# Prime number

positive integer with exactly two divisors, 1 and itself Here is another way to think of prime numbers. The number 12 is not prime, because a rectangle can be made, with sides of lengths 4 and 3. This rectangle has an area of 12, because all 12 blocks are used. This cannot be done with 11. No matter how the rectangle is arranged, there will always be blocks left over, except for the rectangle with sides of lengths 11 and 1. 11 must therefore be a prime number.

A prime number is a natural number of a particular kind. Any natural number is equal to 1 times itself. If the number is equal to any other numbers multiplied, then the number is called a "composite number". The smallest composite number is 4, because 2 x 2 = 4. 1 is not a composite number. Every other number is a prime number. The prime numbers are the numbers other than 1 which are not equal to m x n (except 1 x itself). The smallest prime number is 2. The next prime numbers are 3, 5, 7, 11, and 13. There is no largest prime number.

The way that the prime numbers occur is a difficult problem for mathematicians. When a number is larger, it is more difficult to know if it is a prime number. One of the answers is the prime number theorem. One of the unsolved problems is Goldbach's conjecture.

## How to find small prime numbers

There is a simple method to find a list of prime numbers. Eratosthenes created it. It has the name Sieve of Eratosthenes. It catches numbers that are not prime (like a sieve) and lets the prime numbers pass through.

The method works with a list of numbers, and a special number called b that changes during the method. As you go through the method, you circle some numbers in the list and cross out others. Each circled number is prime and each crossed-out number is composite. At the start, all the numbers are plain: not circled and not crossed out.

The method is always the same:

1. On a sheet of paper, write all the whole numbers from 2 up to the number being tested. Do not write down the number 1. Go to the next step.
2. Start with b equal to 2. Go to the next step.
3. Circle b in the list. Go to the next step.
4. Starting from b, count up b more in the list and cross out that number. Repeat counting up b more numbers and crossing out numbers until the end of the list. Go to the next step.
• (For example: When b is 2, you will circle 2 and cross out 4, 6, 8, and so on. When b is 3, you will circle 3 and cross out 6, 9, 12, and so on. 6 and 12 have already been crossed out. Cross them out again.)
5. Increase b by 1. Go to the next step.
6. If b has been crossed out, go back to the previous step. If b is a number in the list that has not been crossed out, go to the 3rd step. If b is not in the list, go to the final step.
7. (This is the final step.) You are done. All of the prime numbers are circled and all of the composite numbers are crossed out

As an example, you could do this method on a list of the numbers from 2 to 10. At the end, the numbers 2, 3, 5, and 7 will end up circled. They are prime numbers. 4, 6, 8, 9, and 10 will be crossed out. They are composite numbers.

This method or algorithm takes too long to find very large prime numbers. But it is less complicated than methods used for very large primes, like Fermat's primality test (a test to see whether a number is prime or not) or the Miller-Rabin primality test.

## What prime numbers are used for

Prime numbers are very important in mathematics & computer science. Some real-world uses are given below. Very long numbers are hard to solve. It is difficult to find their prime factors, so most of the time, numbers that are probably prime are used for encryption and secret codes.

• Most people have a bank card, where they can get money from their account, using an ATM. This card is protected by a secret access code. Since the code needs to be kept secret, it cannot be stored in cleartext on the card. Encryption is used to store the code in a secret way. This encryption uses multiplications, divisions, and finding remainders of large prime numbers. An algorithm called RSA is often used in practice. It uses the Chinese remainder theorem.
• If someone has a digital signature for their email, encryption is used. This makes sure that no one can fake an email from them. Before signing, a hash value of the message is created. This is then combined with a digital signature to produce a signed message. Methods used are more or less the same as in the first case above.
• Finding the largest prime known so far has become a sport of sorts. Testing if a number is prime can be difficult if the number is large. The largest primes known at any time are usually Mersenne primes because the fastest known test for primality is the Lucas-Lehmer test, which relies on the special form of Mersenne numbers. A group that searches for Mersenne primes is here.