Innumber theory, aprimek-tuple is a finite collection of values representing a repeatable pattern of differences betweenprime numbers. For ak-tuple(a,b, …), the positions where thek-tuple matches a pattern in the prime numbers are given by the set ofintegersn for which all of the values(n +a,n +b, …) are prime. Typically the first value in thek-tuple is 0 and the rest are distinct positiveeven numbers.^[1]

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

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

In order for ak-tuple to have infinitely many positions at which all of its values are prime, there cannot exist a primep such that the tuple includes every different possible valuemodulop. If such a primep existed, then no matter which value ofn was chosen, one of the values formed by addingn to the tuple would be divisible byp, so the only possible placements would have to includep itself, and there are at mostk of those. For example, the numbers in ak-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.

Ak-tuple that includes every possible residue modulop is said to beinadmissible modulop. It should be obvious that this is only possible whenk ≥p. A tuple which is not inadmissible modulop is calledadmissible.

It isconjectured that every admissiblek-tuple matches infinitely many positions in the sequence of prime numbers. However, there is no tuple for which this has beenproven except the trivial 1-tuple (0). In that case, the conjecture is equivalent to the statement that there areinfinitely 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.^[2]

Although(0, 2, 4) is inadmissible modulo 3, it does produce the single set of primes,(3, 5, 7).

Because 3 is the first odd prime, a non-trivial (k ≥ 1)k-tuple matching the prime 3 can only match in one position. If the tuple begins(0, 1, ...) (i.e. is inadmissible modulo 2) then it can only match(2, 3, ...); if the tuple contains only even numbers, it can only match(3, ...).

Inadmissiblek-tuples can have more than one all-prime solution if they are admissible modulo 2 and 3, and inadmissible modulop ≥ 5. 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(0, 2, 8, 14, 26), which has two solutions:(3, 5, 11, 17, 29) and(5, 7, 13, 19, 31), where all values modulo 5 are included in both cases. Examples with three or more solutions also exist.^[3]

Thediameter of ak-tuple is the difference of its largest and smallest elements. An admissible primek-tuple with the smallest possible diameterd (among all admissiblek-tuples) is aprime constellation. For alln ≥k this will always produce consecutive primes.^[4] (Recall that alln are integers for which the values(n +a,n +b, …) are prime.)

This means that, for largen:

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

wherep_n is thenth prime number.

The first few prime constellations are:

The diameterd as a function ofk issequence A008407 in theOEIS.

A prime constellation is sometimes referred to as aprimek-tuplet, but some authors reserve that term for instances that are not part of longerk-tuplets.

Thefirst 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 thesecond Hardy–Littlewood conjecture, in contrast, is false.

A primek-tuple of the form(0,n, 2n, 3n, …, (k − 1)n) is said to be aprime arithmetic progression. In order for such ak-tuple to meet the admissibility test,n must be a multiple of theprimorial ofk.^[6]

TheSkewes numbers for primek-tuples are an extension of the definition ofSkewes's number to primek-tuples based on thefirst Hardy–Littlewood conjecture (Tóth (2019)). Let${\displaystyle P=(p,p+i_1,p+i_2,\dots ,p+i_k)}$ denote a primek-tuple,${\displaystyle {\pi }_P(x)}$ the number of primesp belowx such that${\displaystyle p,p+i_1,p+i_2,\dots ,p+i_k}$ are all prime, let${\mathrm{li}}_P(x)={\int }_2^x\frac{dt}{(\ln t{)}^{k+1}}$ and let${\displaystyle C_P}$ denote its Hardy–Littlewood constant (seefirst Hardy–Littlewood conjecture). Then the first primep that violates the Hardy–Littlewood inequality for thek-tupleP, i.e., such that

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

(if such a prime exists) is theSkewes number forP.

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

Primek-tuple	Skewes number	Found by
⁠${\displaystyle (p,p+2)}$⁠	1369391	Wolf (2011)
⁠${\displaystyle (p,p+4)}$⁠	5206837	Tóth (2019)
⁠${\displaystyle (p,p+2,p+6)}$⁠	87613571	Tóth (2019)
⁠${\displaystyle (p,p+4,p+6)}$⁠	337867	Tóth (2019)
⁠${\displaystyle (p,p+2,p+6,p+8)}$⁠	1172531	Tóth (2019)
⁠${\displaystyle (p,p+4,p+6,p+10)}$⁠	827929093	Tóth (2019)
⁠${\displaystyle (p,p+2,p+6,p+8,p+12)}$⁠	21432401	Tóth (2019)
⁠${\displaystyle (p,p+4,p+6,p+10,p+12)}$⁠	216646267	Tóth (2019)
⁠${\displaystyle (p,p+4,p+6,p+10,p+12,p+16)}$⁠	251331775687	Tóth (2019)
⁠${\displaystyle (p,p+2,p+6,p+8,p+12,p+18,p+20)}$⁠	7572964186421	Pfoertner (2020)
⁠${\displaystyle (p,p+2,p+8,p+12,p+14,p+18,p+20)}$⁠	214159878489239	Pfoertner (2020)
⁠${\displaystyle (p,p+2,p+6,p+8,p+12,p+18,p+20,p+26)}$⁠	1203255673037261	Pfoertner / Luhn (2021)
⁠${\displaystyle (p,p+2,p+6,p+12,p+14,p+20,p+24,p+26)}$⁠	523250002674163757	Pfoertner / Luhn (2021)
⁠${\displaystyle (p,p+6,p+8,p+14,p+18,p+20,p+24,p+26)}$⁠	750247439134737983	Pfoertner / Luhn (2021)

The Skewes number (if it exists) forsexy primes⁠${\displaystyle (p,p+6)}$⁠ is still unknown.

• Prime numbers

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