Prime k-Tuple

In number theory, a prime k-tuple is a set of values (i.e., a vector) representing a repeatable pattern of differences between prime numbers. A k-tuple is represented as (a, b, ...) to represent any set of values (n + a, n + b, ...) for all values of n. In practice, 0 is usually used for the lowest value of the k-tuple.[1]

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

(0, 2)twin primes
(0, 4)cousin primes
(0, 6)sexy primes
(0, 2, 6), (0, 4, 6)prime triplets
(0, 6, 12)sexy prime triplets
(0, 2, 6, 8)prime quadruplets
(0, 6, 12, 18)sexy prime quadruplets
(0, 2, 6, 8, 12), (0, 4, 6, 10, 12)quintuplet primes
(0, 4, 6, 10, 12, 16)sextuplet primes

A prime k-tuple is sometimes referred to as an admissible k-tuple. In order for a k-tuple to be admissible, it must not include the complete modulo set of residue classes (i.e. the values 0 through p − 1) of any prime p less than or equal to k. For example, the complete modulo residue of p = 3 is 0, 1, and 2, so the numbers in a k-tuple modulo 3 would have to include at most two of these values to be admissible; 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. 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. The smallest of these is (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.

Prime constellations

The diameter of k-tuple is the difference of its largest and smallest elements. An admissible prime k-tuple with the smallest possible diameter d is a prime constellation. For all n ≥ k this will always produce consecutive primes.[2]

The first few prime constellations are:

kdConstellation smallest[3]
22(0, 2)(3, 5)
36(0, 2, 6)
(0, 4, 6)
(5, 7, 11)
(7, 11, 13)
48(0, 2, 6, 8)(5, 7, 11, 13)
512(0, 2, 6, 8, 12)
(0, 4, 6, 10, 12)
(5, 7, 11, 13, 17)
(7, 11, 13, 17, 19)
616(0, 4, 6, 10, 12, 16)(7, 11, 13, 17, 19, 23)
720(0, 2, 6, 8, 12, 18, 20)
(0, 2, 8, 12, 14, 18, 20)
(11, 13, 17, 19, 23, 29, 31)
(5639, 5641, 5647, 5651, 5653, 5657, 5659)
826(0, 2, 6, 8, 12, 18, 20, 26)
(0, 2, 6, 12, 14, 20, 24, 26)
(0, 6, 8, 14, 18, 20, 24, 26)
(11, 13, 17, 19, 23, 29, 31, 37)
(17, 19, 23, 29, 31, 37, 41, 43)
(88793, 88799, 88801, 88807, 88811, 88813, 88817, 88819)
930(0, 2, 6, 8, 12, 18, 20, 26, 30)
(0, 4, 6, 10, 16, 18, 24, 28, 30)
(0, 2, 6, 12, 14, 20, 24, 26, 30)
(0, 4, 10, 12, 18, 22, 24, 28, 30)
(11, 13, 17, 19, 23, 29, 31, 37, 41)
(13, 17, 19, 23, 29, 31, 37, 41, 43)
(17, 19, 23, 29, 31, 37, 41, 43, 47)
(88789, 88793, 88799, 88801, 88807, 88811, 88813, 88817, 88819)

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. While the conjecture is unproven it is considered likely to be true.

In 2013, Zhang Yitang proved that for k = 3.5 × 106, any admissible k-tuple will contain at least two prime numbers for infinitely many values of n, establishing that prime gaps bounded by a certain constant occur infinitely often.[4]

Prime arithmetic progressions

A prime k-tuple of the form (0, n, 2n, ...) is said to be a prime arithmetic progression. In order for such a k-tuple to meet the admissibility test, n must be a multiple of the primorial of k.[5]

References

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