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# Prime constellations

(Redirected from Twin primes)

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

## Prime clusters

A prime cluster, also called a prime k-tuple (double, triple, quadruple, ...) is a strictly increasing sequence of ${\displaystyle \scriptstyle k\,}$ 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 ${\displaystyle \scriptstyle k\,}$ primes ${\displaystyle \scriptstyle (p_{1},\,p_{2},\,\ldots ,\,p_{k})\,}$ with ${\displaystyle \scriptstyle p_{k}-p_{1}\,=\,s(k)\,}$, where ${\displaystyle \scriptstyle s(k)\,}$ is not necessarily the smallest number ${\displaystyle \scriptstyle s\,}$ for which there exist ${\displaystyle \scriptstyle k\,}$ integers ${\displaystyle \scriptstyle b_{1}\,<\,b_{2}\,<\,\ldots \,<\,b_{k},\,b_{k}-b_{1}\,=\,s,\,}$ and for every prime ${\displaystyle \scriptstyle q\,}$, not all the residues modulo ${\displaystyle \scriptstyle q\,}$ are represented by ${\displaystyle \scriptstyle b_{1},\,b_{2},\,\ldots ,\,b_{k}\,}$.[1]

A nonadmissible prime cluster, also called a nonadmissible prime k-tuple, is a prime cluster such that for some prime ${\displaystyle \scriptstyle q\,}$, all the residues modulo ${\displaystyle \scriptstyle q\,}$ are represented by ${\displaystyle \scriptstyle b_{1},\,b_{2},\,\ldots ,\,b_{k}\,}$. Only a finite number of nonadmissible prime clusters may appear at the beginning of a prime cluster sequence.

For each ${\displaystyle \scriptstyle k\,}$, 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 ${\displaystyle \scriptstyle k\,}$ 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 ${\displaystyle \scriptstyle k\,}$-tuplet is a sequence of ${\displaystyle \scriptstyle k\,}$ consecutive primes, i.e. ${\displaystyle \scriptstyle (p_{1},\,p_{2},\,\ldots ,\,p_{k})\,}$ with ${\displaystyle \scriptstyle p_{k}-p_{1}\,=\,s(k)\,}$, where ${\displaystyle \scriptstyle s(k)\,}$ is the smallest number ${\displaystyle \scriptstyle s\,}$ for which there exist ${\displaystyle \scriptstyle k\,}$ integers ${\displaystyle \scriptstyle b_{1}\,<\,b_{2}\,<\,\ldots \,<\,b_{k},\,b_{k}-b_{1}\,=\,s\,}$ and, for every prime ${\displaystyle \scriptstyle q\,}$, not all the residues modulo ${\displaystyle \scriptstyle q\,}$ are represented by ${\displaystyle \scriptstyle b_{1},\,b_{2},\,\ldots ,\,b_{k}\,}$.[1]

### Prime pairs

#### Twin primes

 Twin primes: prime pairs ${\displaystyle \scriptstyle p\ +\ (0,2)\,}$ {(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 ${\displaystyle \scriptstyle 6n\,+\,(-1,+1),\,n\,\geq \,1\,}$ 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 ${\displaystyle \scriptstyle \pi _{2}(x)\,}$ of twin primes ${\displaystyle \scriptstyle p\,+\,(0,2)\,}$, with ${\displaystyle \scriptstyle p\leq x\,}$ (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 ${\displaystyle \scriptstyle p\,+\,(0,4)\,}$ {(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 ${\displaystyle \scriptstyle 6n\,+\,(+1,+5),\,n\,\geq \,1\,}$ 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 ${\displaystyle \scriptstyle \pi _{4}(x)\,}$ of cousin primes ${\displaystyle \scriptstyle p\,+\,(0,4)\,}$, with ${\displaystyle \scriptstyle p\,\leq \,x\,}$ (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: ${\displaystyle \scriptstyle p\,+\,(0,2,6)\,}$ 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 ${\displaystyle \scriptstyle p\,+\,(0,2,6)\,}$ are of the form ${\displaystyle \scriptstyle 6n\,+\,(-1,+1,+5),\,n\,\geq \,1\,}$ First member of ${\displaystyle \scriptstyle p\,+\,(0,2,6)\,}$ A022004 {5, 11, 17, 41, 101, 107, 191, 227, 311, 347, 461, 641, 821, 857, 881, 1091, 1277, 1301, 1427, 1481, ...} Second member of ${\displaystyle \scriptstyle p\,+\,(0,2,6)\,}$ A073648 {7, 13, 19, 43, 103, 109, 193, 229, 313, 349, 463, 643, 823, 859, 883, 1093, 1279, 1303, 1429, 1483, ...} Third member of ${\displaystyle \scriptstyle p\,+\,(0,2,6)\,}$ A098412 {11, 17, 23, 47, 107, 113, 197, 233, 317, 353, 467, 647, 827, 863, 887, 1097, 1283, 1307, 1433, 1487, ...}

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

 Prime triplets: ${\displaystyle \scriptstyle p\ +\ (0,2,6)\,}$ or ${\displaystyle \scriptstyle p\ +\ (0,4,6)\,}$ 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 ${\displaystyle \scriptstyle p\,+\,(0,2,6)\,}$ are of the form ${\displaystyle \scriptstyle 6n\,+\,(-1,+1,+5),\,n\,\geq \,1\,}$ (shown in red) All prime triplets ${\displaystyle \scriptstyle p\,+\,(0,4,6)\,}$ are of the form ${\displaystyle \scriptstyle 6n\,+\,(+1,+5,+7),\,n\,\geq \,1\,}$ (shown in green) First member of ${\displaystyle \scriptstyle p\,+\,(0,2,6)\,}$ or ${\displaystyle \scriptstyle p\,+\,(0,4,6)\,}$ 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 ${\displaystyle \scriptstyle p\,+\,(0,2,6)\,}$ or ${\displaystyle \scriptstyle p\,+\,(0,4,6)\,}$ 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 ${\displaystyle \scriptstyle p\,+\,(0,2,6)\,}$ or ${\displaystyle \scriptstyle p\,+\,(0,4,6)\,}$ 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: ${\displaystyle \scriptstyle p\,+\,(0,2,6,8)\,}$ 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 ${\displaystyle \scriptstyle p\,+\,(0,2,6,8)\,}$ are of the form ${\displaystyle \scriptstyle 6n\,+\,(-1,+1,+5,+7),\,n\,\geq \,1\,}$ First member of ${\displaystyle \scriptstyle p\,+\,(0,2,6,8)\,}$ A007530 {5, 11, 101, 191, 821, 1481, 1871, 2081, 3251, 3461, 5651, 9431, 13001, 15641, 15731, 16061, 18041, ...} Second member of ${\displaystyle \scriptstyle p\,+\,(0,2,6,8)\,}$ A136720 {7, 13, 103, 193, 823, 1483, 1873, 2083, 3253, 3463, 5653, 9433, 13003, 15643, 15733, 16063, 18043, ...} Third member of ${\displaystyle \scriptstyle p\,+\,(0,2,6,8)\,}$ A136721 {11, 17, 107, 197, 827, 1487, 1877, 2087, 3257, 3467, 5657, 9437, 13007, 15647, 15737, 16067, 18047, ...} Fourth member of ${\displaystyle \scriptstyle p\,+\,(0,2,6,8)\,}$ A090258 {13, 19, 109, 199, 829, 1489, 1879, 2089, 3259, 3469, 5659, 9439, 13009, 15649, 15739, 16069, 18049, ...}

### Prime 5 tuplets

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

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

## Notes

1. 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.
3. Weisstein, Eric W., k-Tuple Conjecture, from MathWorld—A Wolfram Web Resource.