%I A001122 M2473 N0981
%S A001122 3,5,11,13,19,29,37,53,59,61,67,83,101,107,131,139,149,163,173,179,181,
%T A001122 197,211,227,269,293,317,347,349,373,379,389,419,421,443,461,467,491,509,
%U A001122 523,541,547,557,563,587,613,619,653,659,661,677,701,709,757,773,787,797
%N A001122 Primes with primitive root 2.
%C A001122 Artin conjectured that this sequence is infinite.
%C A001122 Conjecture : sequence contains infinitely many pairs of twin primes. 
               - Benoit Cloitre (benoit7848c(AT)orange.fr), May 08 2003
%C A001122 Pieter Moree writes (Oct 20 2004): Assuming the Generalized Riemann Hypothesis 
               it can be shown that the density of primes p such that a prescribed 
               integer g has order (p-1)/t, with t fixed exists and, moreover, it 
               can be computed. This density will be a rational number times the 
               so called Artin constant. For 2 and 10 the density of primitive roots 
               is A, the Artin constant itself.
%C A001122 It seems that this sequence consists of A050229 \ {1,2}.
%C A001122 Primes p such that 1/p, when written in base 2, has period p-1, which 
               is the greatest period possible for any integer.
%D A001122 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A001122 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001122 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, 
               National Bureau of Standards Applied Math. Series 55, 1964 (and various 
               reprintings), p. 864.
%D A001122 E. Bach and J. O. Shallit, Algorithmic Number Theory, I; see p. 221.
%D A001122 J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, New 
               York, 1996; see p. 169.
%D A001122 R. Gupta and M. R. Murty: A remark on Artin's conjecture, Invent. Math. 
               78 (1984) 127-230.
%D A001122 C. Hooley: On Artin's conjecture, J. Reine Angewandte Math., 225 (1967) 
               209-220.
%D A001122 M. Kraitchik, Recherches sur la Th\'{e}orie des Nombres. Gauthiers-Villars, 
               Paris, Vol. 1, 1924, Vol. 2, 1929, see Vol. 1, p. 56.
%D A001122 Lehmer, D. H. and Lehmer, Emma; Heuristics, anyone? in Studies in mathematical 
               analysis and related topics, pp. 202-210, Stanford Univ. Press, Stanford, 
               Calif., 1962.
%D A001122 F. Pillichshammer, Bounds for the quality parameter of digital shift 
               nets over Z_2, Finite Fields Applic., 8 (2002), 444-454.
%D A001122 D. Shanks, Solved and Unsolved Problems in Number Theory, 2nd. ed., Chelsea, 
               1978, p. 81.
%H A001122 T. D. Noe, <a href="b001122.txt">Table of n, a(n) for n = 1..1000</a>
%H A001122 Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Fxtbook</a>
%H A001122 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/
               abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National 
               Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 
               [alternative scanned copy].
%H A001122 <a href="Sindx_Ar.html#Artin">Index entries for sequences related to 
               Artin's conjecture</a>
%H A001122 <a href="Sindx_Pri.html#primes_root">Index entries for primes by primitive 
               root</a>
%H A001122 P. Moree, <a href="http://arXiv.org/abs/math.NT/0412262">Artin's primitive 
               root conjecture-a survey</a>
%H A001122 D. Williams, <a href="http://www.louisville.edu/~dawill03/crypto/Primitive.html">
               Primitive Roots(Check)</a>
%t A001122 (* first do *) Needs["NumberTheory`NumberTheoryFunctions`"] (* then *) 
               Select[Prime[ Range[137]], PrimitiveRoot[ # ] == 2 & ]
%t A001122 (* first load *) << NumberTheory`NumberTheoryFunctions` (* then *) Select[ 
               Prime@Range@200, PrimitiveRoot@# == 2 &] (from Robert G. Wilson v 
               (rgwv(at)rgwv.com), May 11 2001)
%t A001122 pr=2; Select[Prime[Range[200]], MultiplicativeOrder[pr, # ] == #-1 &]
%o A001122 (PARI) forprime(p=3,1000, if(znprimroot(p)==2,print(p))).
%o A001122 (PARI) { n=0; forprime (p=3, 99999, if (znprimroot(p)==2, n++; write("b001122.txt", 
               n, " ", p); if (n>=1000, break) ) ) } [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), 
               Jun 14 2009]
%Y A001122 Cf. A001123, A001913, A050229.
%Y A001122 Sequence in context: A059644 A059646 A003629 this_sequence A152871 A156221 
               A078971
%Y A001122 Adjacent sequences: A001119 A001120 A001121 this_sequence A001123 A001124 
               A001125
%K A001122 nonn,easy,nice
%O A001122 1,1
%A A001122 N. J. A. Sloane (njas(AT)research.att.com).