A040028 Primes p such that x^3 = 2 has a solution mod p.

2, 3, 5, 11, 17, 23, 29, 31, 41, 43, 47, 53, 59, 71, 83, 89, 101, 107, 109, 113, 127, 131, 137, 149, 157, 167, 173, 179, 191, 197, 223, 227, 229, 233, 239, 251, 257, 263, 269, 277, 281, 283, 293, 307, 311, 317, 347, 353, 359, 383, 389, 397, 401, 419, 431, 433

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

This is the union of {3}, A003627 (primes congruent to 2 mod 3) and A014752 (primes of the form x^2+27y^2). By Thm. 4.15 of [Cox], p is of the form x^2+27y^2 if and only if p is congruent to 1 mod 3 and 2 is a cubic residue mod p. If p is not congruent to 1 mod 3, then every number is a cubic residue mod p, including 2. - Andrew V. Sutherland, Apr 26 2008

Complement of A040034 relative to A000040. - Vincenzo Librandi, Sep 13 2012

References

• David A. Cox, "Primes of the Form x^2+ny^2", 1998, John Wiley & Sons.
• Kenneth Ireland and Michael Rosen, "A Classical Introduction to Modern Number Theory", second ed., 1990, Springer-Verlag.

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• Index entries for related sequences

Crossrefs

Cf. A001132. Number of primes p < 10^n for which 2 is a cubic residue (mod p) is in A097142.

Cf. A000040, A003627, A014752, A040034.

For primes p such that x^m == 2 mod p has a solution for m = 2,3,4,5,6,7,... see A038873, A040028, A040098, A040159, A040992, A042966, ...

Programs

Magma

[ p: p in PrimesUpTo(433) | exists(t){x : x in ResidueClassRing(p) | x^3 eq 2} ]; // Klaus Brockhaus, Dec 02 2008

Mathematica

f[p_] := Block[{k = 2}, While[k < p && Mod[k^3, p] != 2, k++ ]; If[k == p, 0, 1]]; Select[ Prime[ Range[100]], f[ # ] == 1 &] (* Robert G. Wilson v, Jul 26 2004 *)

PARI

select(p->ispower(Mod(2,p),3),primes(100)) \\ Charles R Greathouse IV, Apr 28 2015

Formula

a(n) ~ (3/2) n log n. - Charles R Greathouse IV, Apr 06 2022

Extensions

Typo corrected to A014752 by Paul Landon (paullandon(AT)hotmail.com), Jan 25 2010