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Prime constellations

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Here, we make a distinction between a prime constellation and a prime cluster.[1]

Contents

Prime clusters

A prime cluster, also called a prime k-tuple (double, triple, quadruple, ...) is a strictly increasing sequence of \scriptstyle k \, primes such that the difference between the first and last is not necessarily minimal, i.e. some primes could be skipped over. More precisely, a prime k-tuple is a strictly increasing sequence of \scriptstyle k \, primes \scriptstyle (p_1,\, p_2,\, \ldots,\, p_k) \, with \scriptstyle p_k - p_1 \,=\, s(k) \,, where \scriptstyle s(k) \, is not necessarily the smallest number \scriptstyle s \, for which there exist \scriptstyle k \, integers \scriptstyle b_1 \,<\, b_2 \,<\, \ldots \,<\, b_k,\, b_k - b_1 \,=\, s, \, and for every prime \scriptstyle q \,, not all the residues modulo \scriptstyle q \, are represented by \scriptstyle b_1,\, b_2,\, \ldots,\, b_k \,.[1]

Nonadmissible prime clusters

A nonadmissible prime cluster, also called a nonadmissible prime k-tuple, is a prime cluster such that for some prime \scriptstyle q \,, all the residues modulo \scriptstyle q \, are represented by \scriptstyle b_1,\, b_2,\, \ldots,\, b_k \,. Only a finite number of nonadmissible prime clusters may appear at the beginning of a prime cluster sequence.

For each \scriptstyle 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.[1]

Prime constellations

A prime constellation, also called a prime k-tuplet (doublet, triplet, quadruplet, ...) is a maximally dense prime cluster, i.e. a sequence of \scriptstyle k \, consecutive primes, i.e. such that the difference between the first and last is minimal, i.e. no primes could be skipped over. More precisely, a prime \scriptstyle k \,-tuplet is a sequence of \scriptstyle k \, consecutive primes, i.e. \scriptstyle (p_1,\, p_2,\, \ldots,\, p_k) \, with \scriptstyle p_k - p_1 \,=\, s(k) \,, where \scriptstyle s(k) \, is the smallest number \scriptstyle s \, for which there exist \scriptstyle k \, integers \scriptstyle b_1 \,<\, b_2 \,<\, \ldots \,<\, b_k,\, b_k - b_1 \,=\, s \, and, for every prime \scriptstyle q \,, not all the residues modulo \scriptstyle q \, are represented by \scriptstyle b_1,\, b_2,\, \ldots,\, b_k \,.[1]

Prime pairs

Twin primes

Twin primes: prime pairs \scriptstyle p\ +\ (0, 2)\,
{(3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), (71, 73), (101, 103), (107, 109), (137, 139), (149, 151), ...}
Except for (3, 5), all twin prime pairs are of the form \scriptstyle 6n \,+\, (-1, +1),\, n \,\ge\, 1 \,
5 is the only prime belonging to two twin prime pairs
The twin primes conjecture (and the weak k-tuple conjecture):[2] asserts that there are an infinity of twin primes (not proved yet...)
The strong k-tuple conjecture[3]: predicts the asymptotic number \scriptstyle \pi_2(x) \, of twin primes \scriptstyle p \,+\, (0, 2) \,, with \scriptstyle p \le x\, (not proved yet...)
Twin prime pairs concatenated:
A077800 {3, 5, 5, 7, 11, 13, 17, 19, 29, 31, 41, 43, 59, 61, 71, 73, 101, 103, 107, 109, 137, 139, 149, 151, 179, 181, ...}
Twin prime pairs concatenated (without repetition, thus 5 appears only once):
A001097 {3, 5, 7, 11, 13, 17, 19, 29, 31, 41, 43, 59, 61, 71, 73, 101, 103, 107, 109, 137, 139, 149, 151, 179, 181, ...}
First member of twin prime pair:
A001359 {3, 5, 11, 17, 29, 41, 59, 71, 101, 107, 137, 149, 179, 191, 197, 227, 239, 269, 281, 311, 347, 419, 431, ...}
Second member of twin prime pair:
A006512 {5, 7, 13, 19, 31, 43, 61, 73, 103, 109, 139, 151, 181, 193, 199, 229, 241, 271, 283, 313, 349, 421, 433, ...}

Cousin primes

Cousin primes: prime pairs \scriptstyle p \,+\, (0, 4) \,
{(3, 7), (7, 11), (13, 17), (19, 23), (37, 41), (43, 47), (67, 71), (79, 83), (97, 101), (103, 107), (109, 113), (127, 131), ...}
Except for (3, 7), all cousin prime pairs are of the form \scriptstyle 6n \,+\, (+1, +5),\, n \,\ge\, 1 \,
7 is the only prime belonging to two cousin prime pairs
(3, 7) is the only cousin prime pair with another prime in between (i.e. 5)
The weak k-tuple conjecture: asserts that there are an infinity of cousin primes (not proved yet...)
The strong k-tuple conjecture[3] predicts the asymptotic number \scriptstyle \pi_4(x) \, of cousin primes \scriptstyle p \,+\, (0, 4) \,, with \scriptstyle p \,\le\, x \, (not proved yet...)
Cousin prime pairs concatenated:
A140382 {3, 7, 7, 11, 13, 17, 19, 23, 37, 41, 43, 47, 67, 71, 79, 83, 97, 101, 103, 107, 109, 113, 127, 131, 163, ...}
Cousin prime pairs concatenated (without repetition, thus 7 appears only once):
A094343 {3, 7, 11, 13, 17, 19, 23, 37, 41, 43, 47, 67, 71, 79, 83, 97, 101, 103, 107, 109, 113, 127, 131, 163, ...}
First member of cousin prime pair:
A023200 {3, 7, 13, 19, 37, 43, 67, 79, 97, 103, 109, 127, 163, 193, 223, 229, 277, 307, 313, 349, 379, 397, 439, ...}
Second member of each cousin prime pair:
A046132 {7, 11, 17, 23, 41, 47, 71, 83, 101, 107, 113, 131, 167, 197, 227, 233, 281, 311, 317, 353, 383, 401, 443, ...}

Prime triplets

Prime triplets: \scriptstyle p \,+\, (0, 2, 6) \, are all prime
{(5, 7, 11), (11, 13, 17), (17, 19, 23), (41, 43, 47), (101, 103, 107), (107, 109, 113), (191, 193, 197), (227, 229, 233), (311, 313, 317), (347, 349, 353), ...}
All prime triplets \scriptstyle p \,+\, (0, 2, 6) \, are of the form \scriptstyle 6n \,+\, (-1, +1, +5),\, n \,\ge\, 1 \,
First member of \scriptstyle p \,+\, (0, 2, 6) \,
A022004 {5, 11, 17, 41, 101, 107, 191, 227, 311, 347, 461, 641, 821, 857, 881, 1091, 1277, 1301, 1427, 1481, ...}
Second member of \scriptstyle p \,+\, (0, 2, 6) \,
A073648 {7, 13, 19, 43, 103, 109, 193, 229, 313, 349, 463, 643, 823, 859, 883, 1093, 1279, 1303, 1429, 1483, ...}
Third member of \scriptstyle p \,+\, (0, 2, 6) \,
A098412 {11, 17, 23, 47, 107, 113, 197, 233, 317, 353, 467, 647, 827, 863, 887, 1097, 1283, 1307, 1433, 1487, ...}


Prime triplets: \scriptstyle p\ +\ (0, 4, 6)\, are all prime
{(7, 11, 13), (13, 17, 19), (37, 41, 43), (67, 71, 73), (97, 101, 103), (103, 107, 109), (193, 197, 199), (223, 227, 229), (277, 281, 283), (307, 311, 313), }
All prime triplets \scriptstyle p \,+\, (0, 4, 6) \, are of the form \scriptstyle 6n \,+\, (+1, +5, +7),\, n \,\ge\, 1 \,
First member of \scriptstyle p \,+\, (0, 4, 6) \,
A022005 {7, 13, 37, 67, 97, 103, 193, 223, 277, 307, 457, 613, 823, 853, 877, 1087, 1297, 1423, 1447, 1483, ...}
Second member of \scriptstyle p \,+\, (0, 4, 6) \,
A073649 {11, 17, 41, 71, 101, 107, 197, 227, 281, 311, 461, 617, 827, 857, 881, 1091, 1301, 1427, 1451, 1487, ...}
Third member of \scriptstyle p \,+\, (0, 4, 6) \,
A098413 {13, 19, 43, 73, 103, 109, 199, 229, 283, 313, 463, 619, 829, 859, 883, 1093, 1303, 1429, 1453, 1489, ...}


Prime triplets: \scriptstyle p\ +\ (0, 2, 6)\, or \scriptstyle p\ +\ (0, 4, 6)\, are all prime
{(5, 7, 11), (7, 11, 13), (11, 13, 17), (13, 17, 19), (17, 19, 23), (37, 41, 43), (41, 43, 47), (67, 71, 73), (97, 101, 103), (101, 103, 107), (103, 107, 109), (107, 109, 113), (191, 193, 197), (193, 197, 199), (223, 227, 229), (227, 229, 233), (277, 281, 283), (307, 311, 313), (311, 313, 317), (347, 349, 353), ...}
All prime triplets \scriptstyle p \,+\, (0, 2, 6) \, are of the form \scriptstyle 6n \,+\, (-1, +1, +5),\, n \,\ge\, 1 \, (shown in red)
All prime triplets \scriptstyle p \,+\, (0, 4, 6) \, are of the form \scriptstyle 6n \,+\, (+1, +5, +7),\, n \,\ge\, 1 \, (shown in green)
First member of \scriptstyle p \,+\, (0, 2, 6) \, or \scriptstyle p \,+\, (0, 4, 6) \,
A007529 {5, 7, 11, 13, 17, 37, 41, 67, 97, 101, 103, 107, 191, 193, 223, 227, 277, 307, 311, 347, 457, 461, ...}
Second member of \scriptstyle p \,+\, (0, 2, 6) \, or \scriptstyle p \,+\, (0, 4, 6) \,
A098414 {7, 11, 13, 17, 19, 41, 43, 71, 101, 103, 107, 109, 193, 197, 227, 229, 281, 311, 313, 349, 461, 463, ...}
Third member of \scriptstyle p \,+\, (0, 2, 6) \, or \scriptstyle p \,+\, (0, 4, 6) \,
A098415 {11, 13, 17, 19, 23, 43, 47, 73, 103, 107, 109, 113, 197, 199, 229, 233, 283, 313, 317, 353, 463, 467, ...}

Prime quadruplets

Prime quadruplets: \scriptstyle p \,+\, (0, 2, 6, 8) \, are all prime.
{(5, 7, 11, 13), (11, 13, 17, 19), (101, 103, 107, 109), (191, 193, 197, 199), (821, 823, 827, 829), (1481, 1483, 1487, 1489), (1871, 1873, 1877, 1879), (2081, 2083, 2087, 2089), (3251, 3253, 3257, 3259), (3461, 3463, 3467, 3469), (5651, 5653, 5657, 5659), (9431, 9433, 9437, 9439), ...}
All prime quadruplets \scriptstyle p \,+\, (0, 2, 6, 8) \, are of the form \scriptstyle 6n \,+\, (-1, +1, +5, +7),\, n \,\ge\, 1 \,
First member of \scriptstyle p \,+\, (0, 2, 6, 8) \,
A007530 {5, 11, 101, 191, 821, 1481, 1871, 2081, 3251, 3461, 5651, 9431, 13001, 15641, 15731, 16061, 18041, ...}
Second member of \scriptstyle p \,+\, (0, 2, 6, 8) \,
A136720 {7, 13, 103, 193, 823, 1483, 1873, 2083, 3253, 3463, 5653, 9433, 13003, 15643, 15733, 16063, 18043, ...}
Third member of \scriptstyle p \,+\, (0, 2, 6, 8) \,
A136721 {11, 17, 107, 197, 827, 1487, 1877, 2087, 3257, 3467, 5657, 9437, 13007, 15647, 15737, 16067, 18047, ...}
Fourth member of \scriptstyle p \,+\, (0, 2, 6, 8) \,
A090258 {13, 19, 109, 199, 829, 1489, 1879, 2089, 3259, 3469, 5659, 9439, 13009, 15649, 15739, 16069, 18049, ...}

Prime 5 tuplets

Prime 5 tuplets: \scriptstyle p \,+\, (0, 2, 6, 8, 12) \, are all prime
{(5, 7, 11, 13, 17), (11, 13, 17, 19, 23), (101, 103, 107, 109, 113), ...}
All prime 5 tuplets \scriptstyle p \,+\, (0, 2, 6, 8, 12) \, are of the form \scriptstyle 6n \,+\, (-1, +1, +5, +7, + 11),\, n \,\ge\, 1 \,
First member of \scriptstyle p \,+\, (0, 2, 6, 8, 12) \,
A022006 {5, 11, 101, 1481, 16061, 19421, 21011, 22271, 43781, 55331, 144161, ...}


Prime 5 tuplets: \scriptstyle p\ +\ (0, 4, 6, 10, 12)\, are all prime
{(7, 11, 13, 17, 19), (97, 101, 107, 107, 109), (1867, 1871, 1873, 1877, 1879), ...}
All prime 5 tuplets \scriptstyle p \,+\, (0, 4, 6, 10, 12) \, are of the form \scriptstyle 6n \,+\, (+1, +5, +7, +11, +13),\, n \,\ge\, 1 \,
First member of \scriptstyle p \,+\, (0, 4, 6, 10, 12) \,
A022007 {7, 97, 1867, 3457, 5647, 15727, 16057, 19417, 43777, 79687, 88807, ...}

Notes

  1. 1.0 1.1 1.2 1.3 Tony Forbes, Prime Clusters and Cunningham Chains, Mathematics of Computation, Volume 68, Number 228, pp. 1739-1747.
  2. Weisstein, Eric W., Twin Prime Conjecture, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/TwinPrimeConjecture.html]
  3. 3.0 3.1 Weisstein, Eric W., k-Tuple Conjecture, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/k-TupleConjecture.html]
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