

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
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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 wellknown 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))==n1 \\ 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



