Prime Numbers

 

A prime number can only be divided by itself and 1 (one) to leave a whole number (integer) answer.

A mathematician may say: A prime number is a number that has only two integer divisors: itself and one.

 Some quick facts about prime numbers:

• 1 is NOT a prime number. A prime number, by definition, has to have exactly two positive divisors. 1 only has one positive divisor (1).
• 2 is the only even prime number, because all other even numbers, of course, divide by 2.
• The 1000th prime number is 7,919.
• Euclid, the Greek mathematician, demonstrated in around 300BC that there are an infinite number of prime numbers.
• 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.

• The first 25 prime numbers (all the prime numbers less than 100) are:^[8]

  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 (sequence A000040 in the OEIS).

  No even number ${\displaystyle n}$ greater than 2 is prime because any such number can be expressed as the product ${\displaystyle 2\times n/2}$. Therefore, every prime number other than 2 is an odd number, and is called an odd prime.^[9] Similarly, when written in the usual decimal system, all prime numbers larger than 5 end in 1, 3, 7, or 9. The numbers that end with other digits are all composite: decimal numbers that end in 0, 2, 4, 6, or 8 are even, and decimal numbers that end in 0 or 5 are divisible by 5.^[10]

• Prime numbers are at the heart of Cryptography and generating things such as finite fields. Important applications of prime numbers are their role in producing error correcting codes (via finite fields) which are used in telecommunication to ensure messages can be sent and received with automatic correction if tampered with (within a number of mistakes) and their role in ciphers such as RSA. It is very hard to enumerate every application as there are simply infinitely many of them.

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