Prime number tuple

As prime k-tuple - also prime k-tuples  - are in mathematics , more specifically in the theory of numbers , each other close location primes called. This generalizes the concept of prime twins to tuples of any number of prime numbers. The conditions apply that not all possible residues with respect to a prime number may occur in the tuple and that the difference between the smallest and the largest prime number in the prime number tuple must be the smallest possible value (without violating the first condition). ${\displaystyle \leq k}$ ${\displaystyle s}$

Tuples of prime numbers that do not meet all conditions are not called prime tuples or prime tuples. However, they have so, for example, called tuple of two primes of the form might have different names, prime Cousins (Engl. Cousin primes ) and tuple of two primes of the form also can be sexy primes (Engl. Sexy primes ) called. ${\displaystyle k}$ ${\displaystyle (p,p+4)}$ ${\displaystyle (p,p+6)}$

definition

If the set of all possible constellations (which are themselves tuples) of these tuples is known for the prime number tuples with elements , the following conditions apply: ${\displaystyle (p_{1},\dots ,p_{k})}$ ${\displaystyle k}$ ${\displaystyle B}$ ${\displaystyle b}$ ${\displaystyle k}$

1. Every element of the prime number tuple must be prime:
   ${\displaystyle \forall i\in \{1,\dots ,k\}:p_{i}\in \mathbb {P} }$
2. Not all possible residues relating to a prime number may appear in the tuple. In other words, there must be at least one remainder class for every prime number in which no prime number of the tuple falls. Formally: Read: For all prime modules less than or equal there is a remainder less than that is not congruent to all prime numbers in the prime number tuple with regard to the module . The statements " [...] to all [...] not [...] " and " [...] to none [...] " are equivalent, see quantifier . ${\displaystyle \leq k}$ ${\displaystyle \leq k}$
   ${\displaystyle \forall m\in \left\{x\in \mathbb {P} \mid x\leq k\right\}\exists r\in \left\{x\in \mathbb {N} \mid x<m\right\}\forall i\in \{1,\dots ,k\}:p_{i}\not \equiv r\mod m}$
   ${\displaystyle m}$ ${\displaystyle k}$ ${\displaystyle r}$ ${\displaystyle m}$ ${\displaystyle m}$
   
3. The difference between the smallest and the largest element of the tuple must be the same as the -specific minimum value (which is the smallest value that does not violate condition 2): ${\displaystyle k}$
   ${\displaystyle \max \left(\left\{p_{i}\mid i\in \{1,\dots ,k\}\right\}\right)-\min \left(\left\{p_{i}\mid i\in \{1,\dots ,k\}\right\}\right)=s}$
4. The differences between the elements and the first element must be the same as the values ​​of one (but the same for all elements) constellation: where stands for the -th element from the -tuple . ${\displaystyle p_{1}}$
   ${\displaystyle \exists b\in B\;\forall i\in \{1,\dots ,k\}:a_{i}-a_{1}=b_{i}}$
   ${\displaystyle b_{i}}$ ${\displaystyle i}$ ${\displaystyle k}$ ${\displaystyle b}$

For given, correct constellations , both conditions 2. and 3. are invalid. Analogously the reverse: From 2nd and 3rd is all correct constellations open up . ${\displaystyle b}$ ${\displaystyle b}$

For prime 2-tuple (ie ) - also known as twin primes are known - are and well known. These are: ${\displaystyle k=2}$ ${\displaystyle s}$ ${\displaystyle B}$

${\displaystyle s=2}$

${\displaystyle B=\left\{\left(0,2\right)\right\}}$

The four conditions mentioned above are now for prime 2-tuples : ${\displaystyle (p_{1},p_{2})}$

1. ${\displaystyle p_{1},p_{2}\in \mathbb {P} }$
2. ${\displaystyle \left(p_{1}\equiv p_{2}\equiv 0\mod 2\right)\vee \left(p_{1}\equiv p_{2}\equiv 1\mod 2\right)}$
3. ${\displaystyle \max(p_{1},p_{2})-\min(p_{1},p_{2})=s=2}$
4. ${\displaystyle \left(p_{1}-p_{1}=b_{1}=0\right)\wedge \left(p_{2}-p_{1}=b_{2}=2\right)}$

Condition 2 excludes a finite number of prime tuples for each . In this case , the constellation or the prime number tuple is excluded. This excluded constellation has a maximum difference from, and since all prime tuples must have the same maximum difference, there would only be a single prime 2-tuple without the second condition. The reason is that - if all remainder classes occur with regard to the module - all larger tuples according to a constellation that would have been excluded by condition 2 contain exactly a multiple of the module . In the case of , one can say that all prime 2-tuples according to this constellation can be represented in the form for . Here it becomes quite obvious that for all is either or greater than and divisible by (which violates condition 1). ${\displaystyle k}$ ${\displaystyle k}$ ${\displaystyle k=2}$ ${\displaystyle b_{2}=(0,1)}$ ${\displaystyle (2,3)}$ ${\displaystyle s=1}$ ${\displaystyle k}$ ${\displaystyle m}$ ${\displaystyle b}$ ${\displaystyle m}$ ${\displaystyle b=(0,1)}$ ${\displaystyle b}$ ${\displaystyle (n+0,n+1)}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle n>2}$ ${\displaystyle p_{1}}$ ${\displaystyle p_{2}}$ ${\displaystyle 2}$

special cases

Special terms have been established for the smallest values. The constellations, as well as the smallest and the largest known associated prime tuples are listed below in the section Constellations . ${\displaystyle k}$

Prime twin

→ Main article : Prime twin

A prime twin is a pair of two prime numbers that are 2 apart. The smallest prime twins are (3, 5), (5, 7), and (11, 13).

Prime number triplet

Prime triplets are elements of primer 3-tuples, so it holds . 3-tuples are also called triples , which means that prime triplets can also be called prime triples or prime number triples . All prime triples also contain a pair of prime twins. In the case of prime number triples of the form , the first two, and in the case of those of the form, the last two prime numbers form the pair of prime twins mentioned. According to the second condition of the definition, the constellation is incorrect with regard to the module . ${\displaystyle k=3}$ ${\displaystyle (p,p+2,p+6)}$ ${\displaystyle (p,p+4,p+6)}$ ${\displaystyle (p,p+2,p+4)}$ ${\displaystyle m=3}$

The four prime numbers of two prime number triples with two common prime numbers form a prime number quadruple, are prime number quads.

If a prime number is part of three different prime number triples, then five prime numbers are involved and form a prime number quintuple.

It is not known whether there are an infinite number of prime triplets. However, it is believed that there are infinitely many. In 2013 James Maynard and Terence Tao succeeded in showing that there are an infinite number of triples of prime numbers, the difference between which is at most 400,000. Her proof uses results from Zhang Yitang's work on prime twins. In order to prove the existence of an infinite number of actual prime triplets, this upper limit would have to be reduced to six.

On April 24, 2019, Peter Kaiser found the largest prime number triplet with decimal places to date . It is with . ${\displaystyle 20.008}$ ${\displaystyle 4.111.286.921.397\cdot 2^{66.420}-1+d}$ ${\displaystyle d=0,4,6}$

The following is a list of prime number triplets up to 1000 (sequence A098420 in OEIS ):

p (p + 2) (p + 4) (p + 6) 5 7th 11 7th 11 13 11 13 17th 13 17th 19th 17th 19th 23 37 41 43 41 43 47 67 71 73 97 101 103 101 103 107

p (p + 2) (p + 4) (p + 6) 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

p (p + 2) (p + 4) (p + 6) 457 461 463 461 463 467 613 617 619 641 643 647 821 823 827 823 827 829 853 857 859 857 859 863 877 881 883 881 883 887

Prime quadruples

Prime quadruplets are elements of primer 4-tuples, so it holds . 4-tuples are also called quadruples , which legitimizes the designations prime quadruples or prime number quadruples . There is only one correct constellation for prime quadruples. With the one exception , any prime quadruple can be written in both form and form . The number in the middle ( ) is therefore always divisible by and the sum of the prime numbers of the quadruple is always divisible by. In the decimal system , the numbers always end with and . ${\displaystyle k=4}$ ${\displaystyle (5,7,11,13)}$ ${\displaystyle (p,p+2,p+6,p+8)}$ ${\displaystyle (15n-4,15n-2,15n+2,15n+4)}$ ${\displaystyle 15n}$ ${\displaystyle 15}$ ${\displaystyle 60}$ ${\displaystyle 1,3,7}$ ${\displaystyle 9}$

All prime quadruples contain two pairs of prime twins spaced apart. ${\displaystyle 4}$

All prime quadruples contain two overlapping prime number triples according to different constellations.

It is not known whether there are infinitely many prime quadruplets. However, it is believed that there are infinitely many. In 2013 James Maynard and Terence Tao succeeded in showing that there are infinitely many groups of four prime numbers, the difference between which is at most 25 million. Her proof uses results from Zhang Yitang's work on prime twins. In order to prove the existence of an infinite number of actual prime quadruplets, this upper limit would have to be reduced to 8.

According to the Hardy-Littlewood conjecture , the number of prime quadruplets is less than asymptotically given by the formula ${\displaystyle x}$

${\displaystyle C_{4}\int _{2}^{x}\!{\frac {\mathrm {d} t}{(\ln t)^{4}}}\qquad {\text{ mit }}\qquad C_{4}={\frac {27}{2}}\prod _{p>4 \atop p\;{\text{prim}}}{\frac {p^{3}(p-4)}{(p-1)^{4}}}=4{,}15118{\text{ }}08632{\text{ }}37415{\text{ }}75716...}$

(Follow A061642 in OEIS ).

The largest known prime quadruplet has decimal places, was found on February 27, 2019 by Peter Kaiser and is given by with . ${\displaystyle 10132}$ ${\displaystyle 667.674.063.382.677\cdot 2^{33608}-1+d}$ ${\displaystyle d=0,2,6,8}$

The following is a list of the smallest prime quadruplets to : ${\displaystyle 170.000}$

p (p + 2) (p + 6) (p + 8) n (15n-4) (15n-2) (15n + 2) (15n + 4) - 5 7th 11 13 1 11 13 17th 19th 7th 101 103 107 109 13 191 193 197 199 55 821 823 827 829 99 1481 1483 1487 1489 125 1871 1873 1877 1879 139 2081 2083 2087 2089 217 3251 3253 3257 3259 231 3461 3463 3467 3469 377 5651 5653 5657 5659 629 9431 9433 9437 9439 867 13001 13003 13007 13009 1043 15641 15643 15647 15649 1049 15731 15733 15737 15739 1071 16061 16063 16067 16069

p (p + 2) (p + 6) (p + 8) n (15n-4) (15n-2) (15n + 2) (15n + 4) 1203 18041 18043 18047 18049 1261 18911 18913 18917 18919 1295 19421 19423 19427 19429 1401 21011 21013 21017 21019 1485 22271 22273 22277 22279 1687 25301 25303 25307 25309 2115 31721 31723 31727 31729 2323 34841 34843 34847 34849 2919 43781 43783 43787 43789 3423 51341 51343 51347 51349 3689 55331 55333 55337 55339 4199 62981 62983 62987 62989 4481 67211 67213 67217 67219 4633 69491 69493 69497 69499 4815 72221 72223 72227 72229 5151 77261 77263 77267 77269

p (p + 2) (p + 6) (p + 8) n (15n-4) (15n-2) (15n + 2) (15n + 4) 5313 79691 79693 79697 79699 5403 81041 81043 81047 81049 5515 82721 82723 82727 82729 5921 88811 88813 88817 88819 6523 97841 97843 97847 97849 6609 99131 99133 99137 99139 6741 101111 101113 101117 101119 7323 109841 109843 109847 109849 7769 116531 116533 116537 116539 7953 119291 119293 119297 119299 8147 122201 122203 122207 122209 9031 135461 135463 135467 135469 9611 144161 144163 144167 144169 10485 157271 157273 157277 157279 11047 165701 165703 165707 165709 11123 166841 166843 166847 166849

The first numbers of these prime quadruplets are (sequence A007530 in OEIS ) ${\displaystyle 5,11,101,191,821,1481,1871,2081,\ldots }$

Prime quintuplets

Prime quintuplets are elements of prime 5-tuples, so it holds . 5-tuples are also called quintuples , whereupon tuples of prime quintuples are also called prime quintuples or prime quintuples . There are two constellations for prime quintuplets. Each prime quintuplet can be written either in the form or in the form . ${\displaystyle k=5}$ ${\displaystyle (p,p+2,p+6,p+8,p+12)}$ ${\displaystyle (p,p+4,p+6,p+10,p+12)}$

The numbers in the decimal system (except for the first quintuple ) always end with and or and . ${\displaystyle (5,7,11,13,17)}$ ${\displaystyle 7,1,3,7}$ ${\displaystyle 9}$ ${\displaystyle 1,3,7,9}$ ${\displaystyle 3}$

All prime quintuples contain two pairs of prime twins spaced apart. ${\displaystyle 4}$

All prime quintuples contain three overlapping prime triples.

All prime quintuples contain a prime quadruple.

It is not known whether there are an infinite number of prime quintuplets. However, it is believed that there are infinitely many. Even if one could prove that there are infinitely many prime twins, it has not yet been proven that there are infinitely many prime quintuplets. Likewise, it is not enough to be able to prove that there are an infinite number of prime triplets.

The following is a list of the smallest prime quintuplets up to : ${\displaystyle 200.000}$

p (p + 2) (p + 4) (p + 6) (p + 8) (p + 10) (p + 12) 5 7th 11 13 17th 7th 11 13 17th 19th 11 13 17th 19th 23 97 101 103 107 109 101 103 107 109 113 1481 1483 1487 1489 1493 1867 1871 1873 1877 1879 3457 3461 3463 3467 3469 5647 5651 5653 5657 5659 15727 15731 15733 15737 15739 16057 16061 16063 16067 16069 16061 16063 16067 16069 16073 19417 19421 19423 19427 19429

p (p + 2) (p + 4) (p + 6) (p + 8) (p + 10) (p + 12) 19421 19423 19427 19429 19433 21011 21013 21017 21019 21023 22271 22273 22277 22279 22283 43777 43781 43783 43787 43789 43781 43783 43787 43789 43793 55331 55333 55337 55339 55343 79687 79691 79693 79697 79699 88807 88811 88813 88817 88819 101107 101111 101113 101117 101119 144161 144163 144167 144169 144173 165701 165703 165707 165709 165713 166841 166843 166847 166849 166853 195731 195733 195737 195739 195743

The first numbers of the prime quintuplets of the form are (sequence A022006 in OEIS ) ${\displaystyle (p,p+2,p+6,p+8,p+12)}$ ${\displaystyle 5,11,101,1481,16061,19421,21011,\ldots }$

The first numbers of the prime quintuplets of the form are (sequence A022007 in OEIS ) ${\displaystyle (p,p+4,p+6,p+10,p+12)}$ ${\displaystyle 7,97,1867,3457,5647,15727,16057,\ldots }$

Prime number sixteen

Prime number sixteen are elements of primer 6-tuples, so it holds . 6-tuples are also called sextuples , whereupon tuples of prime number sixteen are also called prime sextuples or prime number sextuples . There is only one correct constellation for prime number six. Every prime number sixteen can be written in both form and form . The number in the middle is therefore always divisible by and the sum of the prime numbers of the sextuple is always divisible by. In the decimal system, the numbers always end with and . ${\displaystyle k=6}$ ${\displaystyle (p,p+4,p+6,p+10,p+12,p+16)}$ ${\displaystyle (15n-8,15n-4,15n-2,15n+2,15n+4,15n+8)}$ ${\displaystyle 15}$ ${\displaystyle 90}$ ${\displaystyle 7,1,3,7,9}$ ${\displaystyle 3}$

All prime sextuples contain two pairs of prime twins spaced apart. ${\displaystyle 4}$

All prime sextuples contain four prime number triples, each with two different constellations.

All prime sextuples contain a prime quadruple in the middle.

All prime sextuples contain two prime number quintuples according to different constellations.

It is not known whether there are an infinite number of prime number sixteen. However, it is believed that there are infinitely many.

The following is a list of the smallest prime number siblings up to : ${\displaystyle 3.000.000}$

	p	(p + 4)	(p + 6)	(p + 10)	(p + 12)	(p + 16)
n	(15n-8)	(15n-4)	(15n-2)	(15n + 2)	(15n + 4)	(15n + 8)
1	7th	11	13	17th	19th	23
7th	97	101	103	107	109	113
1071	16057	16061	16063	16067	16069	16073
1295	19417	19421	19423	19427	19429	19433
2919	43777	43781	43783	43787	43789	43793
72751	1091257	1091261	1091263	1091267	1091269	1091273
107723	1615837	1615841	1615843	1615847	1615849	1615853
130291	1954357	1954361	1954363	1954367	1954369	1954373
188181	2822707	2822711	2822713	2822717	2822719	2822723
189329	2839927	2839931	2839933	2839937	2839939	2839943

The first numbers of these prime number siblings are (sequence A022008 in OEIS ) ${\displaystyle 7,97,16057,19417,43777,1091257,\ldots }$

Prime number sevenling

Prime number sevenlings are elements of primer 7-tuples, so it holds . 7-tuples are also called septuples , which justifies the names prime septuples or prime number septuples for tuples of associated prime number sevenlings . Each prime sevenling can be written in one of the following two constellations: ${\displaystyle k=7}$

${\displaystyle {\begin{aligned}(p,&\quad p+2,&p+6,&\quad p+8,&p+12,&\quad p+18,&p+20)\\(p,&\quad p+2,&p+8,&\quad p+12,&p+14,&\quad p+18,&p+20)\end{aligned}}}$

All prime septuples contain three pairs of prime twins.

All prime septuples contain three prime triples.

The prime septuples of the constellation contain a prime number quadruple at the beginning. ${\displaystyle (p,p+2,p+6,p+8,p+12,p+18,p+20)}$

The prime septuples of the constellation do not contain a prime quadruple. ${\displaystyle (p,p+2,p+8,p+12,p+14,p+18,p+20)}$

The prime septuples of the constellation contain a prime number quintuple of the form at the beginning. ${\displaystyle (p,p+2,p+6,p+8,p+12,p+18,p+20)}$ ${\displaystyle (p,p+2,p+6,p+8,p+12)}$

The prime septuples of the constellation contain a prime number quintuple of the form at the end. ${\displaystyle (p,p+2,p+8,p+12,p+14,p+18,p+20)}$ ${\displaystyle (p,p+4,p+6,p+10,p+12)}$

All prime septuples do not contain a prime number sextuple.

It is not known whether there are an infinite number of prime number sevenlings. However, it is believed that there are infinitely many.

The following is a list of the smallest prime number sevenlings up to : ${\displaystyle 10.000.000}$

p	(p + 2)	(p + 6)	(p + 8)	(p + 12)	(p + 14)	(p + 18)	(p + 20)
11	13	17th	19th	23		29	31
5639	5641		5647	5651	5653	5657	5659
88799	88801		88807	88811	88813	88817	88819
165701	165703	165707	165709	165713		165719	165721
284729	284731		284737	284741	284743	284747	284749
626609	626611		626617	626621	626623	626627	626629
855719	855721		855727	855731	855733	855737	855739
1068701	1068703	1068707	1068709	1068713		1068719	1068721
1146779	1146781		1146787	1146791	1146793	1146797	1146799
6560999	6561001		6561007	6561011	6561013	6561017	6561019
7540439	7540441		7540447	7540451	7540453	7540457	7540459
8573429	8573431		8573437	8573441	8573443	8573447	8573449

The first numbers of these prime number sevenlings of the 1st constellation are (sequence A022009 in OEIS ) ${\displaystyle 11,165701,1068701,11900501,15760091,\ldots }$

The first numbers of these prime number sevenlings of the 2nd constellation are (sequence A022010 in OEIS ) ${\displaystyle 5639,88799,284729,626609,855719,\ldots }$

Prime eightling

Prime eightlings are elements of primer 8-tuples, so it holds . 8-tuples are also called octuples , which justifies the designation prime octuples or prime number octuples for tuples of related prime eightlings . Each prime number eight can be written in one of the three following constellations: ${\displaystyle k=8}$

${\displaystyle {\begin{aligned}(p,&\quad p+2,&p+6,&\quad p+8,&p+12,&\quad p+18,&p+20,&\quad p+26)\\(p,&\quad p+2,&p+6,&\quad p+12,&p+14,&\quad p+20,&p+24,&\quad p+26)\\(p,&\quad p+6,&p+8,&\quad p+14,&p+18,&\quad p+20,&p+24,&\quad p+26)\end{aligned}}}$

All prime octuples contain three pairs of prime twins.

All prime octuples contain three prime triples.

The prime octuples of the constellation contain a prime quadruple at the beginning. ${\displaystyle (p,p+2,p+6,p+8,p+12,p+18,p+20,p+26)}$

The prime octuples of the constellation do not contain a prime quadruple. ${\displaystyle (p,p+2,p+6,p+12,p+14,p+20,p+24,p+26)}$

The prime octuples of the constellation contain a prime quadruple at the end. ${\displaystyle (p,p+6,p+8,p+14,p+18,p+20,p+24,p+26)}$

The prime octuples of the constellation contain a prime number quintuple of the form at the beginning. ${\displaystyle (p,p+2,p+6,p+8,p+12,p+18,p+20,p+26)}$ ${\displaystyle (p,p+2,p+6,p+8,p+12)}$

The prime octuples of the constellation do not contain a prime number quintuple. ${\displaystyle (p,p+2,p+6,p+12,p+14,p+20,p+24,p+26)}$

The prime octuples of the constellation contain a prime number quintuple of the form at the end. ${\displaystyle (p,p+6,p+8,p+14,p+18,p+20,p+24,p+26)}$ ${\displaystyle (p,p+4,p+6,p+10,p+12)}$

All prime octuples do not contain a prime number sextuple.

The prime octuples of the constellation contain a prime number seventh of the form at the beginning. ${\displaystyle (p,p+2,p+6,p+8,p+12,p+18,p+20,p+26)}$ ${\displaystyle (p,p+2,p+6,p+8,p+12,p+18,p+20)}$

The prime octuples of the constellation do not contain a prime number septuple. ${\displaystyle (p,p+2,p+6,p+12,p+14,p+20,p+24,p+26)}$

The prime octuples of the constellation contain a prime number septuple of the form at the end. ${\displaystyle (p,p+6,p+8,p+14,p+18,p+20,p+24,p+26)}$ ${\displaystyle (p,p+2,p+8,p+12,p+14,p+18,p+20)}$

It is not known whether there are an infinite number of prime eightlings. However, it is believed that there are infinitely many.

The following is a list of the smallest prime eightlings up to : ${\displaystyle 100.000.000}$

p	(p + 2)	(p + 6)	(p + 8)	(p + 12)	(p + 14)	(p + 18)	(p + 20)	(p + 24)	(p + 26)
11	13	17th	19th	23		29	31		37
17th	19th	23		29	31		37	41	43
1277	1279	1283		1289	1291		1297	1301	1303
88793		88799	88801		88807	88811	88813	88817	88819
113147	113149	113153		113159	113161		113167	113171	113173
284723		284729	284731		284737	284741	284743	284747	284749
855713		855719	855721		855727	855731	855733	855737	855739
1146773		1146779	1146781		1146787	1146791	1146793	1146797	1146799
2580647	2580649	2580653		2580659	2580661		2580667	2580671	2580673
6560993		6560999	6561001		6561007	6561011	6561013	6561017	6561019
15760091	15760093	15760097	15760099	15760103		15760109	15760111		15760117
20737877	20737879	20737883		20737889	20737891		20737897	20737901	20737903
25658441	25658443	25658447	25658449	25658453		25658459	25658461		25658467
58208387	58208389	58208393		58208399	58208401		58208407	58208411	58208413
69156533		69156539	69156541		69156547	69156551	69156553	69156557	69156559
73373537	73373539	73373543		73373549	73373551		73373557	73373561	73373563
74266253		74266259	74266261		74266267	74266271	74266273	74266277	74266279
76170527	76170529	76170533		76170539	76170541		76170547	76170551	76170553
93625991	93625993	93625997	93625999	93626003		93626009	93626011		93626017

The first numbers of these prime eightlings of the 1st constellation are (sequence A022011 in OEIS ) ${\displaystyle 11,15760091,25658441,93625991,182403491,\ldots }$

The first numbers of these prime number eightlings of the 2nd constellation are (sequence A022012 in OEIS ) ${\displaystyle 17,1277,113147,2580647,20737877,\ldots }$

The first numbers of these prime eightlings of the 3rd constellation are (episode A022013 in OEIS ) ${\displaystyle 88793,284723,855713,1146773,6560993,\ldots }$

Constellations

The following stands for the prime faculty , i.e. the product of all prime numbers . Formally: ${\displaystyle n\#}$ ${\displaystyle \leq n}$ ${\displaystyle n\#=\prod _{p\in [1,n]\cap \mathbb {P} }p}$

With the prime number sevenlings (i.e. at ) the current record number (the starting value of the prime number tuple) has a very large 397-digit factor that has no place in the following table. That is why it should be mentioned here: ${\displaystyle k=7}$

v = 3282186887886020104563334103168841560140170122832878265333984717524446848642006351778066196724473922496202015365392599420232189723690267622904036090100548730918665577766385906339769372916363127576607799875309038457637116938538279395260265064447747742612368890410202171085974848375899782610469497787199182516499466558387976965904497393971453496036241885200541893611077817261813672809971503287259089

Likewise, the current record number for the prime eightlings (i.e. for ) has a very large 309-digit factor that has no place in the following table. It is the following: ${\displaystyle k=8}$

w = 180315603666324941736246499611038008956766654317324393497979420080023419122770753875860484538058745150737924272736595181879630622419937962884400457684262508916903227173539448425672986701352885568601216876574954143683785154052777944853575966721564055977230720210975532569630677164985160348078053972930223236301

The current record number for the prime number newcomers (i.e. at ) also has a very large 214-digit factor that has no place in the following table. It is the following: ${\displaystyle k=9}$

x = 135089395130488839592107939790614694273780519945242859373935990906783020181836989730867402793793099583190261126801577778286057978194674088208035343966415096940731748325144640462796940731748325144640

In the case of the prime number ten-pupils (i.e. at ) the current record value also has a very large factor, which has no place in the following table. The 156-digit value is: ${\displaystyle k=10}$

y = 214827459156724285781326096946472068735663111421143314638264406778127841579031177541867948153048409284186250295821410602471821768366806167983702198639483391

In the case of the prime eleven and twelve primaries (i.e. with and ) the current record number also has a very large 92-digit factor, which is already mentioned here: ${\displaystyle k=11}$ ${\displaystyle k=12}$

z = 13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501

${\displaystyle k}$	${\displaystyle s}$	constellation ${\displaystyle b}$	Smallest prime tuple	Largest known prime tuple (as of July 2, 2020)	Put
2	2	(0, 2)	(3, 5)	2996863034895 · 2 1290000 - 1 + b	388342
3	6th	(0, 2, 6) (0, 4, 6)	(5, 7, 11) (7, 11, 13)	4111286921397 · 2 66420 - 1 + b 6521953289619 · 2 55555 - 5 + b	020008 016737
4th	8th	(0, 2, 6, 8)	(5, 7, 11, 13)	667674063382677 · 2 33608 - 1 + b	010132
5	12	(0, 2, 6, 8, 12) (0, 4, 6, 10, 12)	(5, 7, 11, 13, 17) (7, 11, 13, 17, 19)	2316765173284 3600 # + 16061 + b 394254311495 3733 #: 2 - 8 + b	001543 001606
6th	16	(0, 4, 6, 10, 12, 16)	(7, 11, 13, 17, 19, 23)	28993093368077 2400 # + 19417 + b	001037
7th	20th	(0, 2, 6, 8, 12, 18, 20) (0, 2, 8, 12, 14, 18, 20)	(11, 13, 17, 19, 23, 29, 31) (5639, 5641, 5647, 5651, 5653, 5657, 5659)	v 317 # + 1068701 + b 4733578067069 940 # + 626609 + b	000527 000402
8th	26th	(0, 2, 6, 8, 12, 18, 20, 26) (0, 2, 6, 12, 14, 20, 24, 26) (0, 6, 8, 14, 18, 20, 24, 26 )	(11, 13, 17, 19, 23, 29, 31, 37) (17, 19, 23, 29, 31, 37, 41, 43) (88793, 88799, 88801, 88807, 88811, 88813, 88817, 88819 )	w + b 29995576270632 550 # + 1277 + b 12874261020824 465 # + 88793 + b	000309 000236 000206
9	30th	(0, 2, 6, 8, 12, 18, 20, 26, 30) (0, 2, 6, 12, 14, 20, 24, 26, 30) (0, 4, 6, 10, 16, 18 , 24, 28, 30) (0, 4, 10, 12, 18, 22, 24, 28, 30)	(11, 13, 17, 19, 23, 29, 31, 37, 41) (13, 17, 19, 23, 29, 31, 37, 41, 43) (17, 19, 23, 29, 31, 37 , 41, 43, 47) (88789, 88793, 88799, 88801, 88807, 88811, 88813, 88817, 88819)	x + b 663579549486449 460 # + 1277 + b 106345403186416 300 # + 29247913 + b 68663510211259 337 # + 88789 + b	000214 000203 000135 000150
10	32	(0, 2, 6, 8, 12, 18, 20, 26, 30, 32) (0, 2, 6, 12, 14, 20, 24, 26, 30, 32)	(11, 13, 17, 19, 23, 29, 31, 37, 41, 43) (9853497737, ...)	y + b 772556746441918 300 # + 29247917 + b	000156 000136
11	36	(0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36) (0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 36)	(11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47) (1418575498573, ...)	z + 49376500222690335 * 229 # + b 613176722801194 151 # + 177321217 + 6 + b	000108 000075
12	42	(0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42) (0, 6, 10, 12, 16, 22, 24, 30, 34, 36, 40, 42)	(11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53) (1418575498567, ...)	z + 27407861785763183 * 229 # + b 613176722801194 151 # + 177321217 + b	000108 000075
13	48	(0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48) (0, 2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48) (0, 2, 12, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48) (0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 36, 46, 48) (0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48) (0, 6, 12, 16, 18, 22, 28, 30, 36, 40, 42, 46, 48)	(11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59) (7697168877290909, ...) (10527733922579, ...) (1707898733581273, ...) (13, 17, 19, 23 , 29, 31, 37, 41, 43, 47, 53, 59, 61) (186460616596321, ...)	14815550 107 # + 4385574275277311 + b 381955327397348 80 # + 18393209 + b 4135997219394611 110 # + 117092849 + b 10 30 +8449315049002492743 + b 14815550 107 # + 4385574275 2487311 + 2 + b 381955	000050 000046 000061 000031 000050 000046
14th	50	(0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50) (0, 2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 50)	(11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61) (79287805466244209, ...)	14815550 107 # + 4385574275277311 + b 381955327397348 80 # + 18393209 + b	000050 000046
15th	56	(0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50, 56) (0, 2, 6, 12, 14, 20, 24, 26, 30, 36, 42, 44, 50, 54, 56) (0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56) (0, 6, 8, 14, 20, 24, 26, 30, 36, 38, 44, 48, 50, 54, 56)	(11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67) (17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73) (1158722981124148367, ...) (14094050870111867483, ...)	107173714602413868775303366934621 + b 10004646546202610858599716515809907 + b 33554294028531569 61 # + 57800747 + b 1000543338893999053267943 + b	000033 000035 000040 000025
16	60	(0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56, 60) (0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48, 54, 58, 60)	(47710850533373130107, ...) (13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73)	32225573 # + 1354238543317302647 + b 1003234871202624616703163933853 + b	000035 000031
17th	66	(0, 2, 6, 12, 14, 20, 24, 26, 30, 36, 42, 44, 50, 54, 56, 62, 66) (0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48, 54, 58, 60, 66) (0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60 , 64, 66) (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) (734975534793324512717947, ...) (1620784518619319025971, ...)	100845391935878564991556707107 + b 11413975438568556104209245223 + b 5867208169546174917450987997 + b 5867208169546174917450987997 + 4 + b	000030 000029 000028 000028
18th	70	(0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48, 54, 58, 60, 66, 70) (0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70)	(13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83) (2845372542509911868266807, ...)	183837276562811649018077773 + b 5867208169546174917450987997 + b	000027 000028
19th	76	(0, 4, 6, 10, 12, 16, 24, 30, 34, 40, 42, 46, 52, 54, 60, 66, 70, 72, 76) (0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48, 54, 58, 60, 66, 70, 76) (0, 4, 6, 10, 16, 22, 24, 30, 34, 36, 42 , 46, 52, 60, 64, 66, 70, 72, 76) (0, 6, 10, 16, 18, 22, 28, 30, 36, 42, 46, 48, 52, 58, 60, 66, 70, 72, 76)	(622803914376064301858782434517, ...) (13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89) (37, 41, 43 , 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113) (630134041802574490482213901, ...)	622803914376064301858782434517 + b 13 + b 138433730977092118055599751669 + 8 + b 2406179998282157386567481191 + b	000030 000002 000030 000028
20th	80	(0, 2, 6, 8, 12, 20, 26, 30, 36, 38, 42, 48, 50, 56, 62, 66, 68, 72, 78, 80) (0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80)	(14374153072440029138813893241, ...) (29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109)	999627565307688186459783232931 + b 957278727962618711849051282459 + b	000030 000030
21st	84	(0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80, 84) (0, 4, 6, 10, 12, 16, 24, 30, 34, 40, 42, 46, 52, 54, 60, 66, 70, 72, 76, 82, 84)	(29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113) (622803914376064301858782434517, ...)	248283957683772055928836513589 + b 622803914376064301858782434517 + b	000030 000030

There is at least one associated constellation for every arbitrarily high . These can be found with the help of a computer using a simple brute force algorithm . Finding prime number tuples for given constellations is associated with a high computational effort, especially for higher ones. ${\displaystyle k}$ ${\displaystyle k}$

number

→ Main article : Open question about prime twins

Euclid's theorem, which is trivial to prove, says that there are infinitely many prime numbers. The question that appears very similar, whether there are an infinite number of twins, triplets, etc., has not yet been clarified. So far it has only been possible to prove that an infinite number of prime numbers exist with a maximum distance between them . ${\displaystyle 246}$

→ Main article : Hardy-Littlewood conjecture

According to the unproven first Hardy-Littlewood conjecture , the number of prime tuples up to a limit is asymptotic to a formula established in the conjecture. ${\displaystyle k}$

literature

• Herschel F. Smith: On a generalization of the prime pair problem . (PDF) In: Math. Tables Aids Comput. , 11, 1957, No. 60, pp. 249-254
• Paul Erdős , Hans Riesel : On admissible constellations of consecutive primes . In: BIT (Nordisk Tidskrift for Informationsbehandling), 28, 1988, No. 3, pp. 391-396

Individual evidence

1. ↑ Eric W. Weisstein : Prime Constellation . In: MathWorld (English). Retrieved June 11, 2014.
2. ↑ Eric W. Weisstein : Cousin Primes . In: MathWorld (English). Retrieved June 11, 2014.
3. ↑ Eric W. Weisstein : Sexy Primes . In: MathWorld (English). Retrieved June 11, 2014.
4. ↑ ^{a b} Bounded gaps between primes . Polymath1Wiki; Retrieved June 13, 2014.
5. ^ ^{A b} Yates, Caldwell: The Largest Known Primes . primes.utm.edu
6. ↑ ^{a b c} Prime k-tuplets . forbes.googlepages.com; accessed on July 2, 2020.
7. ↑ Patterns . forbes.googlepages.com; Retrieved June 11, 2014.
8. ↑ Smallest Prime k-tuplets . forbes.googlepages.com; Retrieved June 11, 2014.
9. ^ TJ Engelsma: k-tuple permissible patterns results over very large constellations