TOPICS
Search

Prime Constellation

DOWNLOAD Mathematica NotebookDownload Wolfram Notebook

A prime constellation, also called a prime k-tuple, prime k-tuplet, or prime cluster, is a sequence of k consecutive numbers such that the difference between the first and last is, in some sense, the least possible. More precisely, a prime k-tuplet is a sequence of consecutive primes (p_1, p_2, ..., p_k) with p_k-p_1=s(k), where s(k) is the smallest number s for which there exist k integers b_1<b_2<...<b_k, b_k-b_1=s and, for every prime q, not all the residues modulo q are represented by b_1, b_2, ..., b_k (Forbes). For each k, 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 s(2)=2 is of the form (p, p+2) and is called a pair of twin primes. Prime doubles of the form (p, p+4) are called cousin primes, and prime doubles of the form (p, p+6) are called sexy primes.

A prime triplet has s(3)=6. The constellation (p, p+2, p+4) cannot exist, except for p=3, since one of p, p+2, and p+4 must be divisible by three. However, there are several types of prime triplets which can exist: (p, p+2, p+6), (p, p+4, p+6), (p, p+6, p+12).

A prime quadruplet is a constellation of four successive primes with minimal distance s(4)=8, and is of the form (p, p+2, p+6, p+8). The sequence s(n) therefore begins 2, 6, 8, and continues 12, 16, 20, 26, 30, ... (OEIS A008407). Another quadruplet constellation is (p, p+6, p+12, p+18).

Hardy and Wright (1979, p. 5) conjecture, and it seems almost certain to be true, that there are infinitely many twin primes (p, p+2) and prime triplets of the form (p, p+2, p+6) and (p, p+4, p+6).

The first Hardy-Littlewood conjecture states that the numbers of constellations <=x are asymptotically given by

P_x(p,p+2)∼2product_(p>=3)(p(p-2))/((p-1)^2)int_2^x(dx^')/((lnx^')^2)
(1)
=1.320323632...int_2^x(dx^')/((lnx^')^2)
(2)
P_x(p,p+4)∼2product_(p>=3)(p(p-2))/((p-1)^2)int_2^x(dx^')/((lnx^')^2)
(3)
=1.320323632...int_2^x(dx^')/((lnx^')^2)
(4)
P_x(p,p+6)∼4product_(p>=3)(p(p-2))/((p-1)^2)int_2^x(dx^')/((lnx^')^2)
(5)
=2.640647264...int_2^x(dx^')/((lnx^')^2)
(6)
P_x(p,p+2,p+6)∼9/2product_(p>=5)(p^2(p-3))/((p-1)^3)int_2^x(dx^')/((lnx^')^3)
(7)
=2.858248596...int_2^x(dx^')/((lnx^')^3)
(8)
P_x(p,p+4,p+6)∼9/2product_(p>=5)(p^2(p-3))/((p-1)^3)int_2^x(dx^')/((lnx^')^3)
(9)
=2.858248596...int_2^x(dx^')/((lnx^')^3)
(10)
P_x(p,p+2,p+6,p+8)∼(27)/2product_(p>=5)(p^3(p-4))/((p-1)^4)int_2^x(dx^')/((lnx^')^4)
(11)
=4.151180864...int_2^x(dx^')/((lnx^')^4)
(12)
P_x(p,p+4,p+6,p+10)∼27product_(p>=5)(p^3(p-4))/((p-1)^4)int_2^x(dx^')/((lnx^')^4)
(13)
=8.302361728...int_2^x(dx^')/((lnx^')^4).
(14)

These numbers are sometimes called the Hardy-Littlewood constants, and are OEIS A114907, ....

(◇) is sometimes called the extended twin prime conjecture, and

C_(p,p+2)=2Pi_2,
(15)

where Pi_2 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

int_2^x(dx)/(ln^2x)=Li(x)+2/(ln2)-x/(lnx)
(16)
int_2^x(dx)/(ln^3x)=1/2Li(x)-x/(2ln^2x)-x/(2lnx)+1/(ln2)+1/(ln^22)
(17)
int_2^x(dx)/(ln^4x)=[(Li(x))/6-x/(3ln^3x)-x/(6ln^2x)-x/(6lnx)+2/(3ln^32)+1/(3ln^22)+1/(3ln2)],
(18)

where Li(x) is the logarithmic integral.

The following table gives the number of prime constellations <=10^8, and the second table gives the values predicted by the Hardy-Littlewood formulas.

count10^510^610^710^8
(p,p+2)1224816958980440312
(p,p+4)1216814458622440258
(p,p+6)244716386117207879908
(p,p+2,p+6)2591393854355600
(p,p+4,p+6)2481444867755556
(p,p+2,p+6,p+8)381668994768
(p,p+6,p+12,p+18)7532516959330
Hardy-Littlewood10^510^610^710^8
(p,p+2)1249824858754440368
(p,p+4)1249824858754440368
(p,p+6)249716496117508880736
(p,p+2,p+6)2791446859155491
(p,p+4,p+6)2791446859155491
(p,p+2,p+6,p+8)531848634735
(p,p+6,p+12,p+18)

Consider prime constellations in which each term is of the form n^2+1. Hardy and Littlewood showed that the number of prime constellations of this form <x is given by

P(x)∼Csqrt(x)(lnx)^(-1),
(19)

where

C=product_(p>2; p prime)[1-((-1)^((p-1)/2))/(p-1)]=1.3727...
(20)

(Le Lionnais 1983).

Forbes gives a list of the "top ten" prime k-tuples for 2<=k<=17. The largest known 14-constellations are (11319107721272355839+0, 2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 50), (10756418345074847279+0, 2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 50), (6808488664768715759+0, 2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 50), (6120794469172998449+0, 2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 50), (5009128141636113611+0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50).

The largest known prime 15-constellations are (84244343639633356306067+0, 2, 6, 12, 14, 20, 24, 26, 30, 36, 42, 44, 50, 54, 56), (8985208997951457604337+0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56), (3594585413466972694697+0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56), (3514383375461541232577+0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56), (3493864509985912609487+0, 2, 6, 12, 14, 20, 24, 26, 30, 36, 42, 44, 50, 54, 56).

The largest known prime 16-constellations are (3259125690557440336637+0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56, 60), (1522014304823128379267+0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56, 60), (47710850533373130107+0, 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 (3259125690557440336631+0, 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 {p_n}_(n=5)^(12) (Gardner 1980). K. Conrow and J. J. Devore have found 15 consecutive primes spaced like the cluster {p_n}_(n=5)^(19) given by {1632373745527558118190+p_n}_(n=5)^(19), the first member of which is 1632373745527558118201.

Rivera tabulates the smallest examples of k consecutive primes ending in a given digit d=1, 3, 7, or 9 for k=5 to 11. For example, 216401, 216421, 216431, 216451, 216481 is the smallest set of five consecutive primes ending in the digit 1.

See also

Cluster Prime, Cousin Primes, Prime Arithmetic Progression, Prime Gaps, k-Tuple Conjecture, Prime Products, Prime Quadruplet, Prime Triplet, Sexy Primes, Twin Primes

Explore with Wolfram|Alpha

References

Cohen, H. "High Precision Computation of Hardy-Littlewood Constants." Preprint. http://www.math.u-bordeaux.fr/~cohen/hardylw.dvi.Forbes, T. "Large Prime Quadruplets." 17 Sep 1998. http://listserv.nodak.edu/scripts/wa.exe?A2=ind9809&L=nmbrthry&P=992.Forbes, T. "Prime Clusters and Cunningham Chains." Math. Comput. 68, 1739-1748, 1999.Forbes, T. "Prime k-Tuplets." http://anthony.d.forbes.googlepages.com/ktuplets.htm.Gardner, M. "Mathematical Games." Sci. Amer. 243, Dec. 1980.Guy, R. K. "Patterns of Primes." §A9 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 23-25, 1994.Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, 1979.Rivera, C. "Problems & Puzzles: Puzzle 016-Consecutive Primes and Ending Digits." http://www.primepuzzles.net/puzzles/puzz_016.htm.Smith, H. F. "On a Generalization of the Prime Pair Problem." Math. Tables Aids Comput. 11, 249-254, 1957.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. Sequences A008407 and A114907 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Prime Constellation

Cite this as:

Weisstein, Eric W. "Prime Constellation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PrimeConstellation.html

Subject classifications