2014/02/28 by Dress Francois, Francois, Dress, Bernard Landreau +2
Computer Science · Mathematics · #11C08 #11Y11 #Analytic Number Theory Research #FOS: Mathematics #History and Theory of Mathematics #Number Theory (math.NT) #Numerical Methods and Algorithms #Polynomial and algebraic computation #math.NT #msc:11C08 #msc:11Y11
paper · pdf · doi:10.48550/arxiv.1402.7312
in French
arxiv created 2014/02/28 · openalex publication_date 2014/02/28 · arxiv updated 2014/03/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.