A105875 Primes for which -3 is a primitive root.

2, 5, 11, 17, 23, 29, 47, 53, 59, 71, 83, 89, 101, 107, 113, 131, 137, 149, 167, 173, 179, 191, 197, 227, 233, 239, 251, 257, 263, 269, 281, 293, 311, 317, 347, 353, 359, 383, 389, 401, 419, 443, 449, 461, 467, 479, 503, 509, 521, 557, 563, 569, 587, 593, 599, 617, 641

Offset

1,1

Comments

Also, primes for which -27 is a primitive root. Proof: -27 = (-3)^3, so -27 is a primitive root just when -3 is a primitive root and the prime is not 3k+1. Now if -3 is a primitive root, then -3 is not a quadratic residue and so the prime is not 3k+1. - Don Reble, Sep 15 2007

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Index entries for primes by primitive root

Mathematica

pr=-3; Select[Prime[Range[200]], MultiplicativeOrder[pr, # ] == #-1 &]

Prog

(Python)
from sympy import n_order, nextprime
from itertools import islice
def A105875_gen(startvalue=2): # generator of terms >= startvalue
    p = max(startvalue-1, 1)
    while (p:=nextprime(p)):
        if p!=3 and n_order(-3, p) == p-1:
            yield p
A105875_list = list(islice(A105875_gen(), 20)) # Chai Wah Wu, Aug 11 2023

Crossrefs

Cf. A105874.

Sequence in context: A140556 A003627 A103203 * A031368 A020613 A135478

Adjacent sequences: A105872 A105873 A105874 * A105876 A105877 A105878

Keyword

nonn

Author

N. J. A. Sloane, Apr 24 2005

Status

approved