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Concept
Prime k-tuple
Summary
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:
OEIS sequence 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.
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; subsequent work showed that some 2-tuple exists with values differing by 246 or less that matches infinitely many positions.
Official source
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