OFFSET
1,1
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.
FORMULA
a(n) ~ 6n log n by the Landau prime ideal theorem. - Charles R Greathouse IV, Apr 06 2022
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 *)
PROG
(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
(Magma) [p: p in PrimesUpTo(2500) | NormEquation(27, p) eq true]; // Vincenzo Librandi, Jul 24 2016
CROSSREFS
KEYWORD
nonn
AUTHOR
Klaus Brockhaus, Mar 02 2001
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
STATUS
approved