A $9$-state one-dimensional cellular automaton is
constructed which generates the primes in the following
sense: The content of the $0$-th cell at time $t$ is equal
to \qm{1} if $t$ is a prime, and is equal to \qm{0}
otherwise. The neighbourhood type of this CA is $(-1,0,1)$,
i.e. the most usual one. At time $t=0$ only the $0$-th cell
is in a non-quiescent state (here \qm{0} is not the
quiescent state). Further, a one-dimensional CA is
constructed  with the radius $10$ but with two states only
which also generates the primes. (At time $t=0$ only the
$1$-st cell is in a non-quiescent state.) Also a generalized
Pascal triangle with 75 distinct elements is constructed
which generates the odd primes in a similar sense. Hence the
primes can be real-time generated also by a $75$-state
one-dimensional CA with the neighborhood type $(-1,1)$.
