A prime constellation, also called a prime tuple, prime tuplet, or prime cluster, is a sequence of consecutive numbers such that the difference between the first and last is, in some sense, the least possible. More precisely, a prime tuplet is a sequence of consecutive primes (, , ..., ) with , where is the smallest number for which there exist integers , and, for every prime , not all the residues modulo are represented by , , ..., (Forbes). For each , this definition excludes a finite number of clusters at the beginning of the prime number sequence. For example, (97, 101, 103, 107, 109) satisfies the conditions of the definition of a prime 5tuplet, but (3, 5, 7, 11, 13) does not because all three residues modulo 3 are represented (Forbes).
A prime double with is of the form (, ) and is called a pair of twin primes. Prime doubles of the form (, ) are called cousin primes, and prime doubles of the form (, ) are called sexy primes.
A prime triplet has . The constellation (, , ) cannot exist, except for , since one of , , and must be divisible by three. However, there are several types of prime triplets which can exist: (, , ), (, , ), (, , ).
A prime quadruplet is a constellation of four successive primes with minimal distance , and is of the form (, , , ). The sequence therefore begins 2, 6, 8, and continues 12, 16, 20, 26, 30, ... (OEIS A008407). Another quadruplet constellation is (, , , ).
Hardy and Wright (1979, p. 5) conjecture, and it seems almost certain to be true, that there are infinitely many twin primes (, ) and prime triplets of the form (, , ) and (, , ).
The first HardyLittlewood conjecture states that the numbers of constellations are asymptotically given by
(1)
 
(2)
 
(3)
 
(4)
 
(5)
 
(6)
 
(7)
 
(8)
 
(9)
 
(10)
 
(11)
 
(12)
 
(13)
 
(14)

These numbers are sometimes called the HardyLittlewood constants, and are OEIS A114907, ....
(◇) is sometimes called the extended twin prime conjecture, and
(15)

where is the twin primes constant. Riesel (1994) remarks that the HardyLittlewood constants can be computed to arbitrary accuracy without needing the infinite sequence of primes.
The integrals above have the analytic forms
(16)
 
(17)
 
(18)

where is the logarithmic integral.
The following table gives the number of prime constellations , and the second table gives the values predicted by the HardyLittlewood formulas.
count  
1224  8169  58980  440312  
1216  8144  58622  440258  
2447  16386  117207  879908  
259  1393  8543  55600  
248  1444  8677  55556  
38  166  899  4768  
75  325  1695  9330 
HardyLittlewood  
1249  8248  58754  440368  
1249  8248  58754  440368  
2497  16496  117508  880736  
279  1446  8591  55491  
279  1446  8591  55491  
53  184  863  4735  
Consider prime constellations in which each term is of the form . Hardy and Littlewood showed that the number of prime constellations of this form is given by
(19)

where
(20)

(Le Lionnais 1983).
Forbes gives a list of the "top ten" prime tuples for . The largest known 14constellations are (, 2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 50), (, 2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 50), (, 2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 50), (, 2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 50), (, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50).
The largest known prime 15constellations are (, 2, 6, 12, 14, 20, 24, 26, 30, 36, 42, 44, 50, 54, 56), (, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56), (, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56), (, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56), (, 2, 6, 12, 14, 20, 24, 26, 30, 36, 42, 44, 50, 54, 56).
The largest known prime 16constellations are (, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56, 60), (, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56, 60), (, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56, 60), (13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73).
The largest known prime 17constellations are (, 6, 8, 12, 18, 20, 26, 32, 36, 38, 42, 48, 50, 56, 60, 62, 66), (17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83) (13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79).
Smith (1957) found 8 consecutive primes spaced like the cluster (Gardner 1980). K. Conrow and J. J. Devore have found 15 consecutive primes spaced like the cluster given by , the first member of which is 1632373745527558118201.
Rivera tabulates the smallest examples of consecutive primes ending in a given digit , 3, 7, or 9 for to 11. For example, 216401, 216421, 216431, 216451, 216481 is the smallest set of five consecutive primes ending in the digit 1.