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Prime Constellation

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A prime constellation, in the minimum-span convention, is a sequence of k consecutive primes whose pattern is admissible and has the smallest possible span among admissible patterns, where k>=2 is an integer. A pattern of k integers is admissible if it does not represent all residue classes modulo any prime. Let s(k) be the least span of an admissible pattern of k integers. A prime constellation (p_1, p_2, ..., p_k) therefore satisfies p_k-p_1=s(k) (Forbes 1999, p. 1739). The names prime k-tuple, prime k-tuplet, and prime cluster are also used, sometimes with a broader meaning. This convention excludes certain small exceptional clusters. For example, (97, 101, 103, 107, 109) is a prime 5-tuplet, whereas (3, 5, 7, 11, 13) is not, since the latter represents all three residue classes modulo 3 (Forbes 1999, p. 1739).

A two-prime constellation has s(2)=2 and is of the form (p, p+2) and is called a pair of twin primes. Prime pairs of the form (p, p+4) are called cousin primes, and prime pairs of the form (p, p+6) are called sexy primes.

A three-prime constellation has s(3)=6. The cluster (p, p+2, p+4) consists of primes only for p=3, since one of p, p+2, and p+4 must be divisible by three. However, there are two types of minimum-span prime triplets which can exist: (p, p+2, p+6) and (p, p+4, p+6). The broader prime triplet pattern (p, p+6, p+12) has span 12 and is not a minimum-span constellation.

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). The prime cluster pattern (p, p+6, p+12, p+18) has span 18 and is not a minimum-span constellation.

Hardy and Wright (1979, p. 5) conjecture 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 predicts the numbers of the following prime clusters, which need not have minimum span. Let P_x count clusters whose smallest member is at most x. Their conjectured asymptotic counts are

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

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

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

C_(p,p+2)=2Pi_2,
(8)

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)
(9)
int_2^x(dx)/(ln^3x)=1/2Li(x)-x/(2ln^2x)-x/(2lnx)+1/(ln2)+1/(ln^22)
(10)
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)],
(11)

where Li(x)=li(x)-li(2) is the logarithmic integral in the convention Li(2)=0.

The following table gives the number of prime clusters with smallest member at most the indicated bound, 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)

A separate conjecture concerns individual primes of the form n^2+1, where n is a positive integer. Hardy and Littlewood (1923, p. 48) conjectured that the number P(x) of such primes less than x satisfies

P(x)∼Csqrt(x)(lnx)^(-1),
(12)

where

C=product_(p>2; p prime)[1-((-1)^((p-1)/2))/(p-1)]=1.372813...
(13)

(Hardy and Littlewood 1923, p. 48; Le Lionnais 1983).

Forbes gives a list of the "top ten" prime k-tuples for 2<=k<=17. Examples of 14-constellations include (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).

Examples of prime 15-constellations include (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).

Examples of prime 16-constellations include (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).

Examples of prime 17-constellations include (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, k-Tuple Conjecture, Prime Arithmetic Progression, Prime Gaps, Prime Products, Prime Quadruplet, Prime Triplet, Sexy Primes, Twin Primes

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References

Cohen, H. "High Precision Computation of Hardy-Littlewood Constants." Preprint. https://www.math.u-bordeaux.fr/~hecohen/hardylw.dvi.Forbes, T. "Large Prime Quadruplets." Sept. 17, 1998. https://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind9809&L=nmbrthry&P=992.Forbes, T. "Prime Clusters and Cunningham Chains." Math. Comput. 68, 1739-1747, 1999. https://doi.org/10.1090/S0025-5718-99-01117-5.Forbes, T. "Prime k-Tuplets." https://web.archive.org/web/20090228184801/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 Littlewood, J. E. "Some Problems of 'Partitio Numerorum'; III: On the Expression of a Number as a Sum of Primes." Acta Math. 44, 1-70, 1923. https://doi.org/10.1007/BF02403921.Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, 1979.Le Lionnais, F. Les nombres remarquables. Paris, France: Hermann, p. 38, 1983.Riesel, H. Prime Numbers and Computer Methods for Factorization, 2nd ed. Boston, MA: Birkhäuser, pp. 60-74, 1994.Rivera, C. "Problems & Puzzles: Puzzle 016-Consecutive Primes and Ending Digits." https://www.primepuzzles.net/puzzles/puzz_016.htm.Sloane, N. J. A. Sequences A008407 and A114907 in "The On-Line Encyclopedia of Integer Sequences."Smith, H. F. "On a Generalization of the Prime Pair Problem." Math. Tables Aids Comput. 11, 249-254, 1957.

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Prime Constellation

Cite this as:

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

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