Prime k-tuple

In number theory, a prime Template:Mvar-tuple is a finite collection of values representing a repeatable pattern of differences between prime numbers. For a Template:Mvar-tuple Template:Math, the positions where the Template:Mvar-tuple matches a pattern in the prime numbers are given by the set of integers Template:Mvar such that all of the values Template:Math are prime. Typically the first value in the Template:Mvar-tuple is 0 and the rest are distinct positive even numbers.

Named patterns

Several of the shortest k-tuples are known by other common names:

(0, 2)	twin primes
(0, 4)	cousin primes
(0, 6)	sexy primes
(0, 2, 6), (0, 4, 6)	prime triplets
(0, 6, 12)	sexy prime triplets
(0, 2, 6, 8)	prime quadruplets, prime decade
(0, 6, 12, 18)	sexy prime quadruplets
(0, 2, 6, 8, 12), (0, 4, 6, 10, 12)	prime quintuplets
(0, 4, 6, 10, 12, 16)	prime sextuplets

OEIS sequence A257124 covers 7-tuples (prime septuplets) and contains an overview of related sequences, e.g. the three sequences corresponding to the three admissible 8-tuples (prime octuplets), and the union of all 8-tuples. The first term in these sequences corresponds to the first prime in the smallest prime constellation shown below.

Admissibility

In order for a Template:Mvar-tuple to have infinitely many positions at which all of its values are prime, there cannot exist a prime Template:Mvar such that the tuple includes every different possible value modulo Template:Mvar. If such a prime Template:Mvar existed, then no matter which value of Template:Mvar was chosen, one of the values formed by adding Template:Mvar to the tuple would be divisible by Template:Mvar, so the only possible placements would have to include Template:Mvar itself, and there are at most Template:Mvar of those. For example, the numbers in a Template:Mvar-tuple cannot take on all three values 0, 1, and 2 modulo 3; otherwise the resulting numbers would always include a multiple of 3 and therefore could not all be prime unless one of the numbers is 3 itself.

A Template:Mvar-tuple that includes every possible residue modulo Template:Mvar is said to be inadmissible modulo Template:Mvar. It should be obvious that this is only possible when Template:Math. A tuple which is not inadmissible modulo any prime is called admissible.

It is conjectured that every admissible Template:Mvar-tuple matches infinitely many positions in the sequence of prime numbers. However, there is no tuple for which this has been proven except the trivial 1-tuple (0). In that case, the conjecture is equivalent to the statement that there are infinitely many primes. Nevertheless, Yitang Zhang proved in 2013 that there exists at least one 2-tuple which matches infinitely many positions; subsequent work showed that such a 2-tuple exists with values differing by 246 or less that matches infinitely many positions.

Positions matched by inadmissible patterns

Although Template:Nowrap is inadmissible modulo 3, it does produce the single set of primes, Template:Nowrap.

Because 3 is the first odd prime, a non-trivial (Template:Math) Template:Mvar-tuple matching the prime 3 can only match in one position. If the tuple begins Template:Nowrap (i.e. is inadmissible modulo 2) then it can only match Template:Nowrap if the tuple contains only even numbers, it can only match Template:Nowrap

Inadmissible Template:Mvar-tuples can have more than one all-prime solution if they are admissible modulo 2 and 3, and inadmissible modulo a larger prime Template:Math. This of course implies that there must be at least five numbers in the tuple. The shortest inadmissible tuple with more than one solution is the 5-tuple Template:Nowrap, which has two solutions: Template:Nowrap and Template:Nowrap, where all values mod 5 are included in both cases. Examples with three or more solutions also exist.

Prime constellations

The diameter of a Template:Mvar-tuple is the difference of its largest and smallest elements. An admissible prime Template:Mvar-tuple with the smallest possible diameter Template:Mvar (among all admissible Template:Mvar-tuples) is a prime constellation. For all Template:Math this will always produce consecutive primes. (Recall that all Template:Mvar are integers for which the values Template:Math are prime.)

This means that, for large Template:Mvar:

$p_{n+k-1}-p_n\ge d$

where Template:Mvar is the Template:Mvarth prime number.

The first few prime constellations are:

Template:Mvar	Template:Mvar	Constellation	Smallest
2	2	(0, 2)	(3, 5)
3	6	(0, 2, 6) (0, 4, 6)	(5, 7, 11) (7, 11, 13)
4	8	(0, 2, 6, 8)	(5, 7, 11, 13)
5	12	(0, 2, 6, 8, 12) (0, 4, 6, 10, 12)	(5, 7, 11, 13, 17) (7, 11, 13, 17, 19)
6	16	(0, 4, 6, 10, 12, 16)	(7, 11, 13, 17, 19, 23)
7	20	(0, 2, 6, 8, 12, 18, 20) (0, 2, 8, 12, 14, 18, 20)	(11, 13, 17, 19, 23, 29, 31) (5639, 5641, 5647, 5651, 5653, 5657, 5659)
8	26	(0, 2, 6, 8, 12, 18, 20, 26) (0, 2, 6, 12, 14, 20, 24, 26) (0, 6, 8, 14, 18, 20, 24, 26)	(11, 13, 17, 19, 23, 29, 31, 37) (17, 19, 23, 29, 31, 37, 41, 43) (88793, 88799, 88801, 88807, 88811, 88813, 88817, 88819)
9	30	(0, 2, 6, 8, 12, 18, 20, 26, 30) (0, 4, 6, 10, 16, 18, 24, 28, 30) (0, 2, 6, 12, 14, 20, 24, 26, 30) (0, 4, 10, 12, 18, 22, 24, 28, 30)	(11, 13, 17, 19, 23, 29, 31, 37, 41) (13, 17, 19, 23, 29, 31, 37, 41, 43) (17, 19, 23, 29, 31, 37, 41, 43, 47) (88789, 88793, 88799, 88801, 88807, 88811, 88813, 88817, 88819)

The diameter Template:Mvar as a function of Template:Mvar is sequence A008407 in the OEIS.

A prime constellation is sometimes referred to as a prime Template:Mvar-tuplet, but some authors reserve that term for instances that are not part of longer Template:Mvar-tuplets.

The first Hardy–Littlewood conjecture predicts that the asymptotic frequency of any prime constellation can be calculated. While the conjecture is unproven it is considered likely to be true. If that is the case, it implies that the second Hardy–Littlewood conjecture, in contrast, is false.

Prime arithmetic progressions

Template:Main A prime Template:Mvar-tuple of the form Template:Math is said to be a prime arithmetic progression. In order for such a Template:Mvar-tuple to meet the admissibility test, Template:Mvar must be a multiple of the primorial of Template:Mvar.

Skewes numbers

The Skewes numbers for prime k-tuples are an extension of the definition of Skewes' number to prime k-tuples based on the first Hardy–Littlewood conjecture (Template:Harvtxt). Let $P=(p,p+i_1,p+i_2,\dots ,p+i_k)$ denote a prime Template:Mvar-tuple, ${\pi }_P(x)$ the number of primes Template:Mvar below Template:Mvar such that $p,p+i_1,p+i_2,\dots ,p+i_k$ are all prime, let ${\mathrm{li}}_P(x)={\int }_2^x\frac{dt}{(\ln t{)}^{k+1}}$ and let $C_P$ denote its Hardy–Littlewood constant (see first Hardy–Littlewood conjecture). Then the first prime Template:Mvar that violates the Hardy–Littlewood inequality for the Template:Mvar-tuple Template:Mvar, i.e., such that

${\pi }_P(p)>C_P{\mathrm{li}}_P(p),$

(if such a prime exists) is the Skewes number for Template:Mvar.

The table below shows the currently known Skewes numbers for prime k-tuples:

Prime Template:Mvar-tuple	Skewes number	Found by
Template:Tmath	1369391	Template:Harvtxt
Template:Tmath	5206837	Template:Harvtxt
Template:Tmath	87613571	Tóth (2019)
Template:Tmath	337867	Tóth (2019)
Template:Tmath	1172531	Tóth (2019)
Template:Tmath	827929093	Tóth (2019)
Template:Tmath	21432401	Tóth (2019)
Template:Tmath	216646267	Tóth (2019)
Template:Tmath	251331775687	Tóth (2019)
Template:Tmath	7572964186421	Pfoertner (2020)
Template:Tmath	214159878489239	Pfoertner (2020)
Template:Tmath	1203255673037261	Pfoertner / Luhn (2021)
Template:Tmath	523250002674163757	Pfoertner / Luhn (2021)
Template:Tmath	750247439134737983	Pfoertner / Luhn (2021)

The Skewes number (if it exists) for sexy primes Template:Tmath is still unknown.

References

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