call a cellular automaton a universal pattern generator if from some
finite initial seed it generates all finite patterns over its state set.
A stronger variant requires that all finite patterns get generated in
every position of the space, that is, the forward orbit of the initial
finite seed is dense in the product topology. We demonstrate a
one-dimensional universal pattern generator of the weaker type and
discuss its relation to some open problems in number theory. We
conjecture that strong pattern generators exist, and provide an
automaton that we believe to have this property.