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A014752
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Primes of the form x^2 + 27y^2.
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22
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31, 43, 109, 127, 157, 223, 229, 277, 283, 307, 397, 433, 439, 457, 499, 601, 643, 691, 727, 733, 739, 811, 919, 997, 1021, 1051, 1069, 1093, 1327, 1399, 1423, 1459, 1471, 1579, 1597, 1627, 1657, 1699, 1723, 1753, 1777, 1789, 1801, 1831, 1933, 1999, 2017
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OFFSET
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1,1
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COMMENTS
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Primes p == 1 (mod 3) such that 2 is a cubic residue mod p.
Primes p == 1 (mod 6) such that 2 and -2 are both cubes (one implies the other) mod p. - Warren D. Smith
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
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REFERENCES
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K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, Springer, 1982, Prop. 9.6.2, p. 119.
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LINKS
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FORMULA
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MATHEMATICA
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With[{nn=50}, Take[Select[Union[First[#]^2+27Last[#]^2&/@Tuples[Range[ nn], 2]], PrimeQ], nn]] (* Harvey P. Dale, Jul 28 2014 *)
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 *)
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PROG
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(PARI)
{ fc(a, b, c, M) = my(p, t1, t2, n); t1 = listcreate();
for(n=1, M, p = prime(n);
t2 = qfbsolve(Qfb(a, b, c), p); if(t2 == 0, , listput(t1, p)));
print(t1);
}
fc(1, 0, 27, 1000);
(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
(Magma) [p: p in PrimesUpTo(2500) | NormEquation(27, p) eq true]; // Vincenzo Librandi, Jul 24 2016
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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Definition provided by T. D. Noe, May 08 2005
Defective Mma program replaced with PARI program, b-file recomputed and extended by N. J. A. Sloane, Jun 06 2014
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STATUS
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approved
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