Wikipedia:Reference desk/Archives/Mathematics/2014 March 18
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March 18
editClosest sets of primes
editHow do I know if there is more than 1 type of the prime k tuple of interest? How do I know what those are? Is there a rule for prime k tuples besides that they need to be groups of k primes?Caters1 (talk) 21:39, 18 March 2014 (UTC)
- The usual definition is that a prime k-tuplet is k primes, or a potential pattern for k primes, as closely together as admissible, i.e as closely together as possible for k numbers without small prime factors, specifically prime factors ≤ k. For small k you can work it out by hand to see where the small prime factors can fall. For example, (p, p+2, p+4) is inadmissible because 3 will divide one of the numbers. But (p, p+2, p+6) is an admissible prime triplet because 2 can divide p+1, p+3, p+5, and 3 can divide p+1, p+4. For a computer it's trivial to determine whether a pattern is admissible, but for large k it can be hard to find the closest admissible patterns. The only very simple rule is that if one pattern is an admissible prime k-tuple then the mirror pattern also is. For example (p, p+2, p+6) has mirror (p, p+4, p+6). In some cases, at least for small k, the mirror will be the same, for example for a prime quadruplet (p, p+2, p+6, p+8). Prime k-tuple (which starts with another definition where the primes are not required to be close) shows all admissible patterns up to k=9. [1] shows all up to k=50. See also [2] which says they have been computed up to k=342. For general information, see [3], [4], [5]. I'm Jens Kruse Andersen and found the largest known prime k-tuple for many k at [6]. Well, I haven't searched them for many years and only have the current records for k = 8, 10, 11, but also had 3, 6, 7, 9, 12, 13, 14, 15, 16 earlier. Which k are you interested in? Do you want to find occurrences with k primes, possibly of record size? PrimeHunter (talk) 23:00, 18 March 2014 (UTC)
- Yeah that makes sense. I am trying to figure out where the types of primes intersect and this prime k tuplet up to 16 would help. I mean yes the math gets more complicated but still. What would you call the closest admissable groups of 16 primes? Prime Hexadecuplets maybe? I mean hexadec translated from greek and latin means 16.Caters1 (talk) 06:24, 20 March 2014 (UTC)
- It's just called a prime 16-tuplet. I haven't heard anyone use a name instead of 16 in this context. There are two admissible patterns: {0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48, 54, 58, 60} and its mirror {0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56, 60}. My old 26-digit record from 2004 was 10252256693298561414756287 + d, for d in the second set. The current 31-digit by Roger Thompson in 2012 is 1003234871202624616703163933853 + d, for d in the first set. PrimeHunter (talk) 13:12, 20 March 2014 (UTC)
- Oh I thought it might be called prime hexadecuplet because if you look at the word hexadecane, you know it has 16 carbons that are all single bonds. If you look at the word hexadecagon you know it is a 16 sided polygon. If you look at the word hexadecahedron you know it is a polyhedron with 16 faces. hexadec or hexadeca litterally translates to 16.Caters1 (talk) 19:08, 20 March 2014 (UTC)
- If you wonder whether a name is used then just try a Google search in quotation marks. I get no results on "prime hexadecuplets". You should have a good reason to invent a name, even if it seems to follow a pattern in existing names. If something already has a name then you should have a really good reason to invent a new one. PrimeHunter (talk) 00:20, 21 March 2014 (UTC)
- Hexadec and hexadeca are used a lot in organic chemistry for compounds with a chain or ring of 16 carbons. hexadecagon is used although 16-gon is more common. hexadecahedron is used to mean a solid figure with 16 sides and at least 10 vertices. I have heard of dodecuplets when talking about 12 puppies or kittens born from the same mom and on the same day. Thus it would make sense that hexadecuplet would mean a group of 16 similar things, in this case primes.Caters1 (talk) 00:51, 21 March 2014 (UTC)
- If you wonder whether a name is used then just try a Google search in quotation marks. I get no results on "prime hexadecuplets". You should have a good reason to invent a name, even if it seems to follow a pattern in existing names. If something already has a name then you should have a really good reason to invent a new one. PrimeHunter (talk) 00:20, 21 March 2014 (UTC)
- Oh I thought it might be called prime hexadecuplet because if you look at the word hexadecane, you know it has 16 carbons that are all single bonds. If you look at the word hexadecagon you know it is a 16 sided polygon. If you look at the word hexadecahedron you know it is a polyhedron with 16 faces. hexadec or hexadeca litterally translates to 16.Caters1 (talk) 19:08, 20 March 2014 (UTC)
- It's just called a prime 16-tuplet. I haven't heard anyone use a name instead of 16 in this context. There are two admissible patterns: {0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48, 54, 58, 60} and its mirror {0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56, 60}. My old 26-digit record from 2004 was 10252256693298561414756287 + d, for d in the second set. The current 31-digit by Roger Thompson in 2012 is 1003234871202624616703163933853 + d, for d in the first set. PrimeHunter (talk) 13:12, 20 March 2014 (UTC)