Twin prime - wikipedia Twin prime A twin prime is a prime number that is either 2 less or 2 more than another prime number -- for example, either member of the twin prime pair ( 41, 43 ). In other words, a twin prime is a prime that has a prime gap of two. 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 become increasingly rare as one examines larger ranges, in keeping with the general tendency of gaps between adjacent primes to become larger as the numbers themselves get larger. However, it is unknown if there are infinitely many twin primes or if there is a largest pair. The work of Yitang Zhang in 2013, as well as work by James Maynard, Terence Tao and others, has made substantial progress towards proving that there are infinitely many twin primes, but at present it remains unsolved. Unsolved problem in mathematics : Are there infinitely many twin primes ? ( more unsolved problems in mathematics ) Contents 1 History 2 Properties 2.1 Brun 's theorem 3 Other theorems weaker than the twin prime conjecture 4 Conjectures 4.1 First Hardy -- Littlewood conjecture 4.2 Polignac 's conjecture 5 Large twin primes 6 Other elementary properties 7 Isolated prime 8 See also 9 References 10 Further reading 11 External links History 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 that there are infinitely many primes p such that p + 2 is also prime. In 1849, de Polignac made the more general conjecture that for every natural number k, there are infinitely many primes p such that p + 2k is also prime. The case k = 1 of de Polignac 's conjecture is the twin prime conjecture. A stronger form of the twin prime conjecture, the Hardy -- Littlewood conjecture ( see below ), postulates a distribution law for twin primes akin to the prime number theorem. On April 17, 2013, Yitang Zhang announced a proof that for some integer N that is less than 70 million, there are infinitely many pairs of primes that differ by N. Zhang 's paper was accepted by Annals of Mathematics in early May 2013. Terence Tao subsequently proposed a Polymath Project collaborative effort to optimize Zhang 's bound. As of April 14, 2014, one year after Zhang 's announcement, the bound has been reduced to 246. Further, assuming the Elliott -- Halberstam conjecture and its generalized form, the Polymath project wiki states that the bound has been reduced to 12 and 6, respectively. These improved bounds were discovered using a different approach that was simpler than Zhang 's and was discovered independently by James Maynard and Terence Tao. This second approach also gave bounds for the smallest f ( m ) needed to guarantee that infinitely many intervals of width f ( m ) contain at least m primes. Properties Usually the pair ( 2, 3 ) is not considered to be a pair of twin primes. Since 2 is the only even prime, this pair is the only pair of prime numbers that differ by one ; thus twin primes are as closely spaced as possible for any other two primes. The first few twin prime pairs are : ( 3, 5 ), ( 5, 7 ), ( 11, 13 ), ( 17, 19 ), ( 29, 31 ), ( 41, 43 ), ( 59, 61 ), ( 71, 73 ), ( 101, 103 ), ( 107, 109 ), ( 137, 139 ),... A077800. Five is the only prime in two distinct pairs. Every twin prime pair except ( 3, 5 ) is of the form ( 6n − 1, 6n + 1 ) for some natural number n ; that is, the number between the two primes is a multiple of 6. Brun 's theorem Main article : Brun 's theorem In 1915, Viggo Brun showed that the sum of reciprocals of the twin primes was convergent. 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 C N ( log ⁡ N ) 2 ( displaystyle ( \frac ( CN ) ( ( log N ) ^ ( 2 ) ) ) ) for some absolute constant C > 0. In fact, it is bounded above by C ′ N ( log ⁡ N ) 2 ( 1 + O ( log ⁡ log ⁡ N log ⁡ N ) ) ( displaystyle ( \frac ( C'N ) ( ( log N ) ^ ( 2 ) ) ) left ( 1 + O left ( ( \frac ( log log N ) ( log N ) ) \right ) \right ) ), where C ' = 8C, where C is the twin prime constant, given below. Other theorems weaker than the twin prime conjecture In 1940, Paul Erdős showed that there is a 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, i.e. lim inf n → ∞ p n + 1 − p n log ⁡ p n = 0. ( displaystyle liminf _ ( n \to infty ) ( \frac ( p_ ( n + 1 ) - p_ ( n ) ) ( log p_ ( n ) ) ) = 0. ) 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. The result of Yitang Zhang, lim inf n → ∞ ( p n + 1 − p n ) < N with N = 7 × 10 7, ( displaystyle liminf _ ( n \to infty ) ( p_ ( n + 1 ) - p_ ( n ) ) < N ; ( \text ( with ) ) ; N = 7 \times 10 ^ ( 7 ), ) is a major improvement on the Goldston -- Graham -- Pintz -- Yıldırım result. The Polymath Project optimization of Zhang 's bound and the work of Maynard has reduced the bound to N = 246. Conjectures First Hardy -- Littlewood conjecture The Hardy -- Littlewood conjecture ( named after G.H. Hardy and John Littlewood ) is a generalization of the twin prime conjecture. It is concerned with the distribution of prime constellations, including twin primes, in analogy to the prime number theorem. Let π ( x ) denote the number of primes p ≤ x such that p + 2 is also prime. Define the twin prime constant C as C 2 = ∏ p p r i m e p ≥ 3 ( 1 − 1 ( p − 1 ) 2 ) ≈ 0.660161815846869573927812110014... ( displaystyle C_ ( 2 ) = prod _ ( \textstyle ( p ; ( \rm ( prime ) ) atop p geq 3 ) ) left ( 1 - ( \frac ( 1 ) ( ( p - 1 ) ^ ( 2 ) ) ) \right ) approx 0.660161815846869573927812110014 dots ) A005597 ( here the product extends over all prime numbers p ≥ 3 ). Then the conjecture is that π 2 ( x ) ∼ 2 C 2 x ( ln ⁡ x ) 2 ∼ 2 C 2 ∫ 2 x d t ( ln ⁡ t ) 2 ( displaystyle pi _ ( 2 ) ( x ) sim 2C_ ( 2 ) ( \frac ( x ) ( ( ln x ) ^ ( 2 ) ) ) sim 2C_ ( 2 ) int _ ( 2 ) ^ ( x ) ( dt over ( ln t ) ^ ( 2 ) ) ) in the sense that the quotient of the two expressions tends to 1 as x approaches infinity. ( The second ~ is not part of the conjecture and is proven by integration by parts. ) The 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 and would imply the twin prime conjecture, but remains unresolved. This first Hardy -- Littlewood conjecture on prime k - tuples implies that the second Hardy -- Littlewood conjecture is false. Polignac 's conjecture 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 gaps 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. Indeed, if such a k did not exist, then for any positive even natural number N there are at most finitely many n such that p − p = m for all m < N and so for n large enough we have p − p > N, which would contradict Zhang 's result. Large twin primes Beginning in 2007, two distributed computing projects, Twin Prime Search and PrimeGrid, have produced several record - largest twin primes. As of May 2018, the current largest twin prime pair known is 2996863034895 2 ± 1, with 388,342 decimal digits. It was discovered in September 2016. There are 808,675,888,577,436 twin prime pairs below 10. An empirical analysis of all prime pairs up to 4.35 10 shows that if the number of such pairs less than x is f ( x ) x / ( log x ) then f ( x ) is about 1.7 for small x and decreases towards about 1.3 as x tends to infinity. The limiting value of f ( x ) is conjectured to equal twice the twin prime constant ( A114907 ) ( not to be confused with Brun 's constant ), according to the Hardy -- Littlewood conjecture. Other elementary properties 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 twin prime pairs. The lower member of a pair is by definition a Chen prime. It has been proven that the pair ( m, m + 2 ) is a twin prime if and only if 4 ( ( m − 1 ) ! + 1 ) ≡ − m ( mod m ( m + 2 ) ). ( displaystyle 4 ( ( m - 1 ) ! + 1 ) equiv - m ( pmod ( m ( m + 2 ) ) ). ) If m − 4 or m + 6 is also prime then the three primes are called a prime triplet. For a twin prime pair of the form ( 6n − 1, 6n + 1 ) for some natural number n > 1, n must have units digit 0, 2, 3, 5, 7, or 8 ( A002822 ). The sum of twin primes greater than 3 is divisible by 12. Isolated prime An isolated prime ( also known as single prime or non-twin 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. The first few isolated primes are 2, 23, 37, 47, 53, 67, 79, 83, 89, 97,... A007510 It follows from Brun 's theorem that almost all primes are members of this sequence. See also Cousin prime Prime gap Prime k - tuple Prime quadruplet Prime triplet Sexy prime References Jump up ^ Terry Tao, Small and Large Gaps Between the Primes Jump up ^ McKee, Maggie ( 14 May 2013 ). `` First proof that infinitely many prime numbers come in pairs ''. Nature. doi : 10.1038 / nature. 2013.12989. ISSN 0028 - 0836. Jump up ^ McKee, M. ( 2013 ). `` First proof that prime numbers pair up into infinity ''. Nature. doi : 10.1038 / nature. 2013.12989. Jump up ^ Zhang, Yitang ( 2014 ). `` Bounded gaps between primes ''. Annals of Mathematics. 179 ( 3 ) : 1121 -- 1174. doi : 10.4007 / annals. 2014.179. 3.7. MR 3171761. Jump up ^ Tao, Terence ( June 4, 2013 ). `` Polymath proposal : bounded gaps between primes ''. ^ Jump up to : `` Bounded gaps between primes ''. Polymath. Retrieved 2014 - 03 - 27. Jump up ^ The First 100,000 Twin Primes Jump up ^ Brun, V. ( 1915 ), `` Über das Goldbachsche Gesetz und die Anzahl der Primzahlpaare '', Archiv for Mathematik og Naturvidenskab ( in German ), 34 ( 8 ) : 3 -- 19, ISSN 0365 - 4524, JFM 45.0330. 16 Jump up ^ Bateman & Diamond ( 2004 ) p. 313 Jump up ^ Heini Halberstam, and Hans - Egon Richert, Sieve Methods, p. 117, Dover Publications, 2010 Jump up ^ Goldston, Daniel Alan ; Motohashi, Yoichi ; Pintz, János ; Yıldırım, Cem Yalçın ( 2006 ), `` Small gaps between primes exist '', Japan Academy. Proceedings. Series A. Mathematical Sciences, 82 ( 4 ) : 61 -- 65, arXiv : math. NT / 0505300, doi : 10.3792 / pjaa. 82.61, MR 2222213. Jump up ^ Goldston, D.A. ; Graham, S.W. ; Pintz, J. ; Yıldırım, C.Y. ( 2009 ), `` Small gaps between primes or almost primes '', Transactions of the American Mathematical Society, 361 ( 10 ) : 5285 -- 5330, arXiv : math. NT / 0506067, doi : 10.1090 / S0002 - 9947 - 09 - 04788 - 6, MR 2515812 Jump up ^ Maynard, James ( 2015 ), `` Small gaps between primes '', Annals of Mathematics, Second Series, 181 ( 1 ) : 383 -- 413, arXiv : 1311.4600, doi : 10.4007 / annals. 2015.181. 1.7, MR 3272929 Jump up ^ Polymath, D.H.J. ( 2014 ), `` Variants of the Selberg sieve, and bounded intervals containing many primes '', Research in the Mathematical Sciences, 1 : Art. 12, 83, arXiv : 1407.4897, doi : 10.1186 / s40687 - 014 - 0012 - 7, MR 3373710 Jump up ^ Sloane, N.J.A. ( ed. ). `` Sequence A001692 ( Number of irreducible polynomials of degree n over GF ( 5 ) ; dimensions of free Lie algebras ) ''. The On - Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2011 - 02 - 02. -- A page of number theoretical constants Jump up ^ Bateman & Diamond ( 2004 ) pp. 334 -- 335 Jump up ^ Caldwell, Chris K. `` The Prime Database : 2996863034895 * 2 ^ 1290000 - 1 ''. Jump up ^ `` World Record Twin Primes Found ! ''. Jump up ^ Tomás Oliveira e Silva ( 7 April 2008 ). `` Tables of values of pi ( x ) and of pi2 ( x ) ''. Aveiro University. Retrieved 7 January 2011. Bateman, Paul T. ; Diamond, Harold G. ( 2004 ). Analytic Number Theory. World Scientific. ISBN 981 - 256 - 080 - 7. Zbl 1074.11001. Further reading Sloane, Neil ; Plouffe, Simon ( 1995 ). The Encyclopedia of Integer Sequences. San Diego, CA : Academic Press. ISBN 0 - 12 - 558630 - 2. External links Hazewinkel, Michiel, ed. ( 2001 ) ( 1994 ), `` Twins '', Encyclopedia of Mathematics, Springer Science + Business Media B.V. / Kluwer Academic Publishers, ISBN 978 - 1 - 55608 - 010 - 4 Top - 20 Twin Primes at Chris Caldwell 's Prime Pages Xavier Gourdon, Pascal Sebah : Introduction to Twin Primes and Brun 's Constant `` Official press release '' of 58711 - digit twin prime record Weisstein, Eric W. `` Twin Primes ''. MathWorld. The 20 000 first twin primes Polymath : Bounded gaps between primes Sudden Progress on Prime Number Problem Has Mathematicians Buzzing Prime number classes By formula Fermat ( 2 + 1 ) Mersenne ( 2 − 1 ) Double Mersenne ( 2 − 1 ) Wagstaff ( 2 + 1 ) / 3 Proth ( k 2 + 1 ) Factorial ( n ! ± 1 ) Primorial ( p # ± 1 ) Euclid ( p # + 1 ) Pythagorean ( 4n + 1 ) Pierpont ( 2 3 + 1 ) Quartan ( x + y ) Solinas ( 2 ± 2 ± 1 ) Cullen ( n 2 + 1 ) Woodall ( n 2 − 1 ) Cuban ( x − y ) / ( x − y ) Carol ( 2 − 1 ) − 2 Kynea ( 2 + 1 ) − 2 Leyland ( x + y ) Thabit ( 3 2 − 1 ) Mills ( ⌊ A ⌋ ) By integer sequence Fibonacci Lucas Pell Newman -- Shanks -- Williams Perrin Partitions Bell Motzkin By property Wieferich ( pair ) Wall -- Sun -- Sun Wolstenholme Wilson Lucky Fortunate Ramanujan Pillai Regular Strong Stern Supersingular ( elliptic curve ) Supersingular ( moonshine theory ) Good Super Higgs Highly cototient Base - dependent Happy Dihedral Palindromic Emirp Repunit ( 10 − 1 ) / 9 Permutable Circular Truncatable Strobogrammatic Minimal Weakly Full reptend Unique Primeval Self Smarandache -- Wellin Patterns Twin ( p, p + 2 ) Bi-twin chain ( n − 1, n + 1, 2n − 1, 2n + 1,... ) Triplet ( p, p + 2 or p + 4, p + 6 ) Quadruplet ( p, p + 2, p + 6, p + 8 ) k − Tuple Cousin ( p, p + 4 ) Sexy ( p, p + 6 ) Chen Sophie Germain ( p, 2p + 1 ) Cunningham chain ( p, 2p ± 1,... ) Safe ( p, ( p − 1 ) / 2 ) Arithmetic progression ( p + a n, n = 0, 1,... ) Balanced ( consecutive p − n, p, p + n ) By size Titanic ( 1,000 + digits ) Gigantic ( 10,000 + digits ) Mega ( 1,000,000 + digits ) Largest known Complex numbers Eisenstein prime Gaussian prime Composite numbers Pseudoprime Almost prime Semiprime Interprime Pernicious Related topics Probable prime Industrial - grade prime Illegal prime Formula for primes Prime gap First 60 primes 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281 List of prime numbers Mathematics portal Prime number conjectures Hardy -- Littlewood 1st 2nd Agoh -- Giuga Andrica 's Artin 's Bateman -- Horn Brocard 's Bunyakovsky Chinese hypothesis Cramér 's Dickson 's Elliott -- Halberstam Firoozbakht 's Gilbreath 's Grimm 's Landau 's problems Goldbach 's weak Legendre 's Twin prime Legendre 's constant Lemoine 's Mersenne Oppermann 's Polignac 's Pólya Redmond -- Sun Schinzel 's hypothesis H Waring 's prime number Retrieved from `` https://en.wikipedia.org/w/index.php?title=Twin_prime&oldid=852847942 '' Categories : Classes of prime numbers Unsolved problems in mathematics Hidden categories : CS1 German - language sources ( de ) Articles containing potentially dated statements from May 2018 All articles containing potentially dated statements Talk Contents About Wikipedia বাংলা Bân - lâm - gú Български Català Čeština Dansk Deutsch Ελληνικά Español Esperanto فارسی Français 한국어 Հայերեն Bahasa Indonesia Italiano עברית Magyar Монгол Nederlands 日本 語 Norsk Plattdüütsch Polski Português Ripoarisch Русский Simple English Slovenščina Suomi Svenska தமிழ் Türkçe Українська Tiếng Việt 中文 28 more Edit links This page was last edited on 31 July 2018, at 19 : 45 ( UTC ). 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Question: list of twin prime numbers from 1 to 100
Answer: (3,  5)