Long strings of consecutive composite values of polynomials

2023/10/31 by Ford, Kevin, Mikhail R. Gabdullin, Gabdullin, Mikhail R. · 1 citation
Computer Science · Mathematics · #11B05 #11N32 #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #Primary 11N35 #Stochastic processes and statistical mechanics #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.2310.20449

openalex publication_date 2023/10/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

We show that for any polynomial f from the integers to the integers, with positive leading coefficient and irreducible over the rationals, if x is large enough then there is a string of (log x)(loglog x)^1/835 consecutive integers n ∈ [1,x] for which f(n) is composite. This improves a result of the first author, Konyagin, Maynard, Pomerance and Tao, which states that there are such strings of length (log x)(loglog x)^c_f, where c_f depends on f and c_f is exponentially small in the degree of f for some polynomials.

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• Long strings of composite values of polynomials and a basis of order 2

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