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* 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]].<ref name="post160840">{{LinkForumThread|format=ref|p=160840|title=Re: Oscillator Discussion Thread|author=Carson Cheng|date=April 25, 2023}}</ref> | * 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]].<ref name="post160840">{{LinkForumThread|format=ref|p=160840|title=Re: Oscillator Discussion Thread|author=Carson Cheng|date=April 25, 2023}}</ref> | ||
* 19: Two known: [[cribbage]] and its stator variants, [[goban]] | * 19: Two known: [[cribbage]] and its stator variants, [[goban]] and wick based on it. | ||
* 23: Seven known: [[David Hilbert]], [[p23 honey farm hassler]], [[92P23]], [[70P23]], [[55P23]], a [[p23 R-pentomino hassler]], [[112P23]]. | * 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]], [[Honey farm hasslers#p29|p29 honey farm hassler]]<ref>{{LinkForumThread|format=ref|p=153483|title=Re: Oscillator Discussion Thread}}</ref>, and a p29 unnamed region hassler.<ref name="post160840" /> | * 29: Four known: [[p29 pre-pulsar shuttle]] with many variations, [[p29 traffic-farm hassler]], [[Honey farm hasslers#p29|p29 honey farm hassler]]<ref>{{LinkForumThread|format=ref|p=153483|title=Re: Oscillator Discussion Thread}}</ref>, and a p29 unnamed region hassler.<ref name="post160840" /> | ||
Revision as of 03:00, 26 February 2024
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.
- All Mersenne primes are of the form 2p-1, where p is a prime number.[n 1]
A Fermat prime[4] 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[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
- 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.[6]
- 19: Two known: cribbage and its stator variants, goban and wick based on it.
- 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[7], and a p29 unnamed region hassler.[6]
- 31: Three known: Merzenich's p31 including p31 hasslers based on it, p31 dependent reflector loop, and a p31 TL hassler.[8]
- 37: Six known: Beluchenko's p37, Beluchenko's other p37, 58P37, a p37 traffic light hassler,[9] p37 lumps of muck hassler, p37 wing hassler[10].
- 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)[11], 114P43[12], LOM + block hassler[13]
- 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 dependent reflector loop, p59 pi-heptomino hassler[14].
- 61: Three known: p61 pi-heptomino hassler, a honey farm-based glider gun, and an unnamed object hassler[15].
- 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.[16]
- 97: One known: p97 honey farm hassler.
- 101: One known: 116P101.
- 107: One known: 86P107.
- 109: One known: p109 R-pentomino hassler.[17]
- 113: One known: 86P113 (aside from the conduit-based Nihonium) and a honey-farm hassler supported by it.
- 127: Two known: p127 century hassler and p127 R-pentomino hassler.
- 139: One known: p139 century hassler.
- 149: One known: a p149 G-to-LWSS loop.
- 173: One known: a p173 dependent reflector loop.
- 199: One known: p199 R-pentomino hassler.
Notes
- ↑ More generally, all primes of the form ∑n-1k=0(bk), that are written as a series of 1s in base b, must have n as a prime. If n is not, for any b, for a number f that divides n, it can be expressed as (1+bf)*∑n/f-1k=0(bk) (for instance, in any base b>=2, when n=6, 111111 can be expressed as 1001*111 or 10101*11).
- ↑ Note that by convention, nested exponentiation is right associative, abc means a(bc), not (ab)c, which could be equivalently written ab*c.
- ↑ More generally, all numbers of the form xn+yn (for integers x,y>=1 and n>0) are prime only if n is a power of 2, because if n is odd, xn+yn=(x+y)*∑n-1k=0(xk*(-y)n-1-k). If n is even, it can be divided by 2 and x and y can be replaced with x2 and y2. After repeating this process until n is odd, n=1 is the only value such that the summation returns a factor of 1, which doesn't provide a factorisation. This can be derived using the fact that roots of polynomials in multiple variables are divisible by those of lower degree sharing roots (in this case where x=-y), and the expression on the right side of the equation equivalently factorises xn-yn in terms of x+y when n is even (generalising the commonly-known fact that x2-y2=(x+y)*(x-y)).
References
- ↑ Prime number at Wikipedia
- ↑ Composite number at Wikipedia
- ↑ Mersenne prime at Wikipedia
- ↑ Fermat prime at Wikipedia
- ↑ Twin prime at Wikipedia
- ↑ 6.0 6.1 Carson Cheng (April 25, 2023). Re: Oscillator Discussion Thread (discussion thread) at the ConwayLife.com forums
- ↑ Re: Oscillator Discussion Thread (discussion thread) at the ConwayLife.com forums
- ↑ 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
- ↑ Nico Brown (April 22, 2023). 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
- ↑ https://conwaylife.com/forums/viewtopic.php?p=163279#p163279
- ↑ https://conwaylife.com/forums/viewtopic.php?p=160080#p160080
- ↑ https://conwaylife.com/forums/viewtopic.php?p=163279#p163279
- ↑ Re: Oscillator Discussion Thread (discussion thread) at the ConwayLife.com forums
- ↑ Re: Oscillator Discussion Thread (discussion thread) at the ConwayLife.com forums