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

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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.

Crossrefs

Cf. A000040, A040038. Contains A003627.

Programs

Magma

[p: p in PrimesUpTo(450) | exists(t){x : x in ResidueClassRing(p) | x^3 eq 3}]; // Vincenzo Librandi, Sep 11 2012

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 *)

PARI

isok(p) = isprime(p) && ispower(Mod(3,p), 3); \\ Michel Marcus, Nov 12 2017