Prime Constellation

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 5-tuplet, 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 Hardy-Littlewood 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 Hardy-Littlewood 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 Hardy-Littlewood 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 Hardy-Littlewood 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

Hardy-Littlewood 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 14-constellations 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 15-constellations 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 16-constellations 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 17-constellations 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.