Prime k-tuple

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.

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

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. For, 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 there could only be finitely many prime placements (only those including p itself). For example, the numbers in a k-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 k-tuple that satisfies this condition (i.e. it does not have a p for which it covers all the different values modulo p) is called admissible.

It is conjectured that every admissible k-tuple matches infinitely many positions in the sequence of prime numbers. However, there is no admissible tuple for which this has been proven except the 1-tuple (0). Nevertheless, by Yitang Zhang's famous proof of 2013 it follows that there exists at least one 2-tuple which matches infinitely many positions.

Although (0, 2, 4) is not admissible it does produce the single set of primes, (3, 5, 7).

Some inadmissible k-tuples have more than one all-prime solution. This cannot happen for a k-tuple that includes all values modulo 3, so to have this property a k-tuple must cover all values modulo a larger prime, implying that there are 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 congruences (mod 5) are included in both cases.

The diameter of a k-tuple is the difference of its largest and smallest elements. An admissible prime k-tuple with the smallest possible diameter d (among all admissible k-tuples) is a prime constellation. For all n ≥ k this will always produce consecutive primes.

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