In number theory, a prime k-tuple is an ordered set of values (i.e. a vector) representing a repeatable pattern of 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. A prime k-tuple is one which may be used to represent patterns of prime numbers.[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.
An admissible prime k-tuple with the smallest possible maximum value s is a prime constellation. For all n ≥ k this will always produce consecutive primes.[2]
The first few prime constellations are:
| k | s | Constellation | smallest[3] |
|---|---|---|---|
| 2 | 2 | (0, 2) | (3, 5) |
| 3 | 6 | (0, 2, 6) (0, 4, 6) |
(5, 7, 11) (7, 11, 13) |
| 4 | 8 | (0, 2, 6, 8) | (5, 7, 11, 13) |
| 5 | 12 | (0, 2, 6, 8, 12) (0, 4, 6, 10, 12) |
(5, 7, 11, 13, 17) (7, 11, 13, 17, 19) |
| 6 | 16 | (0, 4, 6, 10, 12, 16) | (7, 11, 13, 17, 19, 23) |
| 7 | 20 | (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) |
| 8 | 26 | (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) |
| 9 | 30 | (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 which 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.
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.[4]