A040036 Primes p such that x^3 = 3 has a solution mod p.

2, 3, 5, 11, 17, 23, 29, 41, 47, 53, 59, 61, 67, 71, 73, 83, 89, 101, 103, 107, 113, 131, 137, 149, 151, 167, 173, 179, 191, 193, 197, 227, 233, 239, 251, 257, 263, 269, 271, 281, 293, 307, 311, 317, 347, 353, 359, 367, 383, 389, 401, 419, 431, 439, 443, 449, 461, 467, 479, 491, 499

Offset

1,1

Comments

Complement of A040038 relative to A000040. - Vincenzo Librandi, Sep 13 2012

Being a cube mod p is a necessary condition for 3 to be a 9th power mod p. See Williams link pp. 1, 8 (warning: term 271 is missed). - Michel Marcus, Nov 12 2017

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Kenneth S. Williams, 3 as a Ninth Power (mod p), Math. Scand. Vol 35 (1974), 309-317.

Maple

select(p -> isprime(p) and numtheory:-mroot(3, 3, p) <> FAIL, [2, seq(i, i=3..1000, 2)]); # Robert Israel, Nov 12 2017

Mathematica

ok [p_]:=Reduce[Mod[x^3 - 3, p] == 0, x, Integers] =!= False; Select[Prime[Range[200]], ok] (* Vincenzo Librandi, Sep 11 2012 *)

Prog

(Magma) [p: p in PrimesUpTo(450) | exists(t){x : x in ResidueClassRing(p) | x^3 eq 3}]; // Vincenzo Librandi, Sep 11 2012
(PARI) isok(p) = isprime(p) && ispower(Mod(3, p), 3); \\ Michel Marcus, Nov 12 2017

Crossrefs

Cf. A000040, A040038. Contains A003627.

Sequence in context: A098058 A040054 A093503 * A040078 A045309 A103664

Adjacent sequences: A040033 A040034 A040035 * A040037 A040038 A040039

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved