If that is the case, it implies that the second Hardy–Littlewood conjecture, in contrast, is false.
While the conjecture is unproven it is considered likely to be true. The first Hardy–Littlewood conjecture predicts that the asymptotic frequency of any prime constellation can be calculated. The diameter d as a function of k is sequence A008407 in OEIS.Ī 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. (Remember that all n are integers for which the values ( n + a, n + b. For all n ≥ k this will always produce consecutive primes. An admissible prime k-tuple with the smallest possible diameter d (among all admissible k-tuples) is a prime constellation. The diameter of a k-tuple is the difference of its largest and smallest elements. 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. 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. Some inadmissible k-tuples have more than one all-prime solution. Positions matched by inadmissible patterns Īlthough (0, 2, 4) is not admissible it does produce the single set of primes, (3, 5, 7). 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. However, there is no admissible tuple for which this has been proven except the 1-tuple (0).
It is conjectured that every admissible k-tuple matches infinitely many positions in the sequence of prime numbers. it does not have a p for which it covers all the different values modulo p) is called admissible. A k-tuple that satisfies this condition (i.e.
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. 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).
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. The first term in these sequences corresponds to the first prime in the smallest prime constellation shown below. the three sequences corresponding to the three admissible 8-tuples ( prime octuplets), and the union of all 8-tuples. OEIS sequence OEIS: A257124 covers 7-tuples ( prime septuplets) and contains an overview of related sequences, e.g. Several of the shortest k-tuples are known by other common names:
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