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

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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

Examples

			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.
		

Links

• Klaus Brockhaus, Table of n, a(n) for n=1..1000
• David Bernier, A strong primality test based on third-order linear recurrences, ResearchGate (2025). See p. 8
• 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.

Programs

Magma

[ p: p in PrimesUpTo(937) | forall(t){x : x in ResidueClassRing(p) | x^3 ne 2} ]; // Klaus Brockhaus, Dec 05 2008

Mathematica

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

PARI

forprime(p=2,10^3,if(#polrootsmod(x^3-2,p)==0,print1(p,", "))) \\ Joerg Arndt, Jul 16 2015

Extensions

More terms from Klaus Brockhaus, Dec 05 2008