Prime Number Formula: Facts and Examples

Prime Number Formula: Numbers has always fascinated humankind for ages. Nowadays, we use numbers from dusk to dawn. We can’t even imagine our life if numbers are not involved. Numbers can be categorised into many types: natural numbers, whole numbers, integers, even numbers, odd numbers, prime numbers, composite numbers, etc.

A prime number is a natural number that is divisible only by \(1\) and the number itself. And, there is a formula to find the prime numbers as well. In this article, we will learn about finding the prime numbers with the help of the prime number formula.

What is a Prime Number?

The natural number that is greater than \(1\) and has only two factors, \(1\) and the number itself is called a prime number. The term “greater than \(1\)” is needed to define a prime number, to exclude \(1\).
Examples of prime numbers are \(2,\,3,\,5,\,7,\,11,\,13,\,15,\,17,\,19,\,23,\,29,\) etc.

Factors of Prime Number

When a number is multiplied with another number and gives the given number as a result, the number we multiplied is known as a factor. Now, the prime factor is the factor of a given number which is a prime number. Factors are the numbers that we multiply together to get another number. In other words, the prime factors are the prime numbers when multiplied together gives the original number.

Example: The prime factors of \(15\) are \(3\) and \(5\). Because \(3 \times 5 = 15\), and \(3\) and \(5\) are prime numbers.

Formula to Find Prime Numbers

A prime number is a number that is greater than \(1\) that can only be divided by \(1\) and itself. So, a prime number has only \(2\) factors. Let us have a view of the properties of prime numbers.

1. Every prime number is greater than \(1\).
2. \(2\) is the smallest prime number.
3. All prime numbers except \(2\) are odd numbers.

Now, the prime number formula helps in checking whether the given number is prime or not. Also, the formula helps us to generate a random prime number. We can write a prime number greater than \(3\) in the form of \(6n + 1\) or \(6n – 1\).

This method excludes the numbers that are multiples of prime numbers. To generate the prime number greater than \(40\), we use the formula \(n^2 + n + 41\) where \(n\) can take the values from \(0\) till \(39\).

Rules for Prime Number Formula

Below are some rules that have to be followed when dealing with prime number and prime number formulas.

1. If the digit at the unit’s place is even, then the number cannot be a prime number.
2. Of all the prime numbers, \(2\) is the only even prime number.
3. A prime number greater than \(3\) can be written in the form of \(6n + 1\) or \(6n – 1\).

Prime Numbers Between 1 to 100

Eratosthenes, a Greek scholar, used the following method to distinguish the prime numbers from among the natural numbers. For this reason, this method is known as Sieve of Eratosthenes or Eratosthenes Sieve.

Let us look at the method step by step.
Step 1: Write the numbers from \(1\) to \(100\) in a \(10 × 10\) grid.
Step 2: Highlight number \(1\) because all prime numbers are greater than \(1\).
Step 3: The first and the smallest prime number is \(2\); encircle \(2\) and strike out all other numbers which are divisible by \(2\).
Step 4: The following prime number is \(3\); encircle \(3\) and strike out all other numbers which are divisible by \(3\).
Step 5: The following prime number is \(5\); encircle \(5\) and strike out all other numbers which are divisible by \(5\).
Step 6: The last number left in the first row is \(7\). Encircle it and cross all its multiples.
Step 7: Keep on doing the same process till you find all the numbers either crossed-out or encircled.

All the encircled numbers are prime numbers. All the crossed-out numbers are composite numbers.
The number \(1\) is neither a prime number nor a composite number.

Learn the Concepts of Prime Factorisation

Use of Prime Numbers

There is some real-life use of prime numbers. Whenever you purchase any item online, that involves a credit card, prime and composite numbers participate in the transaction process.  The computer uses a code to identify a person’s credit card. Before the buyer’s information of the credit card is sent through cyberspace, it is encrypted to protect it from being hacked by someone.

To decode the encrypted information, a key to the code is needed. One of the most popular methods of encryption makes use of prime and composite numbers.

Importance of Prime Numbers

Prime numbers are important because they are considered the building blocks of the whole numbers, and also their odd properties make them of perfect use in various fields. The prime numbers have an exceptional property of factorisation. Most modern computers cryptograph works by using the prime factors of large numbers.

Solved Examples – Prime Number Formula

Q.1. Refer to the below-given Eratosthenes Sieve chart and answer the following questions.

a) How many prime numbers are there in between \(1\) to \(100\).
b) Which is the greatest \(2-\) digit prime number?
c) Which is the only even prime number?
Ans: A prime number is a natural number that is divisible only by \(1\) and itself.
a) The prime numbers between \(1\) and \(100\) are \(2,\,3,\,5,\,7,\,11,\,13,\,17,\,19,\,23,\)
\(29,\,31,\,37,\,41,\,43,\,47,\,53,\,59,\,61,\,67,\,71,\,73,\,79,\,83,\,89,\,97.\)
Therefore, there are \(25\) prime numbers in between \(1\) and \(100\).
b) As we can see in the Eratosthenes Sieve chart, the greatest \(2-\)digit prime number is \(97.\)
c) The only even prime number is \(2\); apart from \(2\) all the prime numbers are odd.

Q.2. Find if \(57\) is a prime number.
Ans: By the division method, we find that \(1,\,3,\,19,\) and \(57\) divide \(57\).
For a number to be a prime number, a number should have only \(2\) factors, i.e., \(1\) and the number itself. We can see that \(57\) has more than \(2\) factors. Therefore, \(57\) is not a prime number.
Also, \(57\) could not be represented in the form of \(6n + 1\) where \(n = 1,\,2,\,3….\)
Therefore, \(57\) is not a prime number.

Q.3. Find out whether \(79\) is a prime number or not with the help of the prime number formula.
Ans: With the help of the division method, we find that \(1\) and \(79\) divide \(79\) completely.
No other number divides \(79\) completely. Thus, \(79\) has only \(2\) factors.
Also, if we divide \(79\) by \(6\), we get the remainder as \(1\).
Thus we can represent it as \(6n + 1 : 6 × 13 + 1 = 78 + 1 = 79\)
Therefore, \(79\) is a prime number.

Q.4. Check whether \(64\) is a prime number or not.
Ans: By the division method, we find that \(1,\,2,\,4,\,8,\,16,\,32,\) and \(64\) divide \(64\) completely.
For a number to be a prime number, a number should have only \(2\) factors, i.e., \(1\) and the number itself. We can see that \(64\) has more than \(2\) factors. Therefore, \(64\) is not a prime number.
Also, \(64\) could not be represented in the form of \(6n + 1\) where \(n = 1,\,2,\,3….\)
Therefore, \(64\) is not a prime number.

Q.5. Check if the number \(29\) is a prime number or not using the prime numbers formula.
Ans: The factors of \(29\) are \(29\) and \(1\).
Let us check if \(29\) can be represented using the prime numbers formula: \(6n – 1\)
Divide \(29\) by \(6\).
We can represent \(29\) as \(6 × 5 – 1\)
Therefore, \(29\) is a prime number.

List of Prime Numbers Between 1 to 100

Summary

This article learned a brief study about prime numbers and then learned about the formula to find the prime numbers. We learned that there are a couple of formulas to find the prime numbers. Also, with the help of the formula \(6n + 1\) or \(6n – 1\), we can check whether a given number is prime or not. Later, to master the concept of the prime number formula, we learned to solve examples based on the prime number formula.

FAQs

Q.1. How to find a prime number formula?
Ans: We can write a prime number greater than \(3\) in the form of \(6n + 1\) or \(6n – 1\). This method excludes the numbers that are multiples of prime numbers. To generate the prime number greater than \(40\), we use the formula \(n^2 + n + 41\) where \(n\) can take the values from \(0\) till \(39\).

Q.2. What is the formula for prime numbers?
Ans: The formula to find prime numbers are \(6n + 1\) or \(6n – 1\). This method excludes the numbers that are multiples of prime numbers. Another formula is, \(n^2 + n + 41\) where \(n\) can take the values from \(0\) till \(39\).

Q.3. How many prime numbers are between \(1\) and \(100\) formula?
Ans: The prime numbers between \(1\) and \(100\) are \(2,\,3,\,5,\,7,\,11,\,13,\,17,\,19,\,23,\,29,\)
\(31,\,37,\,41,\,43,\,47,\,53,\,59,\,61,\,67,\,71,\,73,\,79,\,83,\,89,\,97.\)
Hence, in between \(1\) and \(100\), there are \(25\) prime numbers.

Q.4. Is there a formula for finding prime numbers?
Ans: Yes, there is a formula for finding the prime numbers.

Q.5. How to calculate prime numbers using the prime number formula?
Ans: Let us understand this through an example. Consider \(31\). The factors of \(31\) are \(31\) and \(1\).
Let us check if \(31\) can be represented using the prime numbers formula: \(6n – 1\) or \(6n + 1\)
Divide \(31\) by \(6\). We can represent \(31\) as \(6 × 5 + 1\)
Therefore, \(31\) is a prime number.

Learn About Prime and Composite Numbers

We hope this detailed article on prime number formula helped you in your studies. If you have any doubts, queries or suggestions regarding this article, feel to ask us in the comment section and we will be more than happy to assist you. Happy learning!