A014752 Primes of the form x^2 + 27y^2.

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

Offset: 1

Author

Klaus Brockhaus, Mar 02 2001

Keywords

• nonn

Comments

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

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

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

References

• K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, Springer, 1982, Prop. 9.6.2, p. 119.

Links

• N. J. A. Sloane and T. D. Noe, Table of n, a(n) for n = 1..17753 (The first 1000 terms were computed by T. D. Noe)
• Steven R. Finch, Powers of Euler's q-Series, arXiv:math/0701251 [math.NT], 2007.
• Bishnu Paudel and Chris Pinner, The integer group determinants for the abelian groups of order 18, arXiv:2412.10638 [math.NT], 2024. See p. 3.
• Zak Seidov, Corresponding values of x and y
• Bram van Asch, On the structure of the ring Z[2^(1/3)], Internat. J. Pure Appl. Math., 16 (No. 2, 2004), 243-251. See Prop. 7.

Crossrefs

Cf. A040028, A045309, A059899, A059914, A060122, A060123, A060124, A014753.

Programs

Magma

[p: p in PrimesUpTo(2500) | NormEquation(27, p) eq true]; // Vincenzo Librandi, Jul 24 2016

Mathematica

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 *)

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);
\\ N. J. A. Sloane, Jun 06 2014

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

Formula

a(n) ~ 6n log n by the Landau prime ideal theorem. - Charles R Greathouse IV, Apr 06 2022

Extensions

Definition provided by T. D. Noe, May 08 2005

Entry revised by Michael Somos and N. J. A. Sloane, Jul 28 2006

Defective Mma program replaced with PARI program, b-file recomputed and extended by N. J. A. Sloane, Jun 06 2014