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%I #63 Sep 08 2022 08:44:39
%S 31,43,109,127,157,223,229,277,283,307,397,433,439,457,499,601,643,
%T 691,727,733,739,811,919,997,1021,1051,1069,1093,1327,1399,1423,1459,
%U 1471,1579,1597,1627,1657,1699,1723,1753,1777,1789,1801,1831,1933,1999,2017
%N Primes of the form x^2 + 27y^2.
%C Primes p == 1 (mod 3) such that 2 is a cubic residue mod p.
%C Primes p == 1 (mod 6) such that 2 and -2 are both cubes (one implies the other) mod p. - _Warren D. Smith_
%C Subsequence of A040028, complement of A045309 relative to A040028. For p in this sequence, x^3 == 2 (mod p) has three solutions in integers from 0 to p-1, whose sum is p (A059899) or 2*p (A059914). The solutions are given in A060122, A060123 and A060124. - _Klaus Brockhaus_, Mar 02 2001
%C Primes p = 3m+1 such that 2^m == 1 (mod p). Subsequence of A016108 which also includes composites satisfying this congruence. - _Alzhekeyev Ascar M_, Feb 22 2012
%D K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, Springer, 1982, Prop. 9.6.2, p. 119.
%H N. J. A. Sloane and T. D. Noe, <a href="/A014752/b014752.txt">Table of n, a(n) for n = 1..17753</a> (The first 1000 terms were computed by T. D. Noe)
%H S. R. Finch, <a href="http://arxiv.org/abs/math/0701251">Powers of Euler's q-Series</a>, arXiv:math/0701251 [math.NT], 2007.
%H Zak Seidov, <a href="/A014752/a014752.txt">Corresponding values of x and y</a>
%H Bram van Asch, <a href="http://www.ijpam.eu/contents/2004-16-2/15/15.pdf">On the structure of the ring Z[2^(1/3)]</a>, Internat. J. Pure Appl. Math., 16 (No. 2, 2004), 243-251. See Prop. 7.
%F a(n) ~ 6n log n by the Landau prime ideal theorem. - _Charles R Greathouse IV_, Apr 06 2022
%t With[{nn=50},Take[Select[Union[First[#]^2+27Last[#]^2&/@Tuples[Range[ nn], 2]],PrimeQ],nn]] (* _Harvey P. Dale_, Jul 28 2014 *)
%t nn = 1398781;re = Sort[Reap[Do[Do[If[PrimeQ[p = x^2 + 27*y^2], Sow[{p, x, y}]], {x, Sqrt[nn - 27*y^2]}], {y, Sqrt[nn/27]}]][[2, 1]]]; (* For all 17753 values of a(n), x(n) and y(n). - _Zak Seidov_, May 20 2016 *)
%o (PARI)
%o { fc(a,b,c,M) = my(p,t1,t2,n); t1 = listcreate();
%o for(n=1,M, p = prime(n);
%o t2 = qfbsolve(Qfb(a,b,c),p); if(t2 == 0,, listput(t1,p)));
%o print(t1);
%o }
%o fc(1,0,27,1000);
%o \\ _N. J. A. Sloane_, Jun 06 2014
%o (PARI) list(lim)=my(v=List()); forprimestep(p=31,lim,6, if(Mod(2,p)^(p\3)==1, listput(v,p))); Vec(v) \\ _Charles R Greathouse IV_, Apr 06 2022
%o (Magma) [p: p in PrimesUpTo(2500) | NormEquation(27, p) eq true]; // _Vincenzo Librandi_, Jul 24 2016
%Y Cf. A040028, A045309, A059899, A059914, A060122, A060123, A060124, A014753.
%K nonn
%O 1,1
%A _Klaus Brockhaus_, Mar 02 2001
%E Definition provided by _T. D. Noe_, May 08 2005
%E Entry revised by _Michael Somos_ and _N. J. A. Sloane_, Jul 28 2006
%E Defective Mma program replaced with PARI program, b-file recomputed and extended by _N. J. A. Sloane_, Jun 06 2014
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