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) |
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).
See also Composite Runs, Prime Arithmetic Progression, k-Tuple Conjecture, Prime k-Tuples Conjecture, Prime Quadruplet, Sexy Primes, Twin Primes
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 Eric W. Weisstein