not. What shall you do? How will you know? Do you know how to find prime numbers? So here in this article, we will be discussing what a prime number is, how to find prime numbers easily, and how to check prime numbers.

Prime numbers are the numbers which have only two factors, the number itself and 1. So we have to find such numbers which have only two factors. We will be studying various methods to find prime numbers, how to check prime numbers, and tables for prime numbers 1 to 200. 

What is the Prime Number?

A prime number is an integer greater than one and can be divisible by only itself and one i.e it has only two factors. Zero, one, and numbers less than one are not considered as prime numbers.

A number having more than two factors is referred to as a composite number. The smallest prime number is 2 because it is divisible by itself and 1 only.

Finding Prime Numbers Using Factorization

The most common method used to find prime numbers is by factorization method. 

The steps involved in finding prime numbers using the factorization method are:

  • Step 1: First let us find the factors of the given number( factors are the number which completely divides the given number)
  • Step 2: Then check the total number of factors of that number
  • Step 3: Hence, If the total number of factors is more than two, it is not a prime number but a composite number.

For Example: Take a number 45. Is it a prime number?

Factors of 45 are 5 x 3 x 3

Since the factors of 45 are more than 2 we can say that 45 is not a prime number but a composite number.

Now, if we take the example of 11. The prime factorization of 11 is 1 x 11. You can see here, there are two factors of 11. Hence, 11 is a prime number.

Methods to Find Prime Numbers

Prime numbers can also be found by the other two methods using the general formula. The methods to find prime numbers are:

Method 1:

Two consecutive numbers which are natural numbers and prime numbers are 2 and 3. Apart from 2 and 3, every prime number can be written in the form of 6n + 1 or 6n – 1, where n is a natural number.

For example:

6(1) – 1 = 5

6(1) + 1 = 7

6(2) – 1 = 11

6(2) + 1 = 13

6(3) – 1 = 17

6(3) + 1 = 19…..so on

Method 2:

To find the prime numbers greater than 40,the general formula that can be used is n2+ n + 41, where n are natural numbers  0, 1, 2, ….., 39

For example:

(0)2 + 0 + 0 = 41

(1)2 + 1 + 41 = 43

(2)2 + 2 + 41 = 47

(3)2 + 3 + 41 = 53 

(4)2 + 2 + 41 =  59…..so on 

Note: These both are the general formula to find the prime numbers. But values for some of them will not yield a prime number.

Table of Prime Numbers 1 to 200

Here is the list of prime numbers from 1 to 200. This table will help you to find any prime numbers 1 to 200.

List of Prime numbers from 1 to 200 

Memorize this list of prime numbers from 1 to 200 and you can easily identify whether any number is prime or not.

Points to be Noted

  • Numbers having even numbers in one’ place cannot be a prime number.
  • Only 2 is an even prime number; all the rest prime numbers are odd numbers.
  • To find whether a larger number is prime or not, add all the digits in a number, if the sum is divisible by 3 it is not a prime number.
  • Except 2 and 3, all the other prime numbers can be expressed in the general form as    6n + 1 or 6n – 1, where n is the natural number.

Solved Examples

Example 1: Is 19 a Prime Number or not?

Solution:

We can check if the number is prime or not in two ways.

Method 1:

The formula for the prime number is 6n + 1

Let us write the given number in the form of 6n + 1.

6(3) + 1 = 18 + 1 = 19

Method 2: 

Check for the factors of 19

19 has only two factors 1 and 19. 

Therefore, by both methods, we get 19 as a prime number.

Example 2: Is 53 a prime number or not?

Solution:

Method 1:

To know the prime numbers greater than 40, the below formula can be used.

n2 + n + 41, where n = 0, 1, 2, ….., 39

Put n= 3

32 + 3 + 41 = 9 + 3 + 41 = 53

Method 2:

53 has only factors 1 and 53.

So, 53 is a prime number by both the methods.

FAQs (Frequently Asked Questions)

1. Check if number is prime.

The most common method used to check the prime numbers is by factorization method. 

The steps involved to check prime numbers using the factorization method are:

  • Step 1: First let us find the factors of the given number( factors are the number that completely divides the given number)
  • Step 2: Then check the total number of factors of that number
  • Step 3: Hence, If the total number of factors is more than two, it is not a prime number but a composite number.

2. How to calculate prime numbers?

Prime numbers can also be found by the other two methods using the general formula. The methods to find prime numbers are:

Method 1:

Two consecutive numbers which are natural numbers and prime numbers are 2 and 3. Apart from 2 and 3, every prime number can be written in the form of 6n + 1 or 6n – 1, where n is a natural number.

For example:

6(1) – 1 = 5

6(1) + 1 = 7

6(2) – 1 = 11

6(2) + 1 = 13

6(3) – 1 = 17

6(3) + 1 = 19…..so on

Method 2:

To find the prime numbers greater than 40,the general formula that can be used is n2+ n + 41, where n are natural numbers  0, 1, 2, ….., 39

For example:

(0)2 + 0 + 0 = 41

(1)2 + 1 + 41 = 43

(2)2 + 2 + 41 = 47

(3)2 + 3 + 41 = 53 

(4)2 + 2 + 41 =  59…..so on

3. Is 1 a prime number or composite number?

1 is neither a prime number or a composite number because 1 is divisible by only itself, thus it has only 1 factor. Hence it contradicts both the definition of a prime number and composite number. They both have more than two factors.

4. What is the difference between prime numbers and composite numbers?

A prime number has only two factors that is the number itself and one, while a composite number has more than two factors. 

A prime number is divisible by only 1 and by itself, while the composite number is divisible by all its factors.

For example, 2 is a prime number it is divisible by 1 and 2 itself

9 is a composite number, it has three factors 1, 3, and 9 and it is divisible by all its factors.

5. How is it possible to find the prime numbers among large numbers?

Prime numbers can be easily detected in various ways but finding out the larger values whether they are prime or not can be a bit hard. But you need not worry as here are some of the easiest ways to determine whether a large number is prime or not:

  1. If there is a large number that ends with either the digits 0,2,4,6 and 8 then it is not a prime number as in all cases they are always divisible by 2
  2. Add all the digits present in the large number and then divide it by 3. If it gets divided by 3 then the large number is not a prime number.
  3. If the result from the above two methods goes wrong then you can take the square root of the number.
  4. Make a list of all the prime numbers under the square root value and then divide the number that is provided by all the listed prime numbers.
  5. If there is a number that can be easily divided by prime numbers that are less than its square root value then it cannot be considered as a prime number.
  6. If there is any number that ends with 5 or 0 can never be a prime number as it will always be divisible by 5.

6. Why is it important to learn regarding prime numbers?

Prime numbers play an important role in arithmetic equations and hence it is important not only in Math but also in various subjects and their applications. It was also around 20,000 years ago that people started to learn about the series of prime numbers. These numbers have also been discovered by other species and have been able to adapt them for various evolution and survival reasons. It is seen that a particular species of cicadas in North America tend to follow the prime number cycle for its breeding seasons. A number such as 13 or 17 is indicated where it is after 13 years that they breed. This allows various organisms their periodic cycle to go on. Hence students can see that these are important not only in understanding Math but also in a variety of topics.

7. Is there any relationship between prime numbers and composite numbers?

Yes, there is definitely a relation between a prime number and composite numbers as they both are divisible by a certain number. However the number of factors through which they are divisible differs. For example in the case of the prime number they are divisible by only one number and the other number being itself. While in the case of composite numbers it is seen that not only does 1 and the number itself is divisible but also there are other 2 or more factors that are divisible.

8. Is it possible to use the concept of prime numbers in the real world?

Yes, it is highly possible to use prime numbers in the real world. One of the most important applications is in the cyber security field where information shared on the internet makes it safer. It is also used in various applications such as WhatsApp to encrypt the information that is being shared. It is seen that by multiplying two large numbers, there is a formation of one single large number that is very hard to decode and hence software engineers use these prime numbers to create such big codes.

Prime Number Formula

When a number is divisible by only one and itself, then it is a prime number. The prime numbers cannot be factorised as they do not have factors other than 1 and the number itself. The numbers with more than two factors are called composite numbers. 1 is neither prime nor composite.

Method 1:

Every prime number can be written in the form of 6n + 1 or 6n – 1 (except the multiples of prime numbers, i.e. 2, 3, 5, 7, 11), where n is a natural number.

Method 2:

To know the prime numbers greater than 40, the below formula can be used.

n2 + n + 41, where n = 0, 1, 2, ….., 39

How do we get to know if a number is prime or not?

To identify the prime number, we need to find the factors of the number. For example, take a number; 11, 11 is divisible only by 1 and itself. Since it has only two factors, it is a prime number.

Solved Examples

Question 1: Find if 53 is a prime number or not.

Solution:

The factors of 53 are 1 and 53.
So 53 is only divisible by 1 and 53.
Therefore, 53 is a prime number.

Question 2: Check if 64 is a prime number or not.

Solution:

The factors of 64 are 1, 2, 4, 8, 16, 32, 64.

64 has more than 2 factors.
Hence, it is a composite number but not a prime number.

What is Prime Numbers Formula?

Any whole number greater than 1 that is divisible only by 1 and itself, is defined as a prime number.  To check if a number is prime or not: find all the possible factors of the number. If the number has only two factors, 1 and the number itself only, then it is a prime number. The prime numbers formula helps in generating the prime number or it could be used to test if a given number is prime or not. 

Formula 1: For any positive integer n,

(n+1) is prime if and only if n! ≡ n (mod n+1)

Formula 2: A prime number greater than 3 can be represented in the form: 6n ± 1

Prime number ≡ ± 1 (mod 6)

This method does not include the numbers that are multiples of prime numbers.

Formula 3: Prime numbers greater than 40 can be generated using:

n2 + n + 41

Prime Numbers Formula 

The prime numbers formula helps in generating the prime numbers or testing if the given number is prime.  

Formula 1: 6n ± 1 where, n = natural number >3

Prime number  ≡ ± 1 (mod 6) 

Example: To check if 541 is prime, divide 541 by 6. The remainder is 1. 541 can be represented as 6(90)+1 and thus 541 is prime.

Formula 2: n2 + n + 41 , where n = 0, 1, 2, ….., 39

Example: To generate a random prime number, give values between 0 to 39 to n. Let us key in 5 for n.

We get 52 + 5 + 41  = 71 

71 is prime.

Prime Numbers Formula Rules

Listed below are a few important aspects to be remembered while dealing with prime numbers and their formulas. 

  • Even numbers placed in the unit’s place of any number cannot be a prime number
  • The number 2 is the only even prime number.
  • For large numbers, add all the digits together if the sum is divisible by 3 then it is not a prime number.
  • Except for numbers 2 and 3, all other numbers can be expressed using the prime numbers formula 6n ± 1, where, n = natural number

Examples Using Prime Numbers Formula

Example 1: Find if 57 is a prime number.

Solution:     

By division method, we find that 1, 3,19, and 57, divide 57 completely without any remainder.

57 has 4 factors. Therefore, 57 is not a prime number since it has more than two factors.

Also, 57 could not be represented in the form of 6n+1 where ‘n’ = 1,2,3…

Therefore, 57 is not a prime number

Example 2: Check if 79 is a prime number. 

Solution:     

By division method, we find that 1, and 79 divide 79 completely without any remainder.

No other number divides 79 completely. Thus, 79 has only 2 factors.

Also, divide 79 by 6. We get the remainder 1. 

Thus we can represent it as 6n+1 : 6 × 13 + 1 = 78 + 1 = 79

Therefore, 79 is a prime number.

Example 3: Check if 19 is a prime number or not using the prime numbers formula. 

Solution: 

The factors of 19 are 19 and 1.

Let us check if 19 can be represented using the prime numbers formula: 6n + 1

Divide 19 by 6. We get the remainder 1.

We can represent 19 as 6 × 3 + 1 

Therefore, 19 is a prime number. 

FAQs on Prime Numbers Formula 

What is Meant by Prime Numbers Formula?

A prime number cannot be factorized because it does not have factors other than 1 and the number itself. The numbers which have more than two factors are called composite numbers. The prime numbers formula helps in representing the general form of a prime number. The formula to test if a number is prime or not is 6n ± 1 (i.e. divide and check if the given number leaves the remainder 1 on dividing by 6) and n2 + n + 41 is used to generate a random number, where n can take values from 0 to 39

What is the Formula to Find the General Form of Prime Numbers?

There are two basic methods with respective formulas to find the prime number of any number. The two methods are:

Method 1 or Formula 1: A prime number can be represented in the form:  6n ± 1

Prime number  ≡ ± 1 (mod 6) 

This method does not include the numbers that are multiples of prime numbers.

Method 2 or Formula 2: Prime numbers greater than 40 can be represented as: n2 + n + 41, where n = 0, 1, 2, ….., 39

Method 3: To check if a number is prime or not:

To find if any number is prime or not, do the prime factorization. If the number has only two factors, 1 and the number itself, only then it will be a prime number.

What are the Basic Rules to Remember While Using Prime Numbers Formula?

The basic rules to remember are:

  • Even numbers placed in the unit’s place of any number cannot be a prime number.
  • The number 2 is the only even prime number
  • For large numbers, add all the digits together if the sum is divisible by 3 then it is not a prime number
  • Except for numbers 2 and 3, all other numbers can be expressed in the prime numbers formula 6n ± 1, where, n = natural number >3

Using the Prime Numbers Formula, Find if 43 is a Prime Number or Not. 

The factors of 43 are 1 and 43. Using the prime number formula, we can divide it by 6 and the remainder is 1. It can be represented as 6n + 1 (i.e) 6×7 + 1 = 42 + 1 = 43

Therefore, 43 is a prime number. 

Prime Numbers – Facts, Examples, & Table Of All Up To 1,000

A prime number can be divided, without a remainder, only by itself and by 1. For example, 17 can be divided only by 17 and by 1.

Some facts:

  • The only even prime number is 2. All other even numbers can be divided by 2.
  • If the sum of a number’s digits is a multiple of 3, that number can be divided by 3.
  • No prime number greater than 5 ends in a 5. Any number greater than 5 that ends in a 5 can be divided by 5.
  • Zero and 1 are not considered prime numbers.
  • Except for 0 and 1, a number is either a prime number or a composite number. A composite number is defined as any number, greater than 1, that is not prime.

To prove whether a number is a prime number, first try dividing it by 2, and see if you get a whole number. If you do, it can’t be a prime number. If you don’t get a whole number, next try dividing it by prime numbers: 3, 5, 7, 11 (9 is divisible by 3) and so on, always dividing by a prime number (see table below).

Here is a table of all prime numbers up to 1,000:

 23571113171923
29313741434753596167
717379838997101103107109
113127131137139149151157163167
173179181191193197199211223227
229233239241251257263269271277
281283293307311313317331337347
349353359367373379383389397401
409419421431433439443449457461
463467479487491499503509521523
541547557563569571577587593599
601607613617619631641643647653
659661673677683691701709719727
733739743751757761769773787797
809811821823827829839853857859
863877881883887907911919929937
941947953967971977983991997

Some Applications

Prime numbers are having many applications in science. It is a fact that we can represent all prime numbers, except 2 and 3, in the form 6n+-1). Prime numbers are at the heart of Cryptography and data security and encryption. And generating things such as finite fields. Important applications of prime numbers are their role in producing error-correcting codes using finite fields.

These are useful in telecommunication to ensure that the messages can be sent and received with automatic correction. Also, their role in ciphers such as RSA is very much popular. It is very difficult to enumerate every application as there are simply infinitely many of them.

Solved Examples for Prime Number Formula

Question 1: Find if 73 is a prime number or not?

Solution: The factors of 73 are 1 and 73.

So 73 is only divisible by 1 and 73.

So, 73 is a prime number.

Question 2: Check if 34 is a prime number or not?

Solution: The factors of 1, 2, 17 and 34.

Hence 34 is a composite number and not a prime number.

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