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A105876 Primes for which -4 is a primitive root. 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; refs; listen; history; text; internal format)
OFFSET

1,1

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 another 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.

LINKS

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

MATHEMATICA

pr=-4; Select[Prime[Range[200]], MultiplicativeOrder[pr, # ] == #-1 &] (* OR *)

a[p_, q_]:=Sum[2 Cos[2^n Pi/((2 q+1)(2 p+1))], {n, 1, 2 q p}]

2 Select[Range[500], Rationalize[N[a[#, 2], 20]]==1 &]+1

(* Gerry Martens, Apr 28 2015 *)

PROG

(PARI) is(n)=isprime(n) && n>2 && znorder(Mod(-4, n))==n-1 \\ Charles R Greathouse IV, Apr 30 2015

CROSSREFS

Cf. A114564.

Sequence in context: A292083 A135932 A231847 * A141101 A236632 A098379

Adjacent sequences:  A105873 A105874 A105875 * A105877 A105878 A105879

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Apr 24 2005

EXTENSIONS

Edited by N. J. A. Sloane, Aug 08 2009

STATUS

approved

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Last modified August 20 01:14 EDT 2019. Contains 326136 sequences. (Running on oeis4.)