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=3) (p(p - 2))/(p - 1)^2 integral_2^x (d x')/(ln x')^2
=1.32032... integral_2^x (d x')/(ln x')^2
P_x(p, p + 4)~2 product_(p>=3) (p(p - 2))/(p - 1)^2 integral_2^x (d x')/(ln x')^2
=1.32032... integral_2^x (d x')/(ln x')^2
P_x(p, p + 6)~4 product_(p>=3) (p(p - 2))/(p - 1)^2 integral_2^x (d x')/(ln x')^2
=2.64065... integral_2^x (d x')/(ln x')^2
P_x(p, p + 2, p + 6)~9/2 product_(p>=5) (p^2(p - 3))/(p - 1)^3 integral_2^x (d x')/(ln x')^3
=2.85825... integral_2^x (d x')/(ln x')^3
P_x(p, p + 4, p + 6)~9/2 product_(p>=5) (p^2(p - 3))/(p - 1)^3 integral_2^x (d x')/(ln x')^3
=2.85825... integral_2^x (d x')/(ln x')^3
P_x(p, p + 2, p + 6, p + 8)~27/2 product_(p>=5) (p^3(p - 4))/(p - 1)^4 integral_2^x (d x')/(ln x')^4
=4.15118... integral_2^x (d x')/(ln x')^4
P_x(p, p + 4, p + 6, p + 10)~27 product_(p>=5) (p^3(p - 4))/(p - 1)^4 integral_2^x (d x')/(ln x')^4
=8.30236... integral_2^x (d x')/(ln x')^4.
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) = 2Π_2, where Π_2 is the twin primes constant. Riesel 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
integral_2^x (d x)/(ln^2 x) | = | Li(x) + 2/(ln2) - x/(ln x)
integral_2^x (d x)/(ln^3 x) | = | 1/2 Li(x) - x/(2ln^2 x) - x/(2ln x) + 1/(ln2) + 1/(ln^2 2)
integral_2^x (d x)/(ln^4 x) | = | [(Li(x))/6 - x/(3ln^3 x) - x/(6ln^2 x) - x/(6ln x) auto right match + auto left match 2/(3ln^3 2) + 1/(3ln^2 2) + 1/(3ln2)], 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.
count | 10^5 | 10^6 | 10^7 | 10^8
(p, p + 2) | 1224 | 8169 | 58980 | 440312
(p, p + 4) | 1216 | 8144 | 58622 | 440258
(p, p + 6) | 2447 | 16386 | 117207 | 879908
(p, p + 2, p + 6) | 259 | 1393 | 8543 | 55600
(p, p + 4, p + 6) | 248 | 1444 | 8677 | 55556
(p, p + 2, p + 6, p + 8) | 38 | 166 | 899 | 4768
(p, p + 6, p + 12, p + 18) | 75 | 325 | 1695 | 9330
Hardy-Littlewood | 10^5 | 10^6 | 10^7 | 10^8
(p, p + 2) | 1249 | 8248 | 58754 | 440368
(p, p + 4) | 1249 | 8248 | 58754 | 440368
(p, p + 6) | 2497 | 16496 | 117508 | 880736
(p, p + 2, p + 6) | 279 | 1446 | 8591 | 55491
(p, p + 4, p + 6) | 279 | 1446 | 8591 | 55491
(p, p + 2, p + 6, p + 8) | 53 | 184 | 863 | 4735
(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 2
p prime)[1 - (-1)^((p - 1)/2)/(p - 1)] = 1.3727...
(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 found 8 consecutive primes spaced like the cluster {p_n}_(n = 5)^12. 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.
