Stephens' Constant

Stephens' constant expresses the density of certain subsets of the prime numbers. Let $a$ and $b$ be two multiplicatively independent integers, that is, $a^m b^n \neq 1$ except when both $m$ and $n$ equal zero. Consider the set $T(a,b)$ of prime numbers $p$ such that $p$ evenly divides $a^k - b$ for some power $k$. The density of the set $T(a,b)$ relative to the set of all primes is a rational multiple of
: $C_S = \prod_p \left(1 - \frac \right) = 0.57595996889294543964316337549249669\ldots$

Stephens' constant is closely related to the Artin constant $C_A$ that arises in the study of primitive roots.
:$C_S= \prod_ \left( C_A + \left( \right) \right) \left( \right)$

See also

*Euler product
* Twin prime constant

References

{{numtheory-stub
Algebraic number theory
Infinite products