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A001122
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Primes with primitive root 2.
(Formerly M2473 N0981)
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96
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3, 5, 11, 13, 19, 29, 37, 53, 59, 61, 67, 83, 101, 107, 131, 139, 149, 163, 173, 179, 181, 197, 211, 227, 269, 293, 317, 347, 349, 373, 379, 389, 419, 421, 443, 461, 467, 491, 509, 523, 541, 547, 557, 563, 587, 613, 619, 653, 659, 661, 677, 701, 709, 757, 773, 787, 797
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OFFSET
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1,1
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COMMENTS
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Artin conjectured that this sequence is infinite.
Conjecture : sequence contains infinitely many pairs of twin primes. - Benoit Cloitre, 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.
Positive integer 2*m-1 is in the sequence iff A179382(m)=m-1. [From Vladimir Shevelev, Jul 14 2010]
These are the odd primes p for which the polynomial 1+x+x^2+...+x^(p-1) is irreducible over GF(2). - V. Raman, Sep 17 2012 [Corrected by N. J. A. Sloane, Oct 17 2012]
The prime p belongs to this sequence if and only if A002326((p-1)/2) = (p-1). If A002326((p-1)/2) != (p-1), then the prime p belongs to the sequence A216838. - V. Raman, Dec 01 2012
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REFERENCES
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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 _Jeffrey 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.
Robert Jackson, Dmitriy Rumynin and Oleg V. Zaboronski, An approach to RAID-6 based on cyclic groups, Applied Mathematics & Information Sciences 5(2) (2011), 148-170; http://www.naturalspublishing.com/amis/V5-2/5-2-1.pdf.
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.
Sihem Mesnager and Jean-Pierre Flori, A note on hyper-bent functions via Dillon-like exponents, http://eprint.iacr.org/2012/033.pdf
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.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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T. D. Noe, Table of n, a(n) for n = 1..1000
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].
Joerg Arndt, Fxtbook, pp.876-878
P. Moree, Artin's primitive root conjecture-a survey
D. Williams, Primitive Roots(Check) [link dead as of Jun 27 2011]
Index entries for sequences related to Artin's conjecture
Index entries for primes by primitive root
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MATHEMATICA
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Select[ Prime@Range@200, PrimitiveRoot@# == 2 &] (* from Robert G. Wilson v, May 11 2001 *)
pr = 2; Select[Prime[Range[200]], MultiplicativeOrder[pr, # ] == # - 1 &] (* N. J. A. Sloane, Jun 01 2010 *)
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PROG
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(PARI) forprime(p=3, 1000, if(znorder(Mod(2, p))!=(p-1), print1(p, ", ")));
(PARI) { n=0; forprime (p=3, 99999, if (znorder(Mod(2, p))!=(p-1), n++; write("b001122.txt", n, " ", p); if (n>=1000, break) ) ) } /* Harry J. Smith, Jun 14 2009 */
(PARI) forprime(p=3, 1000, if(factormod((x^p+1)/(x+1), 2, 1)[1, 1]==(p-1), print(p))) /* V. Raman, Sep 17 2012 */
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CROSSREFS
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Cf. A001123, A001913, A005596 (Artin's constant), A050229.
Cf. A002326 for the multiplicative order of 2 mod 2n+1. (Alternatively, the least value of m such that 2n+1 divides with 2^m-1).
Cf. A216838 (Primes for which 2 is not a primitive root).
Sequence in context: A059646 A003629 A175865 * A152871 A156221 A207325
Adjacent sequences: A001119 A001120 A001121 * A001123 A001124 A001125
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane.
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STATUS
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approved
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