Let
be a positive nonsquare integer. Then Artin conjectured that the set of all primes
for which
is a primitive root is infinite. Under the assumption
of the generalized Riemann hypothesis,
Artin's conjecture was solved by Hooley (1967; Finch 2003, p. 105).
Let
be not an th
power for any such the squarefree part
of satisfies (mod 4). Let be the set of all primes for which such an is a primitive root. Then
Artin also conjectured that the density of relative to the primes
is given independently of the choice of by , where
The significance of Artin's constant is more easily seen by describing it as the fraction of primes for which has a maximal period repeating
decimal, i.e.,
is a full reptend prime (Conway and Guy 1996)
corresponding to a cyclic number.
where
is a Lucas number (Ribenboim 1998, Gourdon and Sebah).
Wrench (1961) gave 45 digits of , and Gourdon and Sebah give 60.
If
and
is still restricted not to be an th power, then the density is not itself, but a rational multiple thereof. The explicit
formula for computing the density in this case is conjectured to be
(3)
(Matthews 1976, Finch 2003), where is the Möbius function.
Special cases can be written down explicitly for a prime,
If
is a perfect cube (which is not a perfect square), a perfect fifth power (which is
not a perfect square or perfect cube), etc., other formulas apply (Hooley 1967, Western
and Miller 1968).
Artin, E. Collected Papers (Ed. S. Lang and J. T. Tate). New York: Springer-Verlag,
pp. viii-ix, 1965.Conway, J. H. and Guy, R. K. The
Book of Numbers. New York: Springer-Verlag, p. 169, 1996.Finch,
S. R. "Artin's Constant." §2.4 in Mathematical
Constants. Cambridge, England: Cambridge University Press, pp. 104-110,
2003.Gourdon, X. and Sebah, P. "Some Constants from Number Theory."
http://numbers.computation.free.fr/Constants/Miscellaneous/constantsNumTheory.html.Hooley,
C. "On Artin's Conjecture." J. reine angew. Math.225, 209-220,
1967.Hooley, C. Applications
of Sieve Methods to the Theory of Numbers. Cambridge, England: Cambridge
University Press, 1976.Ireland, K. and Rosen, M. A
Classical Introduction to Modern Number Theory, 2nd ed. New York: Springer-Verlag,
1990.Lehmer, D. H. and Lehmer, E. "Heuristics Anyone?"
In Studies
in Mathematical Analysis and Related Topics: Essays in Honor of George Pólya
(Ed. G. Szegö, C. Loewner, S. Bergman, M. M. Schiffer,
J. Neyman, D. Gilbarg, and H. Solomon). Stanford, CA: Stanford University
Press, pp. 202-210, 1962.Lenstra, H. W. Jr. "On Artin's
Conjecture and Euclid's Algorithm in Global Fields." Invent. Math.42,
201-224, 1977.Matthews, K. R. "A Generalization of Artin's
Conjecture for Primitive Roots." Acta Arith.29, 113-146, 1976.Ram
Murty, M. "Artin's Conjecture for Primitive Roots." Math. Intell.10,
59-67, 1988.Ribenboim, P. The
New Book of Prime Number Records. New York: Springer-Verlag, 1996.Shanks,
D. Solved
and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, pp. 80-83,
1993.Sloane, N. J. A. Sequence A005596/M2608
in "The On-Line Encyclopedia of Integer Sequences."Western,
A. E. and Miller, J. C. P. Tables
of Indices and Primitive Roots. Cambridge, England: Cambridge University
Press, pp. xxxvii-xlii, 1968.Wrench, J. W. "Evaluation
of Artin's Constant and the Twin Prime Constant." Math. Comput.15,
396-398, 1961.
be a positive nonsquare integer. Then Artin conjectured that the set 
of all primes
for which 
is a primitive root is infinite. Under the assumption
of the generalized Riemann hypothesis,
Artin's conjecture was solved by Hooley (1967; Finch 2003, p. 105).
be not an 
th
power for any 
such the squarefree part
of 
satisfies 
(mod 4). Let 
be the set of all primes for which such an 
is a primitive root. Then
Artin also conjectured that the density of 
relative to the primes
is given independently of the choice of 
by 
, where
![C_(Artin)=product_(k=1)^infty[1-1/(p_k(p_k-1))]=0.3739558136...](/images/equations/ArtinsConstant/NumberedEquation1.svg)
is the 
th prime.
for which 
has a maximal period repeating
decimal, i.e., 
is a full reptend prime (Conway and Guy 1996)
corresponding to a cyclic number.
is connected with the prime
zeta function 
by
is a Lucas number (Ribenboim 1998, Gourdon and Sebah).
Wrench (1961) gave 45 digits of 
, and Gourdon and Sebah give 60.
and 
is still restricted not to be an 
th power, then the density is not 
itself, but a rational multiple thereof. The explicit
formula for computing the density in this case is conjectured to be
![C_(Artin)^'=[1-mu(n^')product_(prime q; q|n^('))1/(q^2-q-1)]C_(Artin)](/images/equations/ArtinsConstant/NumberedEquation3.svg)
is the Möbius function.
Special cases can be written down explicitly for 
a prime,
,
where 
are both primes with 
,
is a perfect cube (which is not a perfect square), a perfect fifth power (which is
not a perfect square or perfect cube), etc., other formulas apply (Hooley 1967, Western
and Miller 1968).
