A001097 Twin primes.

3, 5, 7, 11, 13, 17, 19, 29, 31, 41, 43, 59, 61, 71, 73, 101, 103, 107, 109, 137, 139, 149, 151, 179, 181, 191, 193, 197, 199, 227, 229, 239, 241, 269, 271, 281, 283, 311, 313, 347, 349, 419, 421, 431, 433, 461, 463, 521, 523, 569, 571, 599, 601, 617, 619, 641, 643

Offset: 1

Author

N. J. A. Sloane, Mar 15 1996

Keywords

• nonn
• core
• nice

Comments

Union of A001359 and A006512.

The only twin primes that are Fibonacci numbers are 3, 5 and 13 [MacKinnon]. - Emeric Deutsch, Apr 24 2005

(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

A164292(a(n)) = 1. - Reinhard Zumkeller, Mar 29 2010

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

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

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

References

• Tobias Dantzig, Number: The Language of Science, Macmillan, 1930.
• Paulo Ribenboim, The New Book of Prime Number Records, Springer-Verlag, 1996, pp. 259-265.
• David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 132.

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, p. 870.
• 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), 100-104 and 124-128.
• J. P. Delahaye, Premiers jumeaux: frères ennemis? [Twin primes: Enemy Brothers?], Pour la science, No. 260 (Juin 1999), 102-106.
• Harvey Dubner, Twin Prime Statistics, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.2.
• J. C. Evard, Twin primes and their applications. [Cached copy on the Wayback Machine]
• J. C. Evard, Twin primes and their applications. [Local cached copy]
• J. C. Evard, Twin primes and their applications. [Pdf file of cached copy]
• J. W. L. Glaisher, An enumeration of prime-pairs, Messenger of Mathematics, Vol. 8 (1878), pp. 28-33.
• Andrew Granville, Primes in intervals of bounded length, Joint Math Meeting, Current Events Bulletin, Baltimore, Friday, Jan 17 2014.
• G. H. Hardy and J. E. Littlewood, Some problems of 'Partitio numerorum'; III: On the expression of a number as a sum of primes, Acta Mathematica, Vol. 44, No. 1 (1923), pp. 1-70; alternative link.
• Nick MacKinnon, Problem 10844, Amer. Math. Monthly 109, (2002), p. 78.
• James Maynard, Small gaps between primes, Annals of Mathematics, Second series, Vol. 181, No. 1 (2015), pp. 383-413; arXiv preprint, arXiv:1311.4600 [math.NT], 2013-2019.
• Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.
• D. H. J. Polymath, New equidistribution estimates of Zhang type, and bounded gaps between primes, arXiv:1402.0811 [math.NT], 2014.
• D. H. J. Polymath, Variants of the Selberg sieve, and bounded intervals containing many primes, arXiv:1407.4897 [math.NT], 2014.
• Paul Stäckel, Die Darstellung der geraden Zahlen als Summen von zwei Primzahlen, Sitzungsberichte der Heidelberger Akademie der Wissenschaften, Mathematisch-naturwissenschaftliche Klasse (in German), Abt. A, Bd. 10 (1916), pp. 1-47. See p. 22.
• Eric Weisstein's World of Mathematics, Twin Primes.
• Yitang Zhang, Bounded gaps between primes, Annals of Mathematics, Volume 179, Issue 3 (2014), Pages 1121-1174.
• Index entries for "core" sequences.
• Index entries for primes, gaps between.

Crossrefs

Cf. A070076, A001359, A006512, A164292. See A077800 for another version.

Programs

Haskell

a001097 n = a001097_list !! (n-1)
a001097_list = filter ((== 1) . a164292) [1..]
-- Reinhard Zumkeller, Feb 03 2014, Nov 27 2011

Maple

A001097 := proc(n)
    option remember;
    if n = 1 then
        3;
    else
        for a from procname(n-1)+1 do
            if isprime(a) and ( isprime(a-2) or isprime(a+2) ) then
                return a;
            end if;
        end do:
    end if;
end proc: # R. J. Mathar, Feb 19 2015

Mathematica

Select[ Prime[ Range[120]], PrimeQ[ # - 2] || PrimeQ[ # + 2] &] (* Robert G. Wilson v, Jun 09 2005 *)
Union[Flatten[Select[Partition[Prime[Range[200]],2,1],#[[2]]-#[[1]] == 2&]]] (* Harvey P. Dale, Aug 19 2015 *)

PARI

isA001097(n) = (isprime(n) && (isprime(n+2) || isprime(n-2))) \\ Michael B. Porter, Oct 29 2009

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

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

Python

from sympy import nextprime
from itertools import islice
def agen(): # generator of terms
    yield 3
    p, q = 5, 7
    while True:
        if q - p == 2: yield from [p, q]
        p, q = q, nextprime(q)
print(list(islice(agen(), 58))) # Michael S. Branicky, Apr 30 2022