A105880 Primes for which -8 is a primitive root.

5, 23, 29, 47, 53, 71, 101, 149, 167, 173, 191, 197, 239, 263, 269, 293, 311, 317, 359, 383, 389, 461, 479, 503, 509, 557, 599, 647, 653, 677, 701, 719, 743, 773, 797, 821, 839, 863, 887, 941, 983, 1031, 1061, 1109, 1151, 1223, 1229, 1277, 1301, 1319, 1367, 1373, 1439

Offset: 1

Author

N. J. A. Sloane, Apr 24 2005

Keywords

• nonn

Comments

From Jianing Song, May 12 2024: (Start)

Members of A105874 that are not congruent to 1 mod 3. Terms are congruent to 5 or 23 modulo 24.

According to Artin's conjecture, the number of terms <= N is roughly ((3/5)*C)*PrimePi(N), where C is the Artin's constant = A005596, PrimePi = A000720. Compare: the number of terms of A001122 that are no greater than N is roughly C*PrimePi(N). (End)

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Wikipedia, Artin's conjecture on primitive roots
• Index entries for primes by primitive root

Crossrefs

Cf. A105874, A005596, A000720.

Programs

Mathematica

pr=-8; Select[Prime[Range[400]], MultiplicativeOrder[pr, # ] == #-1 &] (* N. J. A. Sloane, Jun 01 2010 *)
a[p_, q_]:= Sum[2 Cos[2^n Pi/((2 q+1)(2 p+1))],{n,1,2 q p}]
2 Select[Range[800], Rationalize[N[a[#, 9], 20]] == 1 &] + 1
(* Gerry Martens, Apr 28 2015 *)

PARI

is(n)=isprime(n) && n>3 && znorder(Mod(-8,n))==n-1 \\ Charles R Greathouse IV, May 21 2015

Formula

Let a(p,q)=sum(n=1,2*p*q,2*cos(2^n*Pi/((2*q+1)*(2*p+1)))). Then 2*p+1 is a prime of this sequence when a(p,9)==1. - Gerry Martens , May 21 2015