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Artin's Constant

If $n\not=-1$ and $n$ is not a Perfect Square, then Artin conjectured that the Set $S(n)$ of all Primes for which $n$ is a Primitive Root is infinite. Under the assumption of the Extended Riemann Hypothesis, Artin's conjecture was solved in 1967 by C. Hooley. If, in addition, $n$ is not an $r$th Power for any $r>1$, then Artin conjectured that the density of $S(n)$ relative to the Primes is $C_{\rm Artin}$ (independent of the choice of $n$), where

C_{\rm Artin}=\prod_{q{\rm\ prime}} \left[{1-{1\over q(q-1)}}\right]= 0.3739558136\ldots,

and the Product is over Primes. The significance of this constant is more easily seen by describing it as the fraction of Primes $p$ for which $1/p$ has a maximal Decimal Expansion (Conway and Guy 1996).


Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, p. 169, 1996.

Finch, S. ``Favorite Mathematical Constants.''

Hooley, C. ``On Artin's Conjecture.'' J. reine angew. Math. 225, 209-220, 1967.

Ireland, K. and Rosen, M. A Classical Introduction to Modern Number Theory, 2nd ed. New York: Springer-Verlag, 1990.

Ribenboim, P. The Book of Prime Number Records. New York: Springer-Verlag, 1989.

Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, pp. 80-83, 1993.

Wrench, J. W. ``Evaluation of Artin's Constant and the Twin Prime Constant.'' Math. Comput. 15, 396-398, 1961.

© 1996-9 Eric W. Weisstein