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

1,1

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

Formula

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

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 *)

Prog

(Magma) [ p: p in PrimesUpTo(433) | exists(t){x : x in ResidueClassRing(p) | x^3 eq 2} ]; // Klaus Brockhaus, Dec 02 2008
(PARI) select(p->ispower(Mod(2, p), 3), primes(100)) \\ Charles R Greathouse IV, Apr 28 2015

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, ...

Sequence in context: A079545 A154755 A040095 * A049589 A049583 A049596

Adjacent sequences: A040025 A040026 A040027 * A040029 A040030 A040031

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

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

Status

approved