# Prime number

A prime number is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number.

Prime numbers come into play in a number of ways in the Game of Life and OCA. Here are some of them:

• Large prime oscillators whose periods are very large prime numbers, right up to the largest known prime number
• Primer, a pattern that produces a stream of lightweight spaceships representing prime numbers
• Prime calculators using guns whose output stream is filtered by primer to generate twin primes, prime quadruplets, cousin primes, and so forth
• Fermat prime calculator, a pattern based on primer that calculates Fermat prime numbers
• Izhora, the largest known cellular automation computer, which can be used to calculate prime numbers

## Classes of prime numbers

A Mersenne prime is a prime number that is one less than a power of two.

• All Mersenne primes are of the form 2p-1, where p is a prime number.[n 1]

A Fermat prime is a prime number of the form 22n+1,[n 2] where n is a nonnegative integer. Only five Fermat primes are known, namely 3, 5, 17, 257, and 65537.

• Together with 2, the Fermat primes are the complete set of prime numbers of the form 2n+1 (n must be 0 or a power of 2).[n 3]

A twin prime is a prime number that is either 2 less or 2 more than another prime number — for example, either member of the twin prime pair (41, 43).