  For degrees $3$ to $6$, we first give numerical results on polynomials which
take many prime values on an interval of consecutive values of the variable. In
particular, we have improved Ruby's record for the "$n$ out of $n$" case, for
$n\ =\ 58$, by using a polynomial of degree $6$. In the theoretical part of
this paper, we describe a heuristic probabilistic model in the "$n$ out of $n$"
case: exactly $n$ (different) prime values on an interval of $n$ consecutive
values of the variable. We find that the heuristic value of the probability of
the event "$n$ out of $n$" for a generic polynomial is equal to the product of
two factors: an arithmetic factor related to global conditions of
non-divisibility, and a size factor determined by the position of the
polynomial in a "well-shaped" domain of the space of coefficients. This leads
to a heuristic estimate for the number of "$n$ out of $n$" polynomials in a
given "well-shaped" domain. Finally, results of extended numerical experiments
show a satisfactory agreement with the heuristic values given by the model.
