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arXiv · 2310.20449

Long strings of consecutive composite values of polynomials

Kevin Ford·Mikhail R. Gabdullin
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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 $(\log x)(\log\log x)^{1/835}$ consecutive integers $n \in [1,x]$ 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 $(\log x)(\log\log x)^{c_f}$, where $c_f$ depends on $f$ and $c_f$ is exponentially small in the degree of $f$ for some polynomials.

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Kevin Ford, Mikhail R. Gabdullin. 2023-10-31. Long strings of consecutive composite values of polynomials. https://arxiv.org/abs/2310.20449

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