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A040034 Primes p such that x^3 = 2 has no solution mod p.

%I A040034 #41 Mar 06 2025 10:16:41
%S A040034 7,13,19,37,61,67,73,79,97,103,139,151,163,181,193,199,211,241,271,
%T A040034 313,331,337,349,367,373,379,409,421,463,487,523,541,547,571,577,607,
%U A040034 613,619,631,661,673,709,751,757,769,787,823,829,853,859,877,883,907,937
%N A040034 Primes p such that x^3 = 2 has no solution mod p.
%C A040034 Primes represented by the quadratic form 4x^2 + 2xy + 7y^2, whose discriminant is -108. - _T. D. Noe_, May 17 2005
%C A040034 Complement of A040028 relative to A000040. - _Vincenzo Librandi_, Sep 17 2012
%H A040034 Klaus Brockhaus, <a href="/A040034/b040034.txt">Table of n, a(n) for n=1..1000</a>
%H A040034 David Bernier, <a href="http://dx.doi.org/10.13140/RG.2.2.15759.09127">A strong primality test based on third-order linear recurrences</a>, ResearchGate (2025). See p. 8
%H A040034 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 A040034 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 A040034 A cube modulo 7 can only be 0, 1 or 6, but not 2, hence the prime 7 is in the sequence.
%e A040034 Because x^3 = 2 mod 11 when x = 7 mod 11, the prime 11 is not in the sequence.
%t A040034 insolublePrimeQ[p_]:= Reduce[Mod[x^3 - 2, p] == 0, x, Integers] == False; Select[Prime[Range[200]], insolublePrimeQ] (* _Vincenzo Librandi_ Sep 17 2012 *)
%o A040034 (Magma) [ p: p in PrimesUpTo(937) | forall(t){x : x in ResidueClassRing(p) | x^3 ne 2} ]; // _Klaus Brockhaus_, Dec 05 2008
%o A040034 (PARI) forprime(p=2,10^3,if(#polrootsmod(x^3-2,p)==0,print1(p,", "))) \\ _Joerg Arndt_, Jul 16 2015
%K A040034 nonn,easy
%O A040034 1,1
%A A040034 _N. J. A. Sloane_
%E A040034 More terms from _Klaus Brockhaus_, Dec 05 2008