|
Search: id:A105876
|
|
|
| A105876 |
|
Primes for which -4 is a primitive root. |
|
+0
3
|
|
| 3, 7, 11, 19, 23, 47, 59, 67, 71, 79, 83, 103, 107, 131, 139, 163, 167, 179, 191, 199, 211, 227, 239, 263, 271, 311, 347, 359, 367, 379, 383, 419, 443, 463, 467, 479, 487, 491, 503, 523, 547, 563, 587, 599, 607, 619, 647, 659, 719, 743, 751, 787, 823, 827, 839, 859, 863
(list; graph; listen)
|
|
|
OFFSET
|
1,1
|
|
|
COMMENT
|
Also, primes for which -16 is a primitive root. For proof see following comments from Michael Somos, Aug 07 2009:
Let p = 8*t + 3 be prime. It is well-known that 2 is a primitive root.
We will use the obvious fact that if a primitive root is a power of an
other element, then that other element is also a primitive root. So,
-1 == 2^(4*t+1) (mod p) because 2 is primitive root.
-2 == 2^(4*t+2) == 4^(2*t+1) (mod p) obvious
2 == (-4)^(2*t+1) (mod p) obvious, therefore -4 is also primitive root.
2 == 2^(8*t+3) (mod p) obviously works not just for 2
4 == 2^(8*t+4) == 16^(2*t+1) (mod p) obvious
-4 == (-16)^(2*t+1) (mod p) obvious, therefore -16 is also primitive root.
The case where p = 8*t + 7 is similar.
|
|
MATHEMATICA
|
pr=-4; Select[Prime[Range[200]], MultiplicativeOrder[pr, # ] == #-1 &]
|
|
CROSSREFS
|
Cf. A114564.
Sequence in context: A118260 A018805 A135932 this_sequence A141101 A098379 A049754
Adjacent sequences: A105873 A105874 A105875 this_sequence A105877 A105878 A105879
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
N. J. A. Sloane (njas(AT)research.att.com), Apr 24 2005
|
|
EXTENSIONS
|
Edited by N. J. A. Sloane, Aug 08 2009
|
|
|
Search completed in 0.002 seconds
|
