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).
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.
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:
- Every element of the prime number tuple must be prime:
- 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 .
- 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):
- 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 .
For given, correct constellations , both conditions 2. and 3. are invalid. Analogously the reverse: From 2nd and 3rd is all correct constellations open up .
For prime 2-tuple (ie ) - also known as twin primes are known - are and well known. These are:
The four conditions mentioned above are now for prime 2-tuples :
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).
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 .
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 .
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 .
The following is a list of prime number triplets up to 1000 (sequence A098420 in OEIS ):
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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 .
All prime quadruples contain two pairs of prime twins spaced apart.
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
The largest known prime quadruplet has decimal places, was found on February 27, 2019 by Peter Kaiser and is given by with .
The following is a list of the smallest prime quadruplets to :
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The first numbers of these prime quadruplets are (sequence A007530 in OEIS )
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 .
The numbers in the decimal system (except for the first quintuple ) always end with and or and .
All prime quintuples contain two pairs of prime twins spaced apart.
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 :
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The first numbers of the prime quintuplets of the form are (sequence A022006 in OEIS )
The first numbers of the prime quintuplets of the form are (sequence A022007 in OEIS )
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 .
All prime sextuples contain two pairs of prime twins spaced apart.
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 :
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 )
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:
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.
The prime septuples of the constellation do not contain a prime quadruple.
The prime septuples of the constellation contain a prime number quintuple of the form at the beginning.
The prime septuples of the constellation contain a prime number quintuple of the form at the end.
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 :
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 )
The first numbers of these prime number sevenlings of the 2nd constellation are (sequence A022010 in OEIS )
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:
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.
The prime octuples of the constellation do not contain a prime quadruple.
The prime octuples of the constellation contain a prime quadruple at the end.
The prime octuples of the constellation contain a prime number quintuple of the form at the beginning.
The prime octuples of the constellation do not contain a prime number quintuple.
The prime octuples of the constellation contain a prime number quintuple of the form at the end.
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.
The prime octuples of the constellation do not contain a prime number septuple.
The prime octuples of the constellation contain a prime number septuple of the form at the end.
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 :
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 )
The first numbers of these prime number eightlings of the 2nd constellation are (sequence A022012 in OEIS )
The first numbers of these prime eightlings of the 3rd constellation are (episode A022013 in OEIS )
Constellations
The following stands for the prime faculty , i.e. the product of all prime numbers . Formally:
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:
- 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:
- 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:
- 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:
- 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:
- z = 13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501
constellation | 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 |
16737 |
20008
4th | 8th | (0, 2, 6, 8) | (5, 7, 11, 13) | 667674063382677 · 2 33608 - 1 + b | 10132 |
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 |
1606 |
1543
6th | 16 | (0, 4, 6, 10, 12, 16) | (7, 11, 13, 17, 19, 23) | 28993093368077 2400 # + 19417 + b | 1037 |
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 |
402 |
527
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 |
236 206 |
309
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 |
203 135 150 |
214
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 |
136 |
156
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 |
75 |
108
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 |
75 |
108
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 |
46 61 31 50 46 |
50
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 |
46 |
50
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 |
35 40 25 |
33
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 |
31 |
35
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 |
29 28 28 |
30
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 |
28 |
27
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 |
2 30 28 |
30
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 |
30 |
30
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 |
30 |
30
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.
number
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 .
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.
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
- ↑ Eric W. Weisstein : Prime Constellation . In: MathWorld (English). Retrieved June 11, 2014.
- ↑ Eric W. Weisstein : Cousin Primes . In: MathWorld (English). Retrieved June 11, 2014.
- ↑ Eric W. Weisstein : Sexy Primes . In: MathWorld (English). Retrieved June 11, 2014.
- ↑ a b Bounded gaps between primes . Polymath1Wiki; Retrieved June 13, 2014.
- ^ A b Yates, Caldwell: The Largest Known Primes . primes.utm.edu
- ↑ a b c Prime k-tuplets . forbes.googlepages.com; accessed on July 2, 2020.
- ↑ Patterns . forbes.googlepages.com; Retrieved June 11, 2014.
- ↑ Smallest Prime k-tuplets . forbes.googlepages.com; Retrieved June 11, 2014.
- ^ TJ Engelsma: k-tuple permissible patterns results over very large constellations