Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #38 Dec 19 2024 10:49:23
%S 7,13,19,37,61,67,73,79,97,103,139,151,163,181,193,199,211,241,271,
%T 313,331,337,349,367,373,379,409,421,463,487,523,541,547,571,577,607,
%U 613,619,631,661,673,709,751,757,769,787,823,829,853,859,877,883,907,937
%N Primes p such that x^3 = 2 has no solution mod p.
%C Primes represented by the quadratic form 4x^2 + 2xy + 7y^2, whose discriminant is -108. - _T. D. Noe_, May 17 2005
%C Complement of A040028 relative to A000040. - _Vincenzo Librandi_, Sep 17 2012
%H Klaus Brockhaus, <a href="/A040034/b040034.txt">Table of n, a(n) for n=1..1000</a>
%H Steven R. Finch, <a href="https://arxiv.org/abs/math/0701251">Powers of Euler's q-Series</a>, arXiv:math/0701251 [math.NT], 2007.
%H Bishnu Paudel and Chris Pinner, <a href="https://arxiv.org/abs/2412.10638">The integer group determinants for the abelian groups of order 18</a>, arXiv:2412.10638 [math.NT], 2024. See p. 3.
%e A cube modulo 7 can only be 0, 1 or 6, but not 2, hence the prime 7 is in the sequence.
%e Because x^3 = 2 mod 11 when x = 7 mod 11, the prime 11 is not in the sequence.
%t insolublePrimeQ[p_]:= Reduce[Mod[x^3 - 2, p] == 0, x, Integers] == False; Select[Prime[Range[200]], insolublePrimeQ] (* _Vincenzo Librandi_ Sep 17 2012 *)
%o (Magma) [ p: p in PrimesUpTo(937) | forall(t){x : x in ResidueClassRing(p) | x^3 ne 2} ]; // _Klaus Brockhaus_, Dec 05 2008
%o (PARI) forprime(p=2,10^3,if(#polrootsmod(x^3-2,p)==0,print1(p,", "))) \\ _Joerg Arndt_, Jul 16 2015
%K nonn,easy,changed
%O 1,1
%A _N. J. A. Sloane_
%E More terms from _Klaus Brockhaus_, Dec 05 2008