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A105875 Primes for which -3 is a primitive root. 4
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 (list; graph; refs; listen; history; text; internal format)
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
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
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Apr 24 2005
STATUS
approved

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Last modified June 19 03:10 EDT 2024. Contains 373492 sequences. (Running on oeis4.)