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