Artin's Constant
If 
and 
is not a Perfect Square, then Artin conjectured that the Set 
of all Primesfor 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 
, thenArtin 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 thefraction 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.
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