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 2^p-1, where p is a prime number.

A Fermat prime is a prime number of the form 2²ⁿ+1, 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).

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

See also category Prime-period oscillators

Not counting Snark loops for p43+ and conduit-based oscillators for p59+, there are relatively 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. See also category "Prime-period oscillators" for oscillators that have dedicated pages.

• 17: Five known: 54P17.1 and 71P17.1 (variations on the same theme), honey thieves, p17 R-pentomino hassler, R2-D2 shifting p5 diamond, and a p17 B-heptomino hassler.
• 19: One known: cribbage and its stator variants
• 23: Seven known: David Hilbert, p23 honey farm hassler, 92P23, 70P23, 55P23, a p23 R-pentomino hassler, 112P23.
• 29: Four known: p29 pre-pulsar shuttle with many variations, p29 traffic-farm hassler, p29 honey farm hassler, and a p29 unnamed region hassler.
• 31: Three known: Merzenich's p31 including p31 hasslers based on it, p31 glider shuttle, and a p31 TL hassler.
• 37: Six known: Beluchenko's p37, Beluchenko's other p37, 58P37, a p37 traffic light hassler, p37 lumps of muck hassler, p37 wing hassler.
• 41: Three known: p41 pi-heptomino hassler, 204P41, and 246P41.
• 43: Five known: capped period-43 glider gun (80P43), p43 pi-heptomino hassler (70P43), a p43 honey farm hassler (88P43), 114P43, LOM + block hassler
• 47: Five known: p47 lumps of muck hassler, an unrelated p47 lumps of muck hassler, p47 pre-pulsar shuttle and several larger oscillators relying on it, p47 honey farm hassler, and p47 pi-heptomino hassler.
• 53: Three known: 94P53 and two p53 pi-heptomino hasslers.
• 59: Four known: 92P59, p59 twirling T-tetsons 2, p59 glider shuttle, p59 pi-heptomino hassler.
• 61: Three known: p61 pi-heptomino hassler, a honey farm-based glider gun, and an unnamed object hassler.
• 67: One known: p67 B-heptomino hassler.
• 71: Two known: p71 honey farm hassler, p71 dependent reflector loop.
• 73: Two known: p73 lumps of muck hassler and a larger oscillator supported by four copies of it; p73 honey farm hassler.
• 79: Five known: three pi-heptomino hasslers, and two p79 glider shuttles.
• 83: Two known: p83 R-pentomino hassler and p83 honey farm hassler.
• 97: One known: p97 honey farm hassler.
• 101: One known: 116P101.
• 107: One known: 86P107.
• 109: One known: p109 R-pentomino hassler.
• 113: One known: 86P113 (aside from the conduit-based Nihonium)
• 127: Two known: p127 century hassler and p127 R-pentomino hassler.
• 139: One known: p139 century hassler.
• 199: One known: p199 R-pentomino hassler.