Prime Constellation

A prime constellation, also called a Prime k-Tuple or Prime k-Tuplet, 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 $b_1 < b_2 < \ldots < b_k$, $b_k - b_1 = s$ 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 ($p$, $p+6$) 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, ... (Sloane's A008407). Another quadruplet constellation is ($p$, $p+6$, $p+12$, $p+18$).

The first First Hardy-Littlewood Conjecture states that the numbers of constellations $\leq x$ are asymptotically given by

$P_x(p,p+2) \sim 2\prod_{p\geq 3} {p(p-2)\over (p-1)^2} \int_2^x {dx'\over (\ln x')^2}$
$\quad = 1.320323632 \int_2^x {dx'\over (\ln x')^2}$	(1)
$P_x(p,p+4) \sim 2\prod_{p\geq 3} {p(p-2)\over (p-1)^2} \int_2^x {dx'\over (\ln x')^2}$
$\quad = 1.320323632 \int_2^x {dx'\over (\ln x')^2}$	(2)
$P_x(p,p+6) \sim 4\prod_{p\geq 3} {p(p-2)\over (p-1)^2} \int_2^x {dx'\over (\ln x')^2}$
$\quad = 2.640647264 \int_2^x {dx'\over (\ln x')^2}$	(3)
$P_x(p,p+2,p+6)\sim {9\over 2} \prod_{p\geq 5} {p^2(p-3)\over (p-1)^3} \int_2^x {dx'\over (\ln x')^3}$
$\quad = 2.858248596 \int_2^x {dx'\over (\ln x')^3}$	(4)
$P_x(p,p+4,p+6)\sim {9\over 2} \prod_{p\geq 5} {p^2(p-3)\over (p-1)^3} \int_2^x {dx'\over (\ln x')^3}$
$\quad = 2.858248596 \int_2^x {dx'\over (\ln x')^3}$	(5)
$P_x(p,p+2,p+6,p+8)\sim {27\over 2} \prod_{p\geq 5} {p^3(p-4)\over (p-1)^4} \int_2^x {dx'\over (\ln x')^4}$
$\quad = 4.151180864\int_2^x {dx'\over (\ln x')^4}$	(6)
$P_x(p,p+4,p+6,p+10)\sim 27 \prod_{p\geq 5} {p^3(p-4)\over (p-1)^4} \int_2^x {dx'\over (\ln x')^4}$
$\quad = 8.302361728\int_2^x {dx'\over (\ln x')^4}.$	(7)

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

$$C_{p,p+2}=2\Pi_2,$$ (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

$$\int_2^x {dx'\over(\ln x')^2} = \mathop{\rm Li}\nolimits (x)+{2\over\ln 2}-{n\over\ln n}$$ (9)

$$\int_2^x {dx'\over(\ln x')^3} = {\textstyle{1\over 2}}\mathop{\rm Li}\nolimits (x)-{x(1+\ln x)\over(\ln x)^2}+{1\over \ln 2}+{1\over(\ln 2)^2}$$ (10)

$$\int_2^x {dx'\over(\ln x')^4} = {1\over 6}\left\{{\mathop{\rm Li}\nolimits (x)+{2[2+\ln 2+(\ln 2)^2]\over(\ln 2)^3}-{n[2+\ln n+(\ln n)^2]\over(\ln n)^3}}\right\},$$ (11)

where $\mathop{\rm Li}\nolimits (x)$ is the Logarithmic Integral.

The following table gives the number of prime constellations $\leq 10^8$, 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 $<x$ is given by

$$P(x)\sim C \sqrt{x}\,(\ln x)^{-1},$$ (12)

where

$$C=\prod_{\scriptstyle p>2\atop \scriptstyle p{\rm\ prime}}\left[{1-{(-1)^{(p-1)/2}\over p-1}}\right]=1.3727\ldots$$ (13)

(Le Lionnais 1983).

Forbes gives a list of the ``top ten'' prime $k$-tuples for $2\leq k\leq 17$. 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).

See also Composite Runs, Prime Arithmetic Progression, k-Tuple Conjecture, Prime k-Tuples Conjecture, Prime Quadruplet, Sexy Primes, Twin Primes

References

Forbes, T. ``Prime $k$-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.