Talk:Prime quadruplet

Practical use?

Do prime quadruplets have any known real-world application? Epastore 03:18, 15 January 2006 (UTC)

Maybe they could be used for cryptography. Other than that they're probably just as useless as any other kind of prime number. PrimeFan 20:59, 17 January 2006 (UTC)

Heh. Then do they, perhaps, reflect some sort of fundamental principle of the universe? They seem like such odd things... that 2 then 15 then 2 wandering through the list of primes. Or is this just us trying to find order in chaos, and therefore finding what we are looking for? Epastore 04:18, 19 January 2006 (UTC)

You're getting too philosophical for my taste. I'll just say that whoever can prove that there are infinitely many prime quadruplets deserves a medal. PrimeFan 00:47, 21 January 2006 (UTC)

Consectutive Prime Quadruplets

Are there any consecutive prime quadruplets, where the first values of the two quadruplets are separated by 30? There are plenty that are separated by 90. If there were any separated by 30, that'd be a prime octuplet, right? And then if there were any prime octuplets separated by 210, that'd be a prime 16-tuplet I suppose. But do any of these higher-order groupings exist? — Preceding unsigned comment added by 71.38.222.24 (talk) 04:18, 13 September 2006 (UTC)

Please sign talk pages with four tildes ~ ~ ~ ~ (without spaces). Two quadruplets separated by 30 is an admissable constellation. This can be tested here. It would not be an octuplet which means something else (8 primes as closely together as possible for primes above 8). There is a plausible conjecture that all admissable constellations have infinitely many occurrences. Up to 10^14 I computed 33480 pairs of two quadruplets 30 apart. The form is not named. Maybe it could be called "twin quadruplets". I found the first "quadruple quadruplet". I'm not adding any of these own results to the article. PrimeHunter 12:50, 13 September 2006 (UTC)

As commented before, it would not be a prime octuplet, but to answer the first question and judging by the list available at primes.utm.edu/lists/small/100000.txt this first occurs at p(78967) = 1006301, 1006303, 1006307, 1006309, 1006331, 1006333, 1006337, 1006339 = p(78974). Sieving 210 consecutive numbers leaves only (210n+11,+13,+17,+19), (210n+101,+103,+107,+109) and (210n+191,+193,+197,+199) as possible prime quadruplets, consecutive prime quadruplets are closest when a (210n+191,+193,+197,+199) quadruplet is followed by (210(n+1)+11,+13,+17,+19) 178.237.234.201 (talk) 11:49, 13 January 2020 (UTC)

On a completely unrelated note, the center of this consecutive pair of quadruplets is 100620 = 210x4792, 4793 = p(645) is a Sophie Germain prime such that p(4793) is twin with p(4794), follows 4789 = p(644) twin with super prime 4787 = p(643), 643 is twin with Sophie Germain prime 641 178.237.234.201 (talk) 12:21, 13 January 2020 (UTC)

Why is {3, 5, 7, 11} not considered as a prime quadruplet?

I believe that {3, 5, 7, 11} should be considered a prime quadruplet for the following reasons:

1. There are 2 close pairs of twin primes (namely {3, 5} and {5, 7}). 2. There are 2 overlapping pairs of prime triplets (namely {3, 5, 7} and {5, 7, 11}). 3. It is of the form {p, p+2, p+4, p+8}, and therefore, in my mind, should be an acceptable prime quadruplet since the first and last numbers of the prime quadruplet differ by 8.

Since it meets the criteria for a prime quadruplet, {3, 5, 7, 11} should be an acccepted prime quadruplet. —Preceding unsigned comment added by PhiEaglesfan712 (talk • contribs) 19:34, 17 June 2007 (UTC)

Wikipedia content should be based on reliable sources. I have seen different definitions but I believe the most common definition is that a prime quadruplet is 4 primes in the closest admissible constellation for 4 numbers. See for example [1] for the meaning of admissible. It's trivial to show that the closest admissible constellation of 4 primes is (p, p+2, p+6, p+8). Some prime quadruplet definitions skip the admissible reason and go straight to defining a prime quadruplet as (p, p+2, p+6, p+8). Note that (p, p+2, p+4, p+8) is not admissible, because at least one of the numbers is always divisible by 3. Apart from being the most common definition, I also personally think it is the most practical definition, because it's part of a systematic definition of a prime k-tuplet as k primes in the closest admissible contellation. Then twin primes are prime 2-tuplets, prime triplets are prime 3-tuplets, prime quadruplets are prime 4-tuplets, prime quintuplets are prime 5-tuplets, and so on. This is also the system used at the record page [2]. PrimeHunter 21:22, 17 June 2007 (UTC)

Prime sextuples

Why is the definition for a prime sextuple (p-4, p, p+2, p+6, p+8, p+12)? (p, p+2, p+6, p+8, p+12, p+14) have a span of only 14 instead of 16. But the span from (p, p+14) has at least one number that is a factor of 5. Is it worth noting? 12.13.126.253 (talk) 18:46, 26 August 2008 (UTC)Tim Forbis

The definition in the article represent the shortest admissible prime sextuple. The one you suggest is inadmissible because it includes the full modulo residue of 5 (i.e. it will always include at least one multiple of 5). It does have one prime solution that includes the number 5 itself, but using it as the definition of a prime sextuplet would be analogous to using the case of (2, 3) to define twin primes as (n, n+1), or (3, 5, 7) to define prime triplets as (n, n+2, n+4). --Mwalimu59 (talk) 20:32, 26 August 2008 (UTC)

Maybe it is a problem of English not being my first language, but I suggest adding an explanation of the specific meaning of "admissible" as meaning "with an infinite number of similar groupings". If you just take the regular (non-mathematic) meaning of admissible, the sentence reads as if "5, 7, 11, 13, 17, 19" would NOT be the closest possible grouping of six primes larger than 3 (which it is), while it is just not the closest possible grouping with an infinite number of similar groupings. — Preceding unsigned comment added by 195.69.113.34 (talk) 12:22, 28 November 2014 (UTC)

JFTR, as of 2017 the "admissibility" business is explained on Prime_k-tuple#Admissibility and mentions "infinite" as you suggested it here. –178.24.246.213 (talk) 13:13, 13 November 2017 (UTC)

Prime decade

@PrimeHunter: (or anybody else willing to login); a suggestion for the slightly obscure prime decade redirection, how about an {{anchor|prime decade}} here, and then tune the redirection to go directly to this anchor? –178.24.246.213 (talk) 13:08, 13 November 2017 (UTC)

No prime quadruplets with the final digits 7, 9, 1, and 3?

This was actually the question that led me here. Checking elsewhere I am pretty sure that the first 177 prime quadruplets (up to the quadruplet starting with 1,043,591) do not include a quadruplet ending with the digits 7, 9, 1, and 3. I figured out that it's the 30 in the constraint equation that make it impossible for the first prime to end with 7 (since it's 30n + 11), but I'm not sure why there isn't a second version of the constraint starting with xn + 17. Also, I'm wondering why the article is entitled "Prime quadruplet" rather than "Prime constellation", since it covers several other types. Shanen (talk) 08:36, 5 March 2020 (UTC)

Okay, I'm an idiot. It's because 0 is even while 5 is odd. But I still don't get the title of the article. Shanen (talk) 08:43, 5 March 2020 (UTC)

@Shanen: 3 divides every third odd number so three consecutive odd numbers cannot be prime (except the numbers 3, 5, 7). The article says: "All prime quadruplets except {5, 7, 11, 13} are of the form {30n + 11, 30n + 13, 30n + 17, 30n + 19} for some integer n. (This structure is necessary to ensure that none of the four primes is divisible by 2, 3 or 5)." Prime quintuplets and prime sextuplets are not considered notable enough for their own article but they contain prime quadruplets so they are mentioned here. We already have a more general article Prime k-tuple with a section about prime constellations. PrimeHunter (talk) 11:34, 5 March 2020 (UTC)

Thanks for the link to Prime k-tuple and I wish I had been smart enough to find it directly. Shanen (talk) 05:49, 20 March 2020 (UTC)