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A172271
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Smaller member p of a twin prime pair (p,p+2) with a cube sum N^3.
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7
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3, 107, 2634011, 29659499, 57395627, 104792291, 271669247, 485149499, 568946591, 588791807, 752530067, 863999999, 2032678367, 2772616499, 2945257307, 3505869971, 4473547487, 4670303507, 5470523999, 6911999999, 7498065347, 8646803027, 8828622431, 8951240447
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OFFSET
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1,1
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COMMENTS
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It is conjectured that the number of twin prime pairs is infinite, one of the great open questions in number theory.
It is conjectured that this sequence is infinite.
Necessarily the cube base is even: N=2n => p = (2n)^3 / 2 - 1.
For n>1: necessarily n=3k since for n=3k+1, p = (2n)^3 / 2 - 1 is divisible by 3, and for n=3k+2, p+2 = (2n)^3 / 2 + 1 is divisible by 3.
It has been proved that the pair (p,p+2) is a twin prime couple iff 4((p-1)! + 1) == -p (mod p*(p+2)).
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REFERENCES
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G. H. Hardy, E. M. Wright, An Introduction to the Theory of Numbers (Fifth Edition), Oxford University Press, 1980.
N. J. A. Sloane, Simon Plouffe: The Encyclopedia of Integer Sequences, Academic Press, 1995.
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LINKS
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EXAMPLE
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3 + 5 = 2^3;
107 + 109 = (2*3)^3;
2634011 + 2634013 = (2*87)^3.
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MAPLE
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select(t -> isprime(t) and isprime(t+2), [seq(4*n^3-1, n=1..2000)]); # Robert Israel, Feb 10 2015
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MATHEMATICA
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lst={}; Do[a=Prime[n]; b=Prime[n+1]; If[b-a==2, c=a+b; If[Mod[c^(1/3), 1]==0, AppendTo[lst, a]]], {n, 11!}]; lst (* Vladimir Joseph Stephan Orlovsky, Feb 13 2010 *)
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PROG
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(PARI) v=List([3]); for(n=1, 1e3, if(isprime(t=108*n^3-1) && isprime(t+2), listput(v, t))); Vec(v) \\ Charles R Greathouse IV, Aug 27 2013
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Jan 30 2010
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EXTENSIONS
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STATUS
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approved
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