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

7, 13, 19, 37, 61, 67, 73, 79, 97, 103, 139, 151, 163, 181, 193, 199, 211, 241, 271, 313, 331, 337, 349, 367, 373, 379, 409, 421, 463, 487, 523, 541, 547, 571, 577, 607, 613, 619, 631, 661, 673, 709, 751, 757, 769, 787, 823, 829, 853, 859, 877, 883, 907, 937

Offset

1,1

Comments

Primes represented by the quadratic form 4x^2 + 2xy + 7y^2, whose discriminant is -108. - T. D. Noe, May 17 2005

Complement of A040028 relative to A000040. - Vincenzo Librandi, Sep 17 2012

Links

Klaus Brockhaus, Table of n, a(n) for n=1..1000

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.

Example

A cube modulo 7 can only be 0, 1 or 6, but not 2, hence the prime 7 is in the sequence.
Because x^3 = 2 mod 11 when x = 7 mod 11, the prime 11 is not in the sequence.

Mathematica

insolublePrimeQ[p_]:= Reduce[Mod[x^3 - 2, p] == 0, x, Integers] == False; Select[Prime[Range[200]], insolublePrimeQ] (* Vincenzo Librandi Sep 17 2012 *)

Prog

(Magma) [ p: p in PrimesUpTo(937) | forall(t){x : x in ResidueClassRing(p) | x^3 ne 2} ]; // Klaus Brockhaus, Dec 05 2008
(PARI) forprime(p=2, 10^3, if(#polrootsmod(x^3-2, p)==0, print1(p, ", "))) \\ Joerg Arndt, Jul 16 2015

Crossrefs

Sequence in context: A059262 A059640 A059643 * A176229 A266268 A110074

Adjacent sequences: A040031 A040032 A040033 * A040035 A040036 A040037

Keyword

nonn,easy,changed

Author

N. J. A. Sloane

Extensions

More terms from Klaus Brockhaus, Dec 05 2008

Status

approved