# How to Find Prime Numbers?

Prime numbers are numbers that are divisible only by 1 and themselves. They are an important concept in number theory and have many applications in computer science and cryptography. In this article, we will explain several methods for finding prime numbers.

Whether you are a student studying mathematics or a teacher, understanding how to find prime numbers can be extremely useful.

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## Method1: Finding Prime Numbers Using Factorization

We can determine whether the number can be considered prime by determining those factorization of the particular number.

To do this, we employ the factorization technique, which is the most efficient method to determine prime numbers. Follow the steps below that provide the method to locate prime numbers.

First step: Locate the variables of the provided number, and then list the factors.
Second Step: Verify the total amount of elements of the number.
Third step: When the given number only has two factors including one, plus the number in itself the number given is an integer. If it is, however, with greater than 2 factors it’s a composite number.

Example 1: Find out if 13 is a prime number or non.
The variables of 13 comprise 1 and 13. since it has only two factors: 1 and the number itself, 13, is a prime number.

Example 2: Find out if 27 is a prime number or non.
The 27 factors are 1, 3, 9 and 27.
Since 27 is comprised of more than two factors, so it’s not a prime number.

## Method2: The Sieve of Eratosthenes

The Sieve of Eratosthenes is an ancient algorithm for finding prime numbers up to a given limit. The algorithm starts by creating a list of numbers from 2 to the given limit. It then marks all multiples of 2 (excluding 2 itself) as not prime. The next unmarked number is then considered to be prime and all of its multiples are marked as not prime. This process is repeated until all numbers up to the given limit have been marked. The remaining unmarked numbers are prime numbers.

Example:

Let’s say we want to find all prime numbers up to 30. We create a list of numbers from 2 to 30.
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
We start by marking all multiples of 2 as not prime.
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
The next unmarked number is 3, so we mark all multiples of 3 as not prime.
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
The next unmarked number is 5, so we mark all multiples of 5 as not prime.
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
The next unmarked number is 7, so we mark all multiples of 7 as not prime.
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
We continue this process until all numbers have been marked. The remaining unmarked numbers are prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.

## Method3: Trial Division

Example:

We divide 37 by all integers from 2 to the square root of 37 (which is 6.086). Since 37 is not divisible by any of these integers, we can conclude that 37 is a prime number.

## Method 4: The Miller-Rabin Primality Test

The Miller-Rabin primality test is a probabilistic algorithm for determining whether a number is prime. It is more efficient than trial division for large numbers, but it only gives a probability of the number being prime. The algorithm involves repeating a series of tests on the number in question, and if the number passes all the tests, it is probably prime. However, if it fails even one test, it is definitely not prime.

Example:

Let’s say we want to find out if the number 61 is prime. We perform the Miller-Rabin primality test on 61, and it passes all the tests. Therefore, we can conclude that 61 is probably prime.