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).

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