A172271 Smaller member p of a twin prime pair (p,p+2) with a cube sum N^3.

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

Author

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Jan 30 2010

Keywords

• nonn

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

Examples

			3 + 5 = 2^3;
107 + 109 = (2*3)^3;
2634011 + 2634013 = (2*87)^3.
		

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

Crossrefs

Cf. A001359, A061308, A069496, A061308.

Programs

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 *)
Select[Partition[Prime[Range[31*10^6]],2,1],#[[2]]-#[[1]]==2&&IntegerQ[CubeRoot[Total[#]]]&][[;;,1]] (* The program generates the first 10 terms of the sequence. *) (* Harvey P. Dale, May 31 2025 *)

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

Extensions

Edits and more terms from Jon E. Schoenfield, Feb 10 2015