“Prime Constellation”的概念、定义、习题解题方法及翻译-数学辞典

单词

Prime Constellation

释义

Prime Constellation

A prime constellation, also called a Prime k-Tuple or Prime k-Tuplet, is a sequence of consecutivenumbers such that the difference between the first and last is, in some sense, the least possible. More precisely, aprime -tuplet is a sequence of consecutive Primes (, , ..., ) with , where is the smallest number for which there exist integers , and, for everyPrime , not all the residues modulo are represented by , , ..., (Forbes). For each , thisdefinition 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 becauseall 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 primetriplets which can exist: (, , ), (, , ), (, , ). A Prime Quadruplet is aconstellation 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 quadrupletconstellation is (, , , ).

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


	(1)

	(2)

	(3)

	(4)

	(5)

	(6)

	(7)

These numbers are sometimes called the Hardy-Littlewood Constants. (1) is sometimes called the extended TwinPrime Conjecture, and

(8)

where

is the Twin Primes Constant. Riesel (1994) remarks that the Hardy-Littlewood Constants can becomputed to arbitrary accuracy without needing the infinite sequence of primes.

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 valuespredicted 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 numberof prime constellations of this form is given by

(12)

where

(13)

(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).

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 A008407in ``The On-Line Version of the Encyclopedia of Integer Sequences.''http://www.research.att.com/~njas/sequences/eisonline.html.

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