A105874 Primes for which -2 is a primitive root.
5, 7, 13, 23, 29, 37, 47, 53, 61, 71, 79, 101, 103, 149, 167, 173, 181, 191, 197, 199, 239, 263, 269, 271, 293, 311, 317, 349, 359, 367, 373, 383, 389, 421, 461, 463, 479, 487, 503, 509, 541, 557, 599, 607, 613, 647, 653, 661, 677, 701, 709, 719, 743, 751, 757, 773, 797
Offset: 1
Keywords
Links
- Joerg Arndt, Table of n, a(n) for n = 1..10000
- L. J. Goldstein, Density questions in algebraic number theory, Amer. Math. Monthly, 78 (1971), 342-349.
- Index entries for primes by primitive root
Programs
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Maple
with(numtheory); f:=proc(n) local t1,i,p; t1:=[]; for i from 1 to 500 do p:=ithprime(i); if order(n,p) = p-1 then t1:=[op(t1),p]; fi; od; t1; end; f(-2);
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Mathematica
pr=-2; Select[Prime[Range[200]], MultiplicativeOrder[pr, # ] == #-1 &] (* N. J. A. Sloane, Jun 01 2010 *) a[p_,q_]:=Sum[2 Cos[2^n Pi/((2 q+1) (2 p+1))], {n,1,2 q p}]; Select[Range[400], Reduce[a[#, 1] == 1, Integers] &]; 2 % + 1 (* Gerry Martens, Apr 28 2015 *)
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PARI
forprime(p=3,10^4,if(p-1==znorder(Mod(-2,p)),print1(p", "))); /* Joerg Arndt, Jun 27 2011 */
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Python
from sympy import n_order, nextprime from itertools import islice def A105874_gen(startvalue=3): # generator of terms >= startvalue p = max(startvalue-1,2) while (p:=nextprime(p)): if n_order(-2,p) == p-1: yield p A105874_list = list(islice(A105874_gen(),20)) # Chai Wah Wu, Aug 11 2023
Formula
Let a(p,q)=sum(n=1,2*p*q,2*cos(2^n*Pi/((2*q+1)*(2*p+1)))). Then 2*p+1 is a prime belonging to this sequence when a(p,1)==1. - Gerry Martens, May 21 2015
Comments