Polynomials taking prime values. (Polynômes prenant des valeurs premières.) (French) Zbl 1005.11042

Summary: We describe two heuristic probabilistic models that give the number \(k\) of prime values of quadratic polynomials on an interval of \(n\) consecutive values of the variable. The first model is devoted to the case \(k=n\) and reveals a “Schinzel barrier” of size \(n^n\): below this barrier, the event “\(n\) prime values for \(n\) consecutive values of the variable” is statistically exceptional, and above, it is statistically frequent provided that there is no arithmetic obstruction. The second model is devoted to the case \(k<n\) and is based on a binomial law with a probability deriving from the Hardy-Littlewood constant. For both models, numerical experiments show a satisfactory similarity. This work is illustrated with a list of the best polynomials found for \(n< 40338\). We also give some numerical results for polynomials of degree 3, 4 and 5.


11N32 Primes represented by polynomials; other multiplicative structures of polynomial values
11Y35 Analytic computations
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