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

%I A172271 #20 May 31 2025 17:27:12
%S A172271 3,107,2634011,29659499,57395627,104792291,271669247,485149499,
%T A172271 568946591,588791807,752530067,863999999,2032678367,2772616499,
%U A172271 2945257307,3505869971,4473547487,4670303507,5470523999,6911999999,7498065347,8646803027,8828622431,8951240447
%N A172271 Smaller member p of a twin prime pair (p,p+2) with a cube sum N^3.
%C A172271 It is conjectured that the number of twin prime pairs is infinite, one of the great open questions in number theory.
%C A172271 It is conjectured that this sequence is infinite.
%C A172271 Necessarily the cube base is even: N=2n => p = (2n)^3 / 2 - 1.
%C A172271 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.
%C A172271 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)).
%C A172271 Equivalently, primes of the form 4n^3-1 such that 4n^3+1 is also prime. - _Charles R Greathouse IV_, Aug 27 2013
%D A172271 G. H. Hardy, E. M. Wright, An Introduction to the Theory of Numbers (Fifth Edition), Oxford University Press, 1980.
%D A172271 N. J. A. Sloane, Simon Plouffe: The Encyclopedia of Integer Sequences, Academic Press, 1995.
%H A172271 Charles R Greathouse IV, <a href="/A172271/b172271.txt">Table of n, a(n) for n = 1..10000</a>
%e A172271 3 + 5 = 2^3;
%e A172271 107 + 109 = (2*3)^3;
%e A172271 2634011 + 2634013 = (2*87)^3.
%p A172271 select(t -> isprime(t) and isprime(t+2), [seq(4*n^3-1, n=1..2000)]); # _Robert Israel_, Feb 10 2015
%t A172271 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 *)
%t A172271 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 *)
%o A172271 (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
%Y A172271 Cf. A001359, A061308, A069496, A061308.
%K A172271 nonn
%O A172271 1,1
%A A172271 Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Jan 30 2010
%E A172271 Edits and more terms from _Jon E. Schoenfield_, Feb 10 2015