### A two variable Artin conjecture, by Pieter Moree and Peter Stevenhagen

Let a and b be non-zero rational numbers that are multiplicatively
independent. We study the natural density of the set of primes p
for which the subgroup of the multiplicative group of the finite
field with p elements generated by (a\mod p) contains (b\mod p).
It is shown that, under assumption of the generalized Riemann
hypothesis (GRH), this density exists and equals a positive
rational multiple of the universal constant
S=\prod_{p prime}(1-p/(p^3-1)). An explicit value of the density is
given under mild conditions on a and b.
This extends and corrects earlier work of P.J. Stephens (1976). Our
result, in combination with earlier work of the second author, allows us
to deduce that any second order linear recurrence with reducible
characteristic polynomial having integer elements, has a positive
density of prime divisors (under GRH).

Pieter Moree and Peter Stevenhagen <moree@math.leidenuniv.nl, psh@wins.uva.nl>