  The recent years have seen remarkable progress in establishing the complexity
of the reachability problem for vector addition systems with states (VASS),
equivalently known as Petri nets. Existing work primarily considers the case in
which both the VASS as well as the initial and target configurations are part
of the input. In this paper, we investigate the reachability problem in the
setting where the VASS is fixed and only the initial configuration is variable.
We show that fixed VASS fully express arithmetic on initial segments of the
natural numbers. It follows that there is a very weak reduction from any fixed
such number-theoretic predicate (e.g. primality or square-freeness) to
reachability in fixed VASS where configurations are presented in unary. If
configurations are given in binary, we show that there is a fixed VASS with
five counters whose reachability problem is PSPACE-hard.
