In number theory, a prime k-tuple is a finite collection of values representing a repeatable pattern of differences between prime numbers.
For a k-tuple (a, b, …), the positions where the k-tuple matches a pattern in the prime numbers are given by the set of integers n such that all of the values (n + a, n + b, …) are prime.
Typically the first value in the k-tuple is 0 and the rest are distinct positive even numbers.
[1] Several of the shortest k-tuples are known by other common names: 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.
In order for a k-tuple to have infinitely many positions at which all of its values are prime, there cannot exist a prime p such that the tuple includes every different possible value modulo p. If such a prime p existed, then no matter which value of n was chosen, one of the values formed by adding n to the tuple would be divisible by p, so the only possible placements would have to include p itself, and there are at most k of those.
It is conjectured that every admissible k-tuple matches infinitely many positions in the sequence of prime numbers.
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.
[2] Although (0, 2, 4) is inadmissible modulo 3, it does produce the single set of primes, (3, 5, 7).
Inadmissible k-tuples can have more than one all-prime solution if they are admissible modulo 2 and 3, and inadmissible modulo a larger prime p ≥ 5.
[3] The diameter of a k-tuple is the difference of its largest and smallest elements.
This means that, for large n: where pn is the nth prime number.
The first few prime constellations are: The diameter d as a function of k is sequence A008407 in the OEIS.
A prime constellation is sometimes referred to as a prime k-tuplet, but some authors reserve that term for instances that are not part of longer k-tuplets.
The first Hardy–Littlewood conjecture predicts that the asymptotic frequency of any prime constellation can be calculated.
If that is the case, it implies that the second Hardy–Littlewood conjecture, in contrast, is false.
In order for such a k-tuple to meet the admissibility test, n must be a multiple of the primorial of k.[6] 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 (Tóth (2019)).