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.

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

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).

Medium-period prime-period oscillators

Not counting Snark loops for p43+ and conduit-based oscillators for p59+, there are very few known prime-period oscillators above 16, although more are starting to be found with symmetric CatForce. Alternative^[which?] forms of the same oscillator are combined into one.

• 17: Three known: 54P17.1 and 71P17.1 (variations on the same theme), honey thieves, and R2-D2 shifting p5 diamond.
• 19: None known.
• 23: Three known: David Hilbert, p23 honey farm hassler, and 92P23.
• 29: Two known: p29 pre-pulsar shuttle with many variations and p29 traffic-farm hassler.
• 31: Three known: Merzenich's p31 including p31 hasslers based on it, p31 glider shuttle, and a p31 TL hassler.
• 37: Four known: Beluchenko's p37, Beluchenko's other p37, 58P37 and a p37 TL hassler.
• 41: None known.
• 43: One known: period-43 glider gun.
• 47: One known: p47 pre-pulsar shuttle and several larger oscillators relying on it.
• 53: Two self-sustaining engines are known, but they both require Snark loops.
• 59: One known: 108P59.
• 71: One known: p71 honey farm hassler.
• 73: One known: p73 lumps of muck hassler and a larger oscillator supported by four copies of it.
• 79: One known: p79 pi-heptomino hassler.
• 83: One known: p83 R-pentomino hassler.
• 109: One known: p109 R-pentomino hassler.
• 113: One known: 86P113 (aside from the conduit-based Nihonium)
• 127: One known: p127 century hassler.
• 199: One known: p199 R-pentomino hassler.