

A001122


Primes with primitive root 2.
(Formerly M2473 N0981)


112



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,1


COMMENTS

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 (p1)/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 p1, which is the greatest period possible for any integer.
Positive integer 2*m1 is in the sequence iff A179382(m)=m1.  Vladimir Shevelev, Jul 14 2010
These are the odd primes p for which the polynomial 1+x+x^2+...+x^(p1) 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((p1)/2) = (p1). If A002326((p1)/2) is not equal to (p1), then the prime p belongs to the sequence A216838.  V. Raman, Dec 01 2012
Prime(n) is in the sequence if (and conjecturally only if) A133954(n) = prime(n).  Vladimir Shevelev, Aug 30 2013
Pollack shows that, on the GRH, that there is some C such that a(n+1)  a(n) < C infinitely often (in fact, 1 can be replaced by any positive integer). Further, for any m, a(n), a(n+1), ..., a(n+m) are consecutive primes infinitely often.  Charles R Greathouse IV, Jan 05 2015


REFERENCES

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. Conde, M. Miller, J. M. Miret, K. Saurav, On the Nonexistence of Almost Moore Digraphs of Degree Four and Five, Mathematics in Computer Science, June 2015, Volume 9, Issue 2, pp. 145149.
J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, New York, 1996; see p. 169.
K. Dilcher, L. Ericksen, Reducibility and irreducibility of Stern (0, 1)polynomials, Communications in Mathematics, Volume 22/2014 , pp. 77102.
R. Gupta and M. R. Murty: A remark on Artin's conjecture, Invent. Math. 78 (1984) 127230.
M. Kraitchik, Recherches sur la ThÃ©orie des Nombres. GauthiersVillars, 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. 202210, Stanford Univ. Press, Stanford, Calif., 1962.
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).


LINKS

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, Matters Computational (The Fxtbook), pp. 876878.
R. Gupta and M. R. Murty, A remark on Artin's conjecture, Invent. Math. 78 (1984) 127230.
C. Hooley, On Artin's conjecture, J. Reine Angewandte Math., 225 (1967) 209220.
Robert Jackson, Dmitriy Rumynin and Oleg V. Zaboronski, An approach to RAID6 based on cyclic groups, Applied Mathematics & Information Sciences 5(2) (2011), 148170.
Sihem Mesnager and JeanPierre Flori, A note on hyperbent functions via Dillonlike exponents
F. Pillichshammer, Bounds for the quality parameter of digital shift nets over Z_2, Finite Fields Applic., 8 (2002), 444454.
P. Moree, Artin's primitive root conjecturea survey, arXiv:math/0412262 [math.NT], 20042012.
Paul Pollack, Bounded gaps between primes with a given primitive root (2014)
V. Shevelev,On the Newman sum over multiples of a prime with a primitive or semiprimitive root 2, arXiv:0710.1354 [math.NT], 2007.
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


FORMULA

Delta(a(n),2^a(n)*x) = a(n)*Delta(a(n),2*x), where Delta(k,x) is the difference between numbers of evil(A001969) and odious(A000069) integers divisible by k in interval [0,x).  Vladimir Shevelev, Aug 30 2013


MATHEMATICA

Select[ Prime@Range@200, PrimitiveRoot@# == 2 &] (* Robert G. Wilson v, May 11 2001 *)
pr = 2; Select[Prime[Range[200]], MultiplicativeOrder[pr, # ] == #  1 &] (* N. J. A. Sloane, Jun 01 2010 *)


PROG

(PARI) forprime(p=3, 1000, if(znorder(Mod(2, p))==(p1), print1(p, ", "))); \\ [corrected by Michel Marcus, Oct 08 2014]
(PARI) n=0; forprime (p=3, 99999, if (znorder(Mod(2, p))!=(p1), 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]==(p1), print(p))) /* V. Raman, Sep 17 2012 */
(PARI) forprime(x=1, 10000, if(lift(znprimroot(x))==2, print1(x", "))) \\ Roderick MacPhee, Jan 10 2017


CROSSREFS

Cf. A001123, A001913, A005596 (Artin's constant), A050229.
Cf. A002326 for the multiplicative order of 2 mod 2n+1. (Alternatively, the least positive value of m such that 2n+1 divides 2^m1).
Cf. A216838 (Odd 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


KEYWORD

nonn,easy,nice,changed


AUTHOR

N. J. A. Sloane, Apr 30 1991


STATUS

approved



