If and is not a Perfect Square, then Artin conjectured that the Set of all Primes
for which 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, is not an th Power for any , then
Artin conjectured that the density of relative to the Primes is (independent of the choice of
), where

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

**References**

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

Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/artin/artin.html

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

1999-05-25