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A172271
Smaller member p of a twin prime pair (p,p+2) with a cube sum N^3.
7
3, 107, 2634011, 29659499, 57395627, 104792291, 271669247, 485149499, 568946591, 588791807, 752530067, 863999999, 2032678367, 2772616499, 2945257307, 3505869971, 4473547487, 4670303507, 5470523999, 6911999999, 7498065347, 8646803027, 8828622431, 8951240447
OFFSET
1,1
COMMENTS
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)).
Equivalently, primes of the form 4n^3-1 such that 4n^3+1 is also prime. - Charles R Greathouse IV, Aug 27 2013
REFERENCES
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.
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
EXAMPLE
3 + 5 = 2^3;
107 + 109 = (2*3)^3;
2634011 + 2634013 = (2*87)^3.
MAPLE
select(t -> isprime(t) and isprime(t+2), [seq(4*n^3-1, n=1..2000)]); # Robert Israel, Feb 10 2015
MATHEMATICA
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Jan 30 2010
EXTENSIONS
Edits and more terms from Jon E. Schoenfield, Feb 10 2015
STATUS
approved