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%I #140 May 20 2023 04:29:16
%S 3,5,7,11,13,17,19,29,31,41,43,59,61,71,73,101,103,107,109,137,139,
%T 149,151,179,181,191,193,197,199,227,229,239,241,269,271,281,283,311,
%U 313,347,349,419,421,431,433,461,463,521,523,569,571,599,601,617,619,641,643
%N Twin primes.
%C Union of A001359 and A006512.
%C The only twin primes that are Fibonacci numbers are 3, 5 and 13 [MacKinnon]. - _Emeric Deutsch_, Apr 24 2005
%C (p, p+2) are twin primes if and only if p + 2 can be represented as the sum of two primes. Brun (1919): Even if there are infinitely many twin primes, the series of all twin prime reciprocals does converges to [Brun's constant] (A065421). Clement (1949): For every n > 1, (n, n+2) are twin primes if and only if 4((n-1)! + 1) == -n (mod n(n+2)). - _Stefan Steinerberger_, Dec 04 2005
%C A164292(a(n)) = 1. - _Reinhard Zumkeller_, Mar 29 2010
%C The 100355-digit numbers 65516468355 * 2^333333 +- 1 are currently the largest known twin prime pair. They were discovered by Twin Prime Search and Primegrid in August 2009. - _Paul Muljadi_, Mar 07 2011
%C For every n > 2, the pair (n, n+2) is a twin prime if and only if ((n-1)!!)^4 == 1 (mod n*(n+2)). - _Thomas Ordowski_, Aug 15 2016
%C The term "twin primes" ("primzahlzwillinge", in German) was coined by the German mathematician Paul Gustav Samuel Stäckel (1862-1919) in 1916. Brun (1919) used the same term in French ("nombres premiers jumeaux"). Glaisher (1878) and Hardy and Littlewood (1923) used the term "prime-pairs". The term "twin primes" in English was used by Dantzig (1930). - _Amiram Eldar_, May 20 2023
%D Tobias Dantzig, Number: The Language of Science, Macmillan, 1930.
%D Paulo Ribenboim, The New Book of Prime Number Records, Springer-Verlag, 1996, pp. 259-265.
%H T. D. Noe, <a href="/A001097/b001097.txt">Table of n, a(n) for n = 1..10000</a>
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, p. 870.
%H Viggo Brun, La série 1/5 + 1/7 + 1/11 + 1/13 + 1/17 + 1/19 + 1/29 + 1/31 + 1/41 + 1/43 + 1/59 + 1/61 + ... où les dénominateurs sont "nombres premiers jumeaux" est convergente ou finie, Bull Sci. Math. 43 (1919), <a href="https://gallica.bnf.fr/ark:/12148/bpt6k96292009/f104.image">100-104</a> and <a href="https://gallica.bnf.fr/ark:/12148/bpt6k96292009/f128.image">124-128</a>.
%H J. P. Delahaye, <a href="http://www.lifl.fr/~delahaye/SIME/JPD/PLS_Nb_premiers_jumeaux.pdf">Premiers jumeaux: frères ennemis?</a> [Twin primes: Enemy Brothers?], Pour la science, No. 260 (Juin 1999), 102-106.
%H Harvey Dubner, <a href="http://www.emis.de/journals/JIS/VOL8/Dubner/dubner71.html">Twin Prime Statistics</a>, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.2.
%H J. C. Evard, <a href="http://web.archive.org/web/20110726012847/http://www.math.utoledo.edu/~jevard/Page012.htm">Twin primes and their applications</a> [Cached copy on the Wayback Machine]
%H J. C. Evard, <a href="/A077800/a077800.html">Twin primes and their applications</a> [Local cached copy]
%H J. C. Evard, <a href="/A077800/a077800.pdf">Twin primes and their applications</a> [Pdf file of cached copy]
%H J. W. L. Glaisher, <a href="https://archive.org/details/messengermathem04glaigoog/page/n38/mode/2up">An enumeration of prime-pairs</a>, Messenger of Mathematics, Vol. 8 (1878), pp. 28-33.
%H Andrew Granville, <a href="http://www.dms.umontreal.ca/~andrew/PDF/BaltimoreLecture.pdf">Primes in intervals of bounded length</a>, Joint Math Meeting, Current Events Bulletin, Baltimore, Friday, Jan 17 2014.
%H G. H. Hardy and J. E. Littlewood, <a href="https://doi.org/10.1007/BF02403921">Some problems of 'Partitio numerorum'; III: On the expression of a number as a sum of primes</a>, Acta Mathematica, Vol. 44, No. 1 (1923), pp. 1-70; <a href="http://archive.ymsc.tsinghua.edu.cn/pacm_download/117/5327-11511_2006_Article_BF02403921.pdf">alternative link</a>.
%H Nick MacKinnon, <a href="http://www.jstor.org/stable/2695779">Problem 10844</a>, Amer. Math. Monthly 109, (2002), p. 78.
%H James Maynard, <a href="https://www.jstor.org/stable/24522956">Small gaps between primes</a>, Annals of Mathematics, Second series, Vol. 181, No. 1 (2015), pp. 383-413; <a href="http://arxiv.org/abs/1311.4600">arXiv preprint</a>, arXiv:1311.4600 [math.NT], 2013-2019.
%H Omar E. Pol, <a href="http://www.polprimos.com">Determinacion geometrica de los numeros primos y perfectos</a>.
%H D. H. J. Polymath, <a href="http://arxiv.org/abs/1402.0811">New equidistribution estimates of Zhang type, and bounded gaps between primes</a>, arXiv:1402.0811 [math.NT], 2014.
%H D. H. J. Polymath, <a href="http://arxiv.org/abs/1407.4897">Variants of the Selberg sieve, and bounded intervals containing many primes</a>, arXiv:1407.4897 [math.NT], 2014.
%H Paul Stäckel, <a href="https://archiv.ub.uni-heidelberg.de/volltextserver/12465/">Die Darstellung der geraden Zahlen als Summen von zwei Primzahlen</a>, Sitzungsberichte der Heidelberger Akademie der Wissenschaften, Mathematisch-naturwissenschaftliche Klasse (in German), Abt. A, Bd. 10 (1916), pp. 1-47. See p. 22.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TwinPrimes.html">Twin Primes</a>.
%H Yitang Zhang, <a href="http://dx.doi.org/10.4007/annals.2014.179.3.7">Bounded gaps between primes</a>, Annals of Mathematics, Volume 179, Issue 3 (2014), Pages 1121-1174.
%H <a href="/index/Cor#core">Index entries for "core" sequences</a>.
%H <a href="/index/Pri#gaps">Index entries for primes, gaps between</a>.
%p A001097 := proc(n)
%p option remember;
%p if n = 1 then
%p 3;
%p else
%p for a from procname(n-1)+1 do
%p if isprime(a) and ( isprime(a-2) or isprime(a+2) ) then
%p return a;
%p end if;
%p end do:
%p end if;
%p end proc: # _R. J. Mathar_, Feb 19 2015
%t Select[ Prime[ Range[120]], PrimeQ[ # - 2] || PrimeQ[ # + 2] &] (* _Robert G. Wilson v_, Jun 09 2005 *)
%t Union[Flatten[Select[Partition[Prime[Range[200]],2,1],#[[2]]-#[[1]] == 2&]]] (* _Harvey P. Dale_, Aug 19 2015 *)
%o (PARI) isA001097(n) = (isprime(n) && (isprime(n+2) || isprime(n-2))) \\ _Michael B. Porter_, Oct 29 2009
%o (PARI) a(n)=if(n==1,return(3));my(p);forprime(q=3,default(primelimit), if(q-p==2 && (n-=2)<0, return(if(n==-1,q,p)));p=q) \\ _Charles R Greathouse IV_, Aug 22 2012
%o (PARI) list(lim)=my(v=List([3]),p=5); forprime(q=7,lim, if(q-p==2, listput(v,p); listput(v,q)); p=q); if(p+2>lim && isprime(p+2), listput(v,p)); Vec(v) \\ _Charles R Greathouse IV_, Mar 17 2017
%o (Haskell)
%o a001097 n = a001097_list !! (n-1)
%o a001097_list = filter ((== 1) . a164292) [1..]
%o -- _Reinhard Zumkeller_, Feb 03 2014, Nov 27 2011
%o (Python)
%o from sympy import nextprime
%o from itertools import islice
%o def agen(): # generator of terms
%o yield 3
%o p, q = 5, 7
%o while True:
%o if q - p == 2: yield from [p, q]
%o p, q = q, nextprime(q)
%o print(list(islice(agen(), 58))) # _Michael S. Branicky_, Apr 30 2022
%Y Cf. A070076, A001359, A006512, A164292. See A077800 for another version.
%K nonn,core,nice
%O 1,1
%A _N. J. A. Sloane_, Mar 15 1996
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