1,1

Artin conjectured that this sequence is infinite.

Conjecture : sequence contains infinitely many pairs of twin primes. - Benoit Cloitre (benoit7848c(AT)orange.fr), May 08 2003

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.

It seems that this sequence consists of A050229 \ {1,2}.

Primes p such that 1/p, when written in base 2, has period p-1, which is the greatest period possible for any integer.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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.

E. Bach and J. O. Shallit, Algorithmic Number Theory, I; see p. 221.

J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, New York, 1996; see p. 169.

R. Gupta and M. R. Murty: A remark on Artin's conjecture, Invent. Math. 78 (1984) 127-230.

C. Hooley: On Artin's conjecture, J. Reine Angewandte Math., 225 (1967) 209-220.

M. Kraitchik, Recherches sur la Th\'{e}orie des Nombres. Gauthiers-Villars, Paris, Vol. 1, 1924, Vol. 2, 1929, see Vol. 1, p. 56.

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.

F. Pillichshammer, Bounds for the quality parameter of digital shift nets over Z_2, Finite Fields Applic., 8 (2002), 444-454.

D. Shanks, Solved and Unsolved Problems in Number Theory, 2nd. ed., Chelsea, 1978, p. 81.

T. D. Noe, Table of n, a(n) for n = 1..1000

Joerg Arndt, Fxtbook

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Index entries for sequences related to Artin's conjecture

Index entries for primes by primitive root

P. Moree, Artin's primitive root conjecture-a survey

D. Williams, Primitive Roots(Check)

(* first do *) Needs["NumberTheory`NumberTheoryFunctions`"] (* then *) Select[Prime[ Range[137]], PrimitiveRoot[ # ] == 2 & ]

(* first load *) << NumberTheory`NumberTheoryFunctions` (* then *) Select[ Prime@Range@200, PrimitiveRoot@# == 2 &] (from Robert G. Wilson v (rgwv(at)rgwv.com), May 11 2001)

pr=2; Select[Prime[Range[200]], MultiplicativeOrder[pr, # ] == #-1 &]

(PARI) forprime(p=3, 1000, if(znprimroot(p)==2, print(p))).

(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]

Cf. A001123, A001913, A050229.

Sequence in context: A059644 A059646 A003629 this_sequence A152871 A156221 A078971

Adjacent sequences: A001119 A001120 A001121 this_sequence A001123 A001124 A001125

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com).