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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 (list; graph; refs; listen; history; text; internal format)
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

Cf. A001359, A061308, A069496, A061308.

Sequence in context: A061308 A302060 A112879 * A053861 A041635 A272572

Adjacent sequences:  A172268 A172269 A172270 * A172272 A172273 A172274

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

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Last modified October 23 23:51 EDT 2019. Contains 328379 sequences. (Running on oeis4.)