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

Author

N. J. A. Sloane, Apr 24 2005

Keywords

• nonn

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

Crossrefs

Cf. A105874.

Programs

Mathematica

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

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