# Prime number

A **prime number**^{[1]} 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**^{[2]}.

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^{[3]} is a prime number that is one less than a power of two.

A Fermat prime^{[4]} 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^{[5]} 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

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: Two known: David Hilbert and p23 honey farm hassler, plus hybrids of the two.
- 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
^{[6]}. - 37: Four known: Beluchenko's p37, Beluchenko's other p37, 58P37 and a p37 TL hassler
^{[7]}. - 41: None known.
- 43: One known excluding Snark loops: period-43 glider gun.
- 47: One known excluding Snark loops: 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 excluding glider or conduit circuitry: 108P59.
- 71: One known excluding glider or conduit circuitry: p71 honey farm hassler.
- 79: One known excluding glider or conduit circuitry: p79 pi-heptomino hassler.
^{[8]} - 83: One known excluding glider or conduit circuitry: p83 R-pentomino hassler.
- 109: One known excluding glider or conduit circuitry: p109 R-pentomino hassler.
^{[9]} - 127: One known excluding glider or conduit circuitry: p127 century hassler.
- 199: One known excluding glider or conduit circuitry: p199 R-pentomino hassler.
- For other prime periods above 59, all known oscillators require glider or conduit circuitry.
^{[citation needed]}

## References

- ↑ Prime number at Wikipedia
- ↑ Composite number at Wikipedia
- ↑ Mersenne prime at Wikipedia
- ↑ Fermat prime at Wikipedia
- ↑ Twin prime at Wikipedia
- ↑ Carson Cheng (Aug 05, 2022). Re: Oscillator Discussion Thread (discussion thread) at the ConwayLife.com forums
- ↑ Mitchell Riley (Aug 02, 2022). Re: Oscillator Discussion Thread (discussion thread) at the ConwayLife.com forums
- ↑ Re: Oscillator Discussion Thread (discussion thread) at the ConwayLife.com forums
- ↑ Re: Oscillator Discussion Thread (discussion thread) at the ConwayLife.com forums