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| A '''twin prime''' is a [[prime number]] that has a [[prime gap]] of [[2 (number)|two]], in other word, differs from another prime number by two, for example the twin prime pair ([[41 (number)|41]], [[43 (number)|43]]). Sometimes the term ''twin prime'' is used for a pair of twin primes; an alternative name for this is '''prime twin''' or '''prime pair'''. Twin primes appear despite the general tendency of gaps between adjacent primes to become larger as the numbers themselves get larger due to the [[prime number theorem]] (the "average gap" between primes less than ''n'' is log(''n'')).
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| {{unsolved|mathematics|Are there infinitely many twin primes?}}
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| ==History==
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| The question of whether there exist infinitely many twin primes has been one of the great open questions in [[number theory]] for many years. This is the content of the '''twin prime conjecture''', which states: ''There are infinitely many primes'' ''p'' ''such that'' ''p'' + 2 ''is also prime.'' In 1849 [[Alphonse de Polignac|de Polignac]] made the more [[de Polignac's conjecture|general conjecture]] that for every [[natural number]] ''k'', there are infinitely many prime pairs ''p'' and ''p''′ such that ''p''′ − ''p'' = 2''k''. The case ''k'' = 1 is the twin prime conjecture.
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| A stronger form of the twin prime conjecture, the Hardy–Littlewood conjecture, postulates a distribution law for twin primes akin to the [[prime number theorem]].
| | Last of all, you ought to make sure to continue on to be inside your economic restrictions. Internet based soccer betting is addictive especially if you received just a few game titles. It could seem quick to bet and remove as there is definitely no bodily parting with the challenging earned pounds. Gamble just for the sheer pleasurable of it. In no way [http://www.adobe.com/cfusion/search/index.cfm?term=&produce&loc=en_us&siteSection=home produce] gambling currently being an dependancy simply because you will just be in the getting rid of conclude.<br><br>[google.com google.com]<br><br>Horse racing and [http://Apuestasdeportivasenlinea.weebly.com/apuestas-online.html ir Ahora] is finding bigger by the day because of the ever growing variety of betting exchanges getting area in excess of the Net. This is exactly where, as opposed to bookmaker, common persons wager against other punters in get to get substantially fairer odds. In this scenario, getting a Ray Edelson Services can be a decisive benefit in opposition to a fellow punter. Possessing a skilled suggestions on your facet will absolutely raise the probabilities of your profitable even though betting on the on the net sporting activities betting web-sites.<br><br>The sports and fitness arena has 1 a lot more choice as the Exercise science.There is a science powering work out and what people require to do in purchase to get rid of body weight and build muscle mass mass.That is why so several workout science majors are working with athletes, fitness centers, famous people and other individuals who are incredibly tuned into the work out earth.There is no dearth of opportunities that just one will get as one particular can decide to work for a huge gymnasium or go as a health and fitness coach to any of the stars.A single other increasing discipline of study is the overall health marketing diploma online.This degree encourages staying wholesome and the health choices driving it.<br><br>To the yet again lawn pool area, the artificial grass signifies no bare spots brought about by the h2o splashed from the pool. After they are about the pool isn't muddy or dusty, the drinking drinking water within just the pool stays cleaner, far more time. Even inside the most well-liked local climate, the grass continues to be a great searching inexperienced color.<br><br>When you're at the racecourse by itself you can either invest the day betting on horses or drinking champagne. Some folks even do equally. If you're the drinking sort, head for the champagne tent. If you're the betting sort then you can wander all around and consider a appear at the horses. See which a single normally takes your fancy. It's often really worth inquiring concerns to the bookies or horse handlers, you in no way know exactly where you could get a valuable suggestion from. Then, once you've put your guess, head to your seat to look at the race but be confident to arrive wonderful and early so you don't get trapped at the back again! There are typically Tv screens up but there is absolutely nothing far more thrilling than looking at your horse appear in initially with your possess eyes.<br><br>First, maintain in thoughts that your overall body is currently developed to be ready to eliminate unwelcome energy and fats, and clean out contaminants by natural means. The obstacle actually is how you can condition oneself in producing your overall body process and organs functionality the way they really should be.<br><br>Betting on soccer is not as straightforward as it appears to be. You will need to study the diverse football betting programs and understand how they function. It is not like you can just wager on a one group and then walk away. Despite the fact that betting on soccer has grow to be a pastime or a way to make more funds, there is much far more to it than meets the eye. Devoid of possessing the simple know-how of how issues perform in the track record, you will not be able to completely improve your likely in earning or raising your winnings. On prime if this, there are usually associated challenges with the handicaps and bets specially if the video game is not a match or when 1 staff is surely an underdog. |
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| On April 17, 2013, [[Yitang Zhang]] announced a proof that for some integer ''N'' that is at most 70 million, there are infinitely many pairs of primes that differ by ''N''.<ref>{{cite journal|last=McKee|first=Maggie|title=First proof that infinitely many prime numbers come in pairs|journal=Nature|date=14 May 2013|url=http://www.nature.com/news/first-proof-that-infinitely-many-prime-numbers-come-in-pairs-1.12989|issn=0028-0836}}</ref><ref>{{cite doi|10.1038/nature.2013.12989}}</ref> Zhang's paper was accepted by ''[[Annals of Mathematics]]'' in early May 2013.<ref>{{cite journal | url = http://annals.math.princeton.edu/articles/7954 | title = Bounded gaps between primes | first = Yitang | last = Zhang | journal = Annals of Mathematics | publisher = Princeton University and the Institute for Advanced Study | accessdate =May 21, 2013}}</ref> [[Terence Tao]] subsequently proposed a [[Polymath project]] collaborative effort to optimize Zhang’s bound;<ref>{{cite web|last=Tao|first=Terence|authorlink=Terence Tao|url=http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/|title=Polymath proposal: bounded gaps between primes|date=June 4, 2013}}</ref> as of November 28, 2013, Thomas Engelsma claims to have reduced the bound to N = 576.<ref>{{cite web|url=http://michaelnielsen.org/polymath1/index.php?title=Bounded_gaps_between_primes|title=Bounded gaps between primes|publisher=Polymath|accessdate=2013-11-28}}</ref>
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| ==Brun's theorem==
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| In 1915, [[Viggo Brun]] showed that the [[Brun's constant|sum of reciprocals of the twin primes]] was convergent.<ref>{{citation | jfm=45.0330.16 | last=Brun | first=V. | authorlink=Viggo Brun | title=Über das Goldbachsche Gesetz und die Anzahl der Primzahlpaare | language=German | journal=Arch. f. Math. og Naturv. | volume=34 | number=8 | pages=3–19 | year=1915 | issn=0365-4524 }}</ref> This famous result, called [[Brun's theorem]], was the first use of the [[Brun sieve]] and helped initiate the development of modern [[sieve theory]]. The modern version of Brun's argument can be used to show that the number of twin primes less than ''N'' does not exceed
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| :<math>\frac{CN}{(\log N)^2}</math>
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| for some absolute constant ''C'' > 0.<ref name=BD313>Bateman & Diamond (2004) p. 313</ref>
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| ==Other theorems weaker than the twin-prime conjecture==
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| In 1940, [[Paul Erdős]] showed that there is a [[Mathematical constant|constant]] ''c'' < 1 and infinitely many primes ''p'' such that (''p''′ − ''p'') < (''c'' ln ''p'') where ''p''′ denotes the next prime after ''p''. This result was successively improved; in 1986 [[Helmut Maier]] showed that a constant ''c'' < 0.25 can be used. In 2004 [[Daniel Goldston]] and [[Cem Yıldırım]] showed that the constant could be improved further to ''c'' = 0.085786… In 2005, Goldston, [[János Pintz]] and Yıldırım established that ''c'' can be chosen to be arbitrarily small<ref>{{citation
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| | last1 = Goldston | first1 = Daniel Alan | author1-link = Daniel Goldston
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| | last2 = Motohashi | first2 = Yoichi
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| | last3 = Pintz | first3 = János | author3-link = János Pintz
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| | last4 = Yıldırım | first4 = Cem Yalçın | author4-link = Cem Yıldırım
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| | arxiv = math.NT/0505300
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| | mr = 2222213
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| | issue = 4
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| | journal = Japan Academy. Proceedings. Series A. Mathematical Sciences
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| | pages = 61–65
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| | title = Small gaps between primes exist
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| | url = http://projecteuclid.org/getRecord?id=euclid.pja/1146576181
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| | volume = 82
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| | year = 2006}}.</ref><ref>{{citation
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| | last1 = Goldston | first1 = D. A. | author1-link = Daniel Goldston
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| | last2 = Graham | first2 = S. W.
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| | last3 = Pintz | first3 = J. | author3-link = János Pintz
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| | last4 = Yıldırım | first4 = C. Y. | author4-link = Cem Yıldırım
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| | arxiv = math.NT/0506067
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| | doi = 10.1090/S0002-9947-09-04788-6
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| | mr = 2515812
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| | issue = 10
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| | journal = Transactions of the American Mathematical Society
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| | pages = 5285–5330
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| | title = Small gaps between primes or almost primes
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| | volume = 361
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| | year = 2009}}.</ref>
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| :<math>\liminf_{n\to\infty}\frac{p_{n+1}-p_n}{\log p_n}=0.</math>
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| By assuming the [[Elliott–Halberstam conjecture]] or a slightly weaker version, they were able to show that there are infinitely many ''n'' such that at least two of ''n'', ''n'' + 2, ''n'' + 6, ''n'' + 8, ''n'' + 12, ''n'' + 18, or ''n'' + 20 are prime. Under a stronger hypothesis they showed that for infinitely many ''n'', at least two of ''n'', ''n'' + 2, ''n'' + 4, and ''n'' + 6 are prime.
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| The result of Zhang,
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| :<math>\liminf_{n\to\infty}(p_{n+1}-p_n)<N\;with\;N=7\times10^7,</math>
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| is a major improvement on the Goldston–Graham–Pintz–Yıldırım result. The Polymath project optimization of Zhang’s bound and Maynard claims to have reduced the bound to N = 600 are farther improvements.
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| Every twin prime pair except (3, 5) is of the form (6''n'' − 1, 6''n'' + 1) for some [[natural number]] ''n'', and with the exception of ''n'' = 1, ''n'' must end in 0, 2, 3, 5, 7, or 8. {{OEIS2C|id=A002822}}
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| It has been proven that the pair (''m'', ''m'' + 2) is a twin prime if and only if
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| :<math>4((m-1)! + 1) \equiv -m \pmod {m(m+2)}.</math>
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| If ''m'' − 4 or ''m'' + 6 is also prime then the three primes are called a [[prime triplet]].
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| ==Largest known twin prime pair==
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| On January 15, 2007 two [[distributed computing]] projects, [[Twin Prime Search]] and [[PrimeGrid]] found the largest known twin primes, 2003663613 · 2<sup>195000</sup> ± 1. The numbers have 58711 [[decimal]] [[Numerical digit|digits]]. Their discoverer was [[Eric Vautier]] of [[France]].
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| On August 6, 2009 those same two projects announced that a new record twin prime had been found.<ref>{{cite web | url = http://www.primegrid.com/all_news.php#188 | title = News Archive | work = [[PrimeGrid]] | accessdate = 2009-08-07 | date = 6 August 2009}}</ref> It is 65516468355 · 2<sup>333333</sup> ± 1.<ref>{{cite web | url = http://primes.utm.edu/primes/page.php?id=89650 | title = The Prime Database: 65516468355*2^333333-1 | work = [[Prime Pages]] | date = 13 August 2009 | accessdate = 2009-08-14}}</ref> The numbers have 100355 decimal digits.
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| On December 25, 2011 PrimeGrid announced that yet another record twin prime had been found.<ref>{{cite web | url = http://www.primegrid.com/forum_thread.php?id=3874&nowrap=true#45670 | title = News Archive | work = [[PrimeGrid]] | accessdate = 2011-12-25 | date = 25 December 2011}}</ref> It is 3756801695685 · 2<sup>666669</sup> ± 1.<ref>{{cite web | url = http://primes.utm.edu/primes/page.php?id=103792 | title = The Prime Database: 3756801695685 · 2666669 - 1 | work = [[Prime Pages]] | date = 25 December 2011 | accessdate = 2011-12-25}}</ref> The numbers have 200700 decimal digits.
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| An empirical analysis of all prime pairs up to 4.35 · 10<sup>15</sup> shows that if the number of such pairs less than <var>x</var> is f(<var>x</var>)·<var>x</var>/(log <var>x</var>)<sup>2</sup> then f(<var>x</var>) is about 1.7 for small <var>x</var> and decreases towards about 1.3 as <var>x</var> tends to infinity.
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| There are 808,675,888,577,436 twin prime pairs below 10<sup>18</sup>.<ref>{{cite web | url = http://www.ieeta.pt/~tos/primes.html | title = Tables of values of pi(x) and of pi2(x) | author = Tomás Oliveira e Silva | publisher = [[Aveiro University]] | date = 7 April 2008 | accessdate = 7 January 2011}}</ref>
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| The limiting value of f(<var>x</var>) is conjectured to equal twice the twin prime constant (not to be confused with [[Brun's constant]])
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| :<math> 2 \prod_{\textstyle{p\;{\rm prime}\atop p \ge 3}} \left(1 - \frac{1}{(p-1)^2}\right) = 1.3203236\ldots;</math>
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| {{OEIS2C|id=A114907}}
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| this conjecture would imply the twin prime conjecture, but remains unresolved.
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| The twin prime conjecture would give a better approximation, as with the [[prime counting function]], by
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| :<math>\pi_2(x) \approx 2C_2\; \operatorname{li}_2(x) = 2C_2 \int_2^x \frac{dt}{\left(\log_e t \right)^2}.</math>
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| == Properties ==
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| The first few twin prime pairs are:
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| :(3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), (71, 73), (101, 103), (107, 109), (137, 139), … {{OEIS2C|id=A077800}}.
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| The only even prime is 2; except for the pair (2, 3), twin primes are as closely spaced as possible for two primes.
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| Every third odd number is divisible by 3, which requires that no three successive odd numbers can be prime unless one of them is 3. Five is therefore the only prime that is part of two pairs. Along the same lines, other than the first pair, the number centered between the twin primes must always be divisible by 6. The lower member of a pair is by definition a [[Chen prime]].
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| ==First Hardy–Littlewood conjecture==
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| <!-- [[Hardy–Littlewood conjecture]] redirects here-->
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| The '''Hardy–Littlewood conjecture''' (after [[G. H. Hardy]] and [[John Edensor Littlewood|John Littlewood]]) is a generalization of the twin prime conjecture. It is concerned with the distribution of [[prime constellation]]s, including twin primes, in analogy to the [[prime number theorem]]. Let π<sub>2</sub>(''x'') denote the number of primes ''p'' ≤ ''x'' such that ''p'' + 2 is also prime. Define the '''twin prime constant''' ''C''<sub>2</sub> as<ref>{{cite web
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| | url = http://oeis.org/A001692/a001692.html
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| | title = A page of number theoretical constants
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| | accessdate = 2011-02-02
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| | year = 2007
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| }}</ref>
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| :<math>C_2 = \prod_{p\ge 3} \frac{p(p-2)}{(p-1)^2} \approx 0.66016 18158 46869 57392 78121 10014\dots</math>
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| {{OEIS2C|id=A005597}}
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| (here the product extends over all prime numbers ''p'' ≥ 3). Then the conjecture is that
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| :<math>\pi_2(n) \sim 2 C_2 \frac{n}{(\ln n)^2} \sim 2 C_2 \int_2^n {dt \over (\ln t)^2}</math>
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| in the sense that the quotient of the two expressions [[limit of a function|tends to]] 1 as ''n'' approaches infinity.<ref name=BD3345>Bateman & Diamond (2004) pp.334–335</ref> (The second ~ is not part of the conjecture and is proved by [[integration by parts]].) | |
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| This conjecture can be justified (but not proven) by assuming that 1 / ln ''t'' describes the [[density function]] of the prime distribution, an assumption suggested by the prime number theorem.
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| ==Polignac's conjecture==
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| [[Polignac's conjecture]] from 1849 states that for every positive even natural number ''k'', there are infinitely many consecutive prime pairs ''p'' and ''p′'' such that ''p′ − p'' = ''k'' (i.e. there are infinitely many [[prime gap]]s of size ''k''). The case ''k'' = 2 is the twin prime conjecture. The conjecture has not yet been proven or disproven for any specific value of ''k'', but Zhang's result proves that it is true for at least one (currently unknown) value of k.
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| ==Isolated prime==
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| An '''isolated prime''' is a prime number ''p'' such that neither ''p'' − 2 nor ''p'' + 2 is prime. In other words, ''p'' is not part of a twin prime pair. For example, 23 is an isolated prime since 21 and 25 are both [[composite number|composite]].
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| The first few isolated primes are
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| :[[2 (number)|2]], [[23 (number)|23]], [[37 (number)|37]], [[47 (number)|47]], [[53 (number)|53]], [[67 (number)|67]], [[79 (number)|79]], [[83 (number)|83]], [[89 (number)|89]], [[97 (number)|97]], … {{OEIS2C|id=A007510}}.
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| ==See also==
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| * [[Cousin prime]]
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| * [[Prime gap]]
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| * [[Prime k-tuple|Prime ''k''-tuple]]
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| * [[Prime quadruplet]]
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| * [[Prime triplet]]
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| * [[Sexy prime]]
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| ==References==
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| {{reflist|colwidth=30em}}
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| * {{cite book | first1=Paul T. | last1=Bateman | author1-link=Paul T. Bateman | first2=Harold G. | last2=Diamond | title=Analytic Number Theory | publisher=World Scientific | year=2004 | isbn=981-256-080-7 | zbl=1074.11001 }}
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| ==Further reading==
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| *{{cite book |authorlink=Neil Sloane |first=Neil |last=Sloane |authorlink2=Simon Plouffe |first2=Simon |last2=Plouffe |title=The Encyclopedia of Integer Sequences |publisher=Academic Press |location=San Diego, CA |year=1995 |isbn=0-12-558630-2 }}
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| == External links ==
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| * {{springer|title=Twins|id=p/t094470}}
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| * [http://primes.utm.edu/top20/page.php?id=1 Top-20 Twin Primes] at Chris Caldwell's [[Prime Pages]].
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| * Xavier Gourdon, Pascal Sebah: [http://numbers.computation.free.fr/Constants/Primes/twin.html ''Introduction to Twin Primes and Brun's Constant'']
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| *[http://mersenneforum.org/showpost.php?p=96237&postcount=51 "Official press release"] of 58711-digit twin prime record.
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| * {{MathWorld | urlname=TwinPrimes | title=Twin Primes}}
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| *[http://arnflo.se/~site_files/Other/twinprimes The 20 000 first twin primes]
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| *[http://michaelnielsen.org/polymath1/index.php?title=Bounded_gaps_between_primes#World_records Polymath: Bounded gaps between primes]
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| *[http://www.wired.com/wiredscience/2013/11/prime/ Sudden Progress on Prime Number Problem Has Mathematicians Buzzing]
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| {{Prime number classes}}
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| [[Category:Classes of prime numbers]]
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| [[Category:Unsolved problems in mathematics]]
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