Polyn\^omes de degr\'e sup\'erieur \`a 2 prenant beaucoup de valeurs premi\`eres

Abstract

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.