A040028 - OEIS

%I #31 Jul 18 2024 04:54:09

%S 2,3,5,11,17,23,29,31,41,43,47,53,59,71,83,89,101,107,109,113,127,131,

%T 137,149,157,167,173,179,191,197,223,227,229,233,239,251,257,263,269,

%U 277,281,283,293,307,311,317,347,353,359,383,389,397,401,419,431,433

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

%C 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

%C Complement of A040034 relative to A000040. - _Vincenzo Librandi_, Sep 13 2012

%D David A. Cox, "Primes of the Form x^2+ny^2", 1998, John Wiley & Sons.

%D Kenneth Ireland and Michael Rosen, "A Classical Introduction to Modern Number Theory", second ed., 1990, Springer-Verlag.

%H T. D. Noe, <a href="/A040028/b040028.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Pri#smp">Index entries for related sequences</a>

%F a(n) ~ (3/2) n log n. - _Charles R Greathouse IV_, Apr 06 2022

%t 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 *)

%o (Magma) [ p: p in PrimesUpTo(433) | exists(t){x : x in ResidueClassRing(p) | x^3 eq 2} ]; // _Klaus Brockhaus_, Dec 02 2008

%o (PARI) select(p->ispower(Mod(2,p),3),primes(100)) \\ _Charles R Greathouse IV_, Apr 28 2015

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

%Y Cf. A000040, A003627, A014752, A040034.

%Y 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, ...

%K nonn,easy

%O 1,1

%A _N. J. A. Sloane_

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