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A105876
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Primes for which -4 is a primitive root.
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2
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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; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| 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.
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MATHEMATICA
| pr=-4; Select[Prime[Range[200]], MultiplicativeOrder[pr, # ] == #-1 &]
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CROSSREFS
| Cf. A114564.
Sequence in context: A018805 A191037 A135932 * A141101 A098379 A049754
Adjacent sequences: A105873 A105874 A105875 * A105877 A105878 A105879
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Apr 24 2005
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EXTENSIONS
| Edited by N. J. A. Sloane, Aug 08 2009
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