  We consider an analogue of Artin's primitive root conjecture for units in
real quadratic fields. Given such a nontrivial unit, for a rational prime p
which is inert in the field the maximal order of the unit modulo p is p+1. An
extension of Artin's conjecture is that there are infinitely many such inert
primes for which this order is maximal. This is known at present only under the
Generalized Riemann Hypothesis. Unconditionally, we show that for any choice of
7 units in different real quadratic fields satisfying a certain simple
restriction, there is at least one of the units which satisfies the above
version of Artin's conjecture.
