A050229 -id:A050229 - cp's OEIS frontend

A001122 Primes with primitive root 2.

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

Offset: 1

N. J. A. Sloane

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. - 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]
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
From Jianing Song, Apr 27 2019: (Start)
All terms are congruent to 3 or 5 modulo 8. If we define
  Pi(N,b) = # {p prime, p <= N, p == b (mod 8)};
     Q(N) = # {p prime, p <= N, p in this sequence},
then by Artin's conjecture, Q(N) ~ C*N/log(N) ~ 2*C*(Pi(N,3) + Pi(N,5)), where C = A005596 is Artin's constant.
Conjecture: if we further define
   Q(N,b) = # {p prime, p <= N, p == b (mod 8), p in this sequence},
then we have:
   Q(N,3) ~ (1/2)*Q(N) ~ C*Pi(N,3);
   Q(N,5) ~ (1/2)*Q(N) ~ C*Pi(N,5). (End)
Conjecture: for a prime p > 5, p has primitive root 2 iff p == +-3 (mod 8) divides 2^k + 3 for some k < p - 1 and divides 2^m + 5 for some m < p - 1. It seems that all primes of the form 2^k + 3 for k <> 2 (A057732) have primitive root 2. - Thomas Ordowski, Nov 27 2023

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.
M. Kraitchik, Recherches sur la Théorie 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.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 20.
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).

Antti Karttunen, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
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. 876-878.
Richard Bartels, Generalized Loewy Length of Cohen-Macaulay Local and Graded Rings, arXiv:2308.14932 [math.AC], 2023. See p. 10.
J. Conde, M. Miller, J. M. Miret, and K. Saurav, On the Nonexistence of Almost Moore Digraphs of Degree Four and Five, Mathematics in Computer Science, 9(2) (2015), 145-149.
Jonathan Detchart and Jérôme Lacan, Improving the coding speed of erasure codes with polynomial ring transforms, arXiv:1709.00178 [cs.IT], 2017.
K. Dilcher and L. Ericksen, Reducibility and irreducibility of Stern (0, 1)-polynomials, Communications in Mathematics, 22 (2014), 77-102.
Gabriele Fici and Estéban Gabory, Generalized De Bruijn Words, Invertible Necklaces, and the Burrows-Wheeler Transform, arXiv:2502.12844 [math.CO], 2025. See p. 12.
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.
Florian Ingels, Anaïs Denis, and Bastien Cazaux, Decomposing Words for Enhanced Compression: Exploring the Number of Runs in the Extended Burrows-Wheeler Transform, arXiv:2506.04926 [cs.DS], 2025. See p. 15.
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.
Jonas Kaiser, On the relationship between the Collatz conjecture and Mersenne prime numbers, arXiv:1608.00862 [math.GM], 2016.
Sihem Mesnager and Jean-Pierre Flori, A note on hyper-bent functions via Dillon-like exponents, IACR, Report 2012/033, 2012.
F. Pillichshammer, Bounds for the quality parameter of digital shift nets over Z_2, Finite Fields Applic., 8 (2002), 444-454.
Pieter Moree, Artin's primitive root conjecture-a survey, arXiv:math/0412262 [math.NT], 2004-2012.
Paul Pollack, Bounded gaps between primes with a given primitive root, arXiv:1404.4007 [math.NT], 2014.
Vladimir Shevelev, On the Newman sum over multiples of a prime with a primitive or semiprimitive root 2, arXiv:0710.1354 [math.NT], 2007.
Stephan Tornier, Groups Acting on Trees With Prescribed Local Action, arXiv:2002.09876 [math.GR], 2020.
Qifu Tyler Sun, Hanqi Tang, Zongpeng Li, Xiaolong Yang, and Keping Long, Circular-shift Linear Network Codes with Arbitrary Odd Block Lengths, arXiv:1806.04635 [cs.IT], 2018.
Eric Weisstein's World of Mathematics, Artin's constant.
Wikipedia, Artin's conjecture on primitive roots.
Index entries for sequences related to Artin's conjecture
Index entries for primes by primitive root

Cf. A001123, A001913, A001917, A005596 (Artin's constant), A050229, A071642, A127209, A163782, A292270.
Cf. A002326 for the multiplicative order of 2 mod 2n+1. (Alternatively, the least positive value of m such that 2n+1 divides 2^m-1).
Cf. A216838 (Odd primes for which 2 is not a primitive root).

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 *)

forprime(p=3, 1000, if(znorder(Mod(2, p))==(p-1), print1(p,", "))); \\ [corrected by Michel Marcus, Oct 08 2014]

from itertools import islice
from sympy import nextprime, is_primitive_root
def A001122_gen(): # generator of terms
    p = 2
    while (p:=nextprime(p)):
        if is_primitive_root(2,p):
            yield p
A001122_list = list(islice(A001122_gen(),30)) # Chai Wah Wu, Feb 13 2023

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

For n >= 2, a(n) = 1 + 2*A163782(n-1). - Antti Karttunen, Oct 07 2017

A242595 a(n) is the primitive period length for the sequence 2^k (mod n), k = 1, 2, ...

Original entry on oeis.org

1, 1, 2, 0, 4, 2, 3, 0, 6, 4, 10, 0, 12, 3, 4, 0, 8, 6, 18, 0, 6, 10, 11, 0, 20, 12, 18, 0, 28, 4, 5, 0, 10, 8, 12, 0, 36, 18, 12, 0, 20, 6, 14, 0, 12, 11, 23, 0, 21, 20, 8, 0, 52, 18, 20, 0, 18, 28, 58, 0, 60, 5, 6, 0, 12, 10, 66, 0, 22, 12, 35, 0, 9, 36, 20, 0, 30, 12, 39, 0, 54, 20, 82, 0, 8, 14, 28, 0

Offset: 1

Wolfdieter Lang, May 18 2014

The computation of this sequence was inspired by Gary Detlefs's May 15 2014 comment on A050229.
It is clear that 2^k (mod 4*m), for k >= 1 is not periodic because otherwise 4*m would divide 2^k*(2^P - 1) for all k >= 1, with P >= 1 the period length. But this is false for k = 1. Therefore, a(4*m) = 0.
a(2*(2*m+1)) = a(2*m+1), m = (0), 1, 2, ... because 2*(2*m+1) has to divide 2^k*(2^a(2*(2*m+1) - 1) for each k >= 1, which means that (2*m+1) has to divide (2^a(2*(2*m+1)) - 1), and a(2*(2*m+1)) has to be the smallest such number. But the smallest number P such that (2*m+1) divides (2^P - 1) is P = a(2*m+1).
a(prime) = phi(prime) = prime - 1 (phi is given in A000010) is equivalent to: prime divides 2^k*(2^(prime-1) - 1), for all k >= 1, and prime-1 is the smallest exponent. For the even prime 2 this is trivial, and for an odd prime p this means that p divides 2^phi(prime) - 1, but not with a smaller exponent; that is 2 is a primitive root modulo this odd p. See A001122 for the primes with primitive root 2. This means that a(prime) = prime - 1 exactly for 2 and the odd primes of A001122. The odd primes with no primitive root 2 are given in A216838.
For composite odd numbers m one has: m divides (2^a(m) - 1) with the smallest such a(m).

			a(1) = 1 because 2^1 == 0 == 1 (mod 1), therefore 2^k (mod 1) is the 0-sequence with primitive period length 1.
a(2) = 1 because 2^k == 0 (mod 2) for k >= 1, hence also the 0-sequence with primitive period length 1. Note that 2 is not a primitive root of 2 even though a(2) = 2-1 = 1 (see the comment above).
a(3) = 3-1 = 2 because 3 is odd and 2 is a primitive root modulo 3. See A001122(1).
a(7) = 3 because the sequence 2^k (mod 7) starts 2, 4, 1, ... therefore the primitive period is 2, 4, 1 of length 3, because 2^(k+3) = 2^k*8 == 2^k*1 (mod 7) == 2^k (mod 7) for all k >= 1. The prime 7 belongs to A216838.
a(4) = 0 because a(4*m) = 0 for all m >= 1 (see the comment above).
a(6) = 2 because the sequence starts with 2, 4, 2, ... and
  6 = 2*3 divides 2^k*(2^2 - 1) = 2^k*3 for all k >= 1. That is a(6) = a(3); see a comment above.
a(9) = 6 from the sequence start 2, 4, 8, 7, 5, 1,... Note that a(3^2) = (3-1)*3. a(5^2) = 20 = (4-1)*5. But a(7^2) = 21 = (7-1)*7/2.

Cf. A000010, A050229, A060749, A001122, A216838.

a(n) is the primitive (smallest) period length of the sequence 2^k (mod n), for k >=1, and n >= 1.

cp's OEIS Frontend

A001122 Primes with primitive root 2.

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