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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
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,1
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COMMENTS
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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
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
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REFERENCES
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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.
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LINKS
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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),100-104and124-128.
Nick MacKinnon,Problem 10844,Amer. Math. Monthly 109, (2002), p. 78.
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MAPLE
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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;
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MATHEMATICA
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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 *)
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PROG
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(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
(Haskell)
a001097 n = a001097_list!! (n-1)
a001097_list = filter ((== 1). a164292) [1..]
(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)
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CROSSREFS
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KEYWORD
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nonn,core,nice,changed
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AUTHOR
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STATUS
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approved
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