A prime constellation, also called a Prime *k*-Tuple or Prime *k*-Tuplet, 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 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, ... (Sloane's A008407). Another quadruplet constellation is (, , , ).

The first First Hardy-Littlewood Conjecture states that the numbers of constellations are asymptotically given by

(1) | |

(2) | |

(3) | |

(4) | |

(5) | |

(6) | |

(7) |

(8) |

The integrals above have the analytic forms

(9) | |||

(10) | |||

(11) |

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

(12) |

(13) |

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).

**References**

Forbes, T. ``Prime -tuplets.'' http://www.ltkz.demon.co.uk/ktuplets.htm.

Guy, R. K. ``Patterns of Primes.'' §A9 in
*Unsolved Problems in Number Theory, 2nd ed.* New York: Springer-Verlag, pp. 23-25, 1994.

Le Lionnais, F. *Les nombres remarquables.* Paris: Hermann, p. 38, 1983.

Riesel, H. *Prime Numbers and Computer Methods for Factorization, 2nd ed.* Boston, MA: Birkhäuser, pp. 60-74, 1994.

Sloane, N. J. A. Sequence A008407 in ``The On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html.

© 1996-9

1999-05-26