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.

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, December 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)

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

Cf. A001123.

Sequence in context: A059644 A059646 A003629 this_sequence A078971 A129096 A079448

Adjacent sequences: A001119 A001120 A001121 this_sequence A001123 A001124 A001125

nonn,easy,nice

njas